{A Ls K 3 4-C '/:1Qs

'/:1qS {a lS K 3 4-c

Published by the PressSyndicate of the University of Cambridge The Pitt Building, Trumpington Street,C ambridgeC B2 IRP 40 West 20th Stre!t,N ew York, NY 10011-4211U, SA 10 Stamford Road, Oakleigh,M elbourne 3166,A ustralia

@ CambridgeU niversityP ress1 995

First published1 995

Printed in Great Britain at the University Press,Cambrid-gE1

A cataloguerecord of this book is aoailablefrom the British Library

Library of Congresscataloguing in publication ilata auailable ISBN 0 521 551773 hardback

I l r ^ l [ \ t''\n ./ lUniv..Biblioth,e,/ld l P r r s aIu l:j Contents

Foreword page ix 1 Linear spaces 1 2 Real and complex algebras 9 3 Exact sequences 18 4 Real quadratic spaces 22 5 The classificationof real quadratic spaces 32 6 Anti-involutions of R(n) 43 7 Anti-involutions of C(n) 49 8 Quaternions 57 9 Quaternioniclinear spaces 67 10 Anti-involutions of H(n) 72 11 Tensorproducts of algebras 81 12 Anti-involutions of 2K(n) 9T 13 The classicalgroups 100 14 QuadricGrassmannians 110 15 Clifford algebras r23 16 Spin groups 140 l7 Conjugation 148 18 2x2Clitrordmatrices t67 19 The Cayley algebra r78 20 Topologicalspaces 19l 2l Manifolds 202 22 Lie groups 22s 23 Conformal groups 245 24 Triality 256 References 285 Index 289

vii

Foreword

This book's parent TopologicalGeometry (Porteous (1969)), originally written in the 1960's to make propaganda for a basis-freeapproach to the differential calculus of functions of severalvariables, contained, almostby accident,a centralsection on Clifford algebras,a generalisation of quaternions that was at that time little known. This section was strengthenedin the secondedition (Porteous(1981)) by an additional chapteron the triality outer automorphismof the group Spln(8),a feature which illuminates the structure of severalof the other Spin groups and which is related to a property of six-dimensionalprojective quadrics first noticed almost a hundred years ago by Study in work on the rigid motions of three-dimensionalspace. In recent years Clifford algebrashave become a more popular tool in theoreticalphysics and it seemstherefore appropriate to rework the original book, summarisingthe linear algebraand calculusrequired but expandingthe Clifford algebramaterial.This seemsthe more worth while sinceit is clear that the central result of the old book. the classification of the conjugationanti-involution of the Clifford algebrasR ,n and their complexifications,was dealt with too briefly to be readily understood, and some of the more recent treatmentsof it elsewherehave been less than complete. As in the previous version, the opportunity has been taken to give an exhaustivetreatment of all the generalisationsof the orthogonal and unitary groups known as the classicalgroups, since the full set plays a part in the Clifford algebrastory. In particular, perhapssurprisingly, one learnsto think ofthe generallinear groupsas unitary groups.Toward the end ofthe book the classicalgroups are presentedas Lie groupsand their Lie algebrasare introduced. The exceptionalLie group G2 also makes an appearanceas the group of automorphismsof the Cayley algebra,a x Foreword non-associativeanalogue of the quaternionsthat plays an essentialrole in the discussionof trialitY. I owe a great debt not only to colleaguesand studentsat the university of Liverpool over the yearsbut also to new found friends at the by now regular internationalmeetings on clifford algebrasand their applications pub- to problemsof mathematicalphysics, whose Proceedings have been lished as chisholm and common (1986),Micali, Boudet and Helmstetter (1993)' (1991)' and Brackx,Delanghe and Serras My interestin Clifford algebrasand their usein physicswas originally stimulated by discussionswith my colleagueat Liverpool Bob Boyer' tragically killed by a madman'sbullet on the campus of the university of Austin, Texas,on August l, 1966.Explicit classificationsof both the conjugation and the reversionanti-involutions in the tables of clifford algebrasin Chapter l7 arc in a Liverpool M'Sc' thesisby Tony Hampson (fgOq).On obtaining the answersHampson and I wrote to my colleague TerryWall,whowasatthat'timeonavisittoMexico'Hereplied by drawing our attention to his paper (1968) which we had not read, and which presentedthe entire theory very succinctly! For the classical groups my main debt is to Prof. E. Artin's classic GeometricAlgebta can be derived from OgSil. The observationthat the Cayley algebra one of the clifford algebrasI owe to Michael Atiyah, while the method group adopted in chapter 22 for constructingthe Lie algebrasof a Lie was outlined to me bY Frank Adams' In preparing this fresh version of the material I am hugely indebted to Pertti Lounestowho has read much of the book in draft and over the yearshas kept me right on many points of detail. His knowledgeof the history of the subjectis unsurpassed.Much more recently,chapter 23 on the conformal groupsowes much to Jan cnops, as is there acknowledged, and to the hospitality of Julius l-awrynowicz and the BanachInstitute in warsaw in 1994.An early version of someof the material of chapter 17 has appearedas Porteous(1993). finatty a disclaimer! I am no physicist,and thereforethe reader will searchin vain for particular applicationsto physics.on the other hand, works that are strongly biasedtoward applicationsfrequently give only a fragmentedand partial view of the subject. It is my belief that the subjectonly makessense when the full picture is unfolded, and some of the otherwiseconfusing details are seennaturally to fall into place. Ian Porteous,Liverpool, January 1995 I Linear spaces

In this chapter we recall briefly some salient facts about linear spaces and linear maps.Proofs for the most part are omitted.

Maps Let X and,Y be sets and f : X'-+Y a map. Then, for each x € X an element/(x) e Y is defined,the subsetof Y consistingof all such elementsbeing called the image of /, denoted by im/. More generally f : X>--+Y will denote a map of an unspecifiedsubset of X to Y, X being called lhe sourceof the map and the subset of X consistingof those points x e X for which /(x) is definedbeing called the domain of /, denotedby dom/. In either casethe set Y is the target of f . Givena mapf : X>Y and a point ! € Y,the subsetf-r{y\ of X consistingof those points x e X such that f(x) :y is called thefibre of / over y, this being non-null if and only if y e im f . The set of non-null fibres of / is called the coimageof / and the map

dom/ --+coim/; x t- /-t{/(x)}

the partition of dom/ inducedby /. The fibres of a map f are sometimes called the leuel setsor the contoursof /, especiallywhen the taryet of f is the field of real numbersR. The compositegf of maps/ : X>-+Y and g :Y >--+Z (read'g following/') is the mapX>Z;x- g("f(x)),with domg/ :1-t(domg).

Propositionl.l Forany maps f :W --+X,g:X-+Y and.h:Y +Z h(gf) : hgf : (hg)f. 2 1 Lineqr sqaces proof we traversethe rebracketingpentagon as follows:- for any w e w : h((efl(w)): h(g(f(w))) : (he)(f(w)): ((hg)/Xy)' ((/r(g/)Xw) -/n

For any set X the identity map X --+X : x r-r x will be denotedby |x, maps/ iX-.y andg: Y --+Xsuchthat gf :lx andfg:lv being : said to be inuersesot'each other, with g : f-r and f g-1. Given : invertiblemaps / :X--+Y and g : Y'-+Z then(g/)-1 f-rg-l' : y' The To any map f : X -'+Y there is associatedan equation"f(x) if, for each €Y' thereis m p f is said Lobe suriectiueot asuriection ! iniection somex e x suchthat /(x) : y.It is said to be injectit)eof an possibly if, for eachy eY,there is at most oneelement xeX, though none,suchthat/(x):y'Themapfailstobesurjectiveifthereexistsan elementy eY suchthat the equationf(x): y hasno solutionxeX' and fails to be injectiveif there exist x, x' e X such that f(x'): f(x)' a A map that is both injective and surjectiveis said to be bijectiueor bijection.A map is bijective if and only if it is invertible' -'+X g: lv then' by If maps f : X'-Y and g : Y are such that f injection g is said to Exercise1.1, / is surjectiveand g is injective. The be asectionofthe surjection/.It selectsfor eachy eY asinglexeX : X--+ Y has a such that f(x): y. It is assumedthat any surjectionf sectiong t Y -- X, this assumptionbeing known as the axiom of choice'

Linear spscesand maPs Alinearspace(ofvectors),X,overafield(ofscalars),K,isanadditive abelian group X, with zero elemento (or 0) and furnishedwith a scalar multiplicationK x x --+x;(x,1)r-+ xA: ,l,xsatisfying both distributive Moreover' laws, with ),(px): Q'trlx' for any )",1t e K and any x e X' -x' 1.x: x, for any x € X, implying that 0x : O and (-1)x: Also' for any )" eK, ).o: o. For us the field K will normally be either the real field R or the complexfield c and the linear spaceswill normally be X being a finite set of vectors finite-dimensional,-that a basisfor such a space arelinearly independentand that spanX. The number of vectorsin any basisis independentof the basisand is called the dimensionof x. A linear map is a map betweenlinear spacesthat respectsthe linear structures;thatisf:X-+YbetweenlinearspacesXandYislinear if, for anya,b e X, 1, p eK, f(Aa+ pb):Lf(a)+ pf(b)'Sucha map of on any basis fot X, f : x --+Y is uniquely determinedby the action / 1 Linear spaces 3 and any assignmentof / on the elementsof a basis for X extendsto a linear map of the whole of X to Y . It is easily verified that the compositeof any two composablelinear maps is lineaq while it follows as a corollary of Exercise1.2 that the inverseof, a linear bijection is linear. A lineqr subspaceof a linear spaceX is a subset W that acquiresa linear structureby the restriction to W of the linear structurefor X. -+Y The kernelof a linear map f : X is the set {x e X : /(x) : 0}, written ker/.

Proposition1.2 For any linearmap f :X-+Y, ker/ is alinear subspace of X and im/ is a linear subspaceof Y.

Proposition1,3 A linear mapf is injectiueif and only if kerf : {0}. The rank of a linear map between finite-dimensionallinear spaces is the dimension of the image of the map, this image being a linear subspaceof the target space.The kernelrank or nullity of the map is the dimensionof its kernel. The rank of the linear map f : X --+Y will be denotedby rk/ and the kernel rank by kr/.

Proposition1.4 Let f : X--+Y be a linear map,X and Y beingfinite- dimensionallinear spaces. Then rk/ * kr f : dimX. In the casethat X is a finite-dimensionallinear spacea linear map f : X --+X is injective if and only if it is surjective,it then being an automorphismor self-isomorphismof X. More generally if X and,Y are linear spacesof the samefinite dimensionthen any linear injection X --+Y is an isomorphism. A linear spaceX is said to be the direct sumXo@Xr of linear subspaces Xs andXr if XoOXI : {0} and Xol Xt: X, eachof the subspacesthen beinga linear complementof the other.Then dimXs * dimXl : dimX. Associatedto any direct sum decompositionX : Xo @Xr there are projectionmaps X --+Xs with kernel X1 and X --+Xy with kernel Xs. The direct product of linear spacesX and Y is the Cartesianproduct X x Y, with the sum (X x Y )2 -->X x Y ; ((x, y),(x', y')) - (x + x', y * yt) and scalarproduct Kx(X xY)--+X xY;Q", (x, y)) r-+(1x,,ty), the linear spaceKn being the n-fold direct power of the field K. The choice of a basis for an n-dimensionalK-linear spaceinduces an isomorphismwith the linear spaceKn, any linear map Kn-+Kn being representedby its matrix, an array of real numbers with m rows and n 4 1 Linear sqaces columns,whose columns are the imagesof the vectors of theJstandard orderedbasis for Kn, vectorsin this context being representedby column matrices. -+K', 'is' A non-zerokernel vector of a linear map Kn a linear depen- dencerelation betweenthe columns of the matrix of the map' The transposeof a matrix with m rows and n columns is the matrix with n rows and m columns whose ith row is the ith column of the original matrix, for eachi. Transpositionwill be denotedby t, the transposeof a matrix a being denotedbY a'. --+Z a e X and A map B : X xY is said to be bilinear if for any b eY themaps X+Z;(x,b)- A(x,b)and Y'-+Z;yg P(a,y)are both linear. scalar multiplication is an exampleof a bilinear map. The set L(X,Y) of linear maps betweenlinear spacesX and Y, of dimensionsn and m say,has a natural linear structureof dimensionrrn. In particular the linear spaceL(X,K) of linear maps from X to K, also of dimensionn, is called the dual of X and will be denotedby Xr. The map X -- (XL)L;x> ex,

where e,(/) : f (x), is easily proved to be injective and so is an isomorphism. The duat of a linear mapf : x -+ Y betweenfinite-dimensional linear spacesX and Y is the linear maP f" ,Y" -- XL; 0Je @f, --+ where arl denotesthe compositeof the map @ : Y R following the : X --+Y and map f : X'-+ Y. Clearly, for composablelinear maps / g : W -+X we have(fg)" : g"f"-

proposition 1.5 Let u and B be elementsof the dual spaceXL of a finite- dimensionalreal or complexlinear spaceX suchthat ket a : ker f . Then there existsa non'zeroscalar ). suchthat fl : )'u.

Proof Eitherkera :ketfr:x,in whichcase o: F:0 and 't can be any not-zeto element of R, or both a and B are surjective. Then any : elementof X is of the form a * Pb, wherea(a) :0 and a(b) 1, and is, )'u' n B@+ ub): pP(b): u(a* pb))",wheteA:0(b); that 9: The dual annihilator W@ of a linear subspaceW of a finite-dimen- '-WL sional linear spaceX is the kernel of the map * : XL dual to the - inclusionmap r : W '+X.Its dimensionis equalto dimX dimW ' 1 Linesr spaces 5

A linear map f : X --+X is said to be an endomorphisrnof the linear spaceX and the linear spaceL(X, X) of all suchendomorphisms of X is also denotedby End X. As previouslymentioned , a linear isomorphism f : X -- X is said to be an automorphismof the linear spaceX and the set of all such automorphismsof X is denoted either by GL(X) ('GL' 'general standing for linear') or by AutX. The linear spaceof all n x n matricesrepresenting the elementsof L(K", K') will be denotedby K(n). A linear inuolutionof a linear spaceX is a linear map t :X -+X such that t2 : Ix.

2K-modiles and maps Let A be a commutative and associativering with unit element. Then a |t-module X is a linear spaceover the ring A, the terminology linear spacebeing reservedmainly for the casethat A is a field. 2K Consider the case that A is the doublefield consistingof the K- linear spaceK2 assignedthe product (a, b)(c, d) : (ac, bd). A direct sum decompositionX0 @Xr of a K-linear spaceX may be regarded as a 2K-modulestructure for X by setting

(1, p)x: lxo I ltxr, for all x e X andQ., p) e2K. Conversely,any 2K-module structure for X determinesa direct sum decompositionXs @Xt of X as a Klinear spacein which X0 : (1, 0)X (: {(1, 0)x : x e X}) and X1 : (0, 1)X.

Proposition1.6 Let t : X --+X be a linear inuolutionof the K-linear space X. Then azK-modulestructure, and thereforea direct sum decomposition, is definedforX by setting,forany xeX,

(1,0)x : l(, + r(x))and (0, t;x: l(x - (x)). zK-module*oo, uni'*-submodulesur.o.nn.l in theobvious ways. In working with a 2K-modulemap t : X --+Y it is often convenientto representX and Y each as the product of its componentsand then to usenotations associatedwith maps betweenproducts, as, for example,in the next proposition.

2K-module Proposition1.7 Let t : X --+Y be a map. Then t is of theform /aoo\-- ( ;" l, whereas € L(Xs,Ys)and a1 € L(XyY). Conuerselyany map \u at/ 2K-module of thisform is a map. 6 1 Linear spaces

A 2K-moduleX suchthat Xe : (1,0)X andX1: (0, l)X are isomorphic will be called a 2K-linear space and a 2K-module map X--+y between2Klinear spacesX and Y will be called a2K-linear map.

Affine spacesand maps An ffine spaceis a linear spacewith its origin deleted- it acquiresa unique linear structure so soon as a point is chosenas origin, and the transfer from any one linear structure to any other is by a translation. An ffine map is a map f : X --+Y betweenaffine spacesX and Y that becomeslinear so soon as a point a of X is chosenas origin for X and f (a) is chosenas origin for Y . An ffine map betweenlinear spacesis the sum of a linear map and a constantmap. An ffine subspaceof a linear spaceis a translateor parallel of a linear subspace.

Determinants We assumethat the readeris familiar with the basicproperties of determinants. Briefly, to any elementa of K(n) there is a unique real number, the determinantof a, deta, such that

(i) if any column of a matrix a is multiplied by a scalar). then the determinantis multiplied by,1, (ii) if any column of a matrix c is added to another then the determinant remainsunaltered, (iii) the determinantof the identity is 1. The map is defined,for all a e K(n), by the formula deta: !sgnnfIonro,,, u€n! j€n where n! denotesthe set of permutations of the set n of all natural numberslrr such that 0 < m < n. Moreover, (iv) for any a, b e K(n),detb a : delb deta, (v) for any invertiblea eK(n),data.-r : (deta\-r, (vi) for any a eK(n), a is invertible if and only if deta is invertible, that is, if and only if deta : 0.

Any linear isomorphisma :Kn --+X inducesa map

EndX : L(X,X)--K; f v->det(a-tf a) Exercises

called the determinanton EndX and also denoted by det. This map is easilyseen to be independentof the isomorphisma.

Proposition1.8 Any inuertiblemqtrix a € K(n) is reducibleby a seriesof columnoperations to a matrix with ail entrieson the main diagonalequar to I exceptfor one which is equal to deta, and all entries of the main diagonalequal to zero.

Note that each of the column operationsmay be performed by mul_ tiplying the matrix on the right by amatrix all of whose entrieson the main diagonal are equal to 1 and a1lentries ofl the main diagonalexcept one are equal to zero.

Linesr grcups For any linear spacex the set Autx has a naturar group structure. The group Aut Kn is usually denotedby GL(n;K) and cailedth e generar Iinear group of degreen. The subgroup of GL(n;K) consistinf of all automorphismsof Kn of determinant 1 is called the specialrinear group of degreen, denotedby SZ(zr;K). For any real linear spaceX __+ there is amap ( :End,X {_1,0,1}, taking the value 1 if the determinant is positive, the value -1 if the determinantis negativeand the value 0 if the determinantis zero.Au- tomorphisms for which the value of ( is equal to 1 are said to be orientation-preseruing,while those for which the value is equal to -1 are said to be orientation-reuersing.For any finite-dimensionallinear space X the iestriction of ( to Autx is a group isomorphism with the multiplicative group 5o : {*1}. The orientation-preservingauto- morphisms of X form a subgroup of AutX which we shail denote bv Aut+x. The question of orientation does not arise for complex linear spaces sincethere is no notion of a positiuecomplex number.

Exercises 1.1 Let --+Y --+X f : X and g : Y be such that fg: ly. provethat / is surjectiveand that g is injective. 1.2 Let W,X and Y be linear spacesand let t : X__+y and u: W--+X be mapswhose composite /u: W__+yis linear. 1 Linear sp&ces

Then (a) if t is a linear injection, a is linear, (b) if a is a linear surjection,r is linear. t.3 Let X and Y be KJinear spaces.Prove that xxY:(xx{o})o({0}xr).

1.4 Let K be eitherthe field R or the field C and considerthe product K2n xK2n --+K; (x, y) * x n y definedby - - x Ay I(t,.v,*, xn+i!).

verify that the product ll"orrrnru, and that, for every x, ! € Kzn,yAx:-xA1l. Now define0 :K(2n)--+K by the formula

o(a): #Dsgnnff oni,) Aan.,+i), "'- ne2nl i€n where 2n! denotesthe set of permutations of all the natural numbersra suchthat 0 < m <2n. Verify that this is an alternating 2n-linearmap on the columns 2'1 of the matrix a, with e(2"1): 1, where denotesthe unit 2n x 2n matrix, and thereforethal' 0(a) : deta. This exercisewill be of usein Proposition6.11.

1- 2 Real and complexalgebras

A linear algebraover the fie1dof real numbersR is, by definition, a linear spaceA over R togetherwith a bilinear mapA2 -> A, the algebraproiluct. Examplesinclude R itself, the field of complexnumbers, C, consisting of the linear spaceR2 with the product (a,b)(c,d): (ac- bd,ad+ bc), 2R the doublefield consisting of the linear spaceR2 with the product (a,b)(c,d) : (ac,bd),and thefull matrix algebraR(n) of all n x n matrices with real entries,with matrix multiplication as the product. An algebraA may, or may not, have a unit element,and the product needbe neithercommutative nor associative,though it is usualto mention explicitly any failure of associativity.The unit element,if it exists,will normally be denoted by IUI or simply, where no confusion need arise, by 1, the map R --+A;)"v+ LLUI being injective. (The notation 17 is reservedfor the identity map on A.) All the above examplesare associativeand have a unit element,and all arecommutative, with the exceptionof the matrix algebraR(n), with n > I. The double field 2R is often identified with the subalgebraof R(2) consistingof the diagonal 2 x 2 matrices,the unit elementbeing denoted by 'I. Likewise, for any n the n-fold power nR of R may be identified with the subalgebraof the algebra R(n) consistingof the diagonal n x n matrices,the unit matrix in R(n) similarly being denoted by "1. Examplesof non-associativealgebras include the Cayley algebra and Lie algebras,discussed in later chapters. Concepts defined in the obvious ways include not only subalgebras buI algebramaps, algebra-reuersing maps, and in particular algebra iso- morphismsand algebraanti-isomorphisms, the latter for examplebeing a linear isomorphism/ of one algebra,4 to another B that reversesthe )

10 2 Realand comPlexalgebras order of multiplication, that is, for all x'y eA, and all )', p'eR,

f(Ax+ pv): Af(x)* ttf(v), and f(xv): f(v)f(x)' The centre of a real algebra ,4 is the set of all those elementsof .4 that commute with each elementof A. For examplethe centre of R(n) '1. consistsof all real multiples of the identity The centre of an algebra ,4 is a subalgebraof ,4'. Analogousdefinitions hold for linear algebrasnot only over any field, in particular the field c of complex numbers,but also over the double 2RJinear fiel,als2R and 2C. The part of a spaceis played by an RJinear space/ with a prescribeddirect sum decomposition'40 @At, wherc Ao ind Ar are isomorphic linear subspacesof ,4 (so, in the casethat .4 is finite-dimensional,dim,Ao : dim'4r) with scalar multiplication defined by ((2,P), (xo * x1))r--+ (Lxo * Px)' 2R12) For example,the elementsof may be representedas matriceseither l^o co0\ (Y':9 9\ o,or,r'. n"'- 'o,:, orthe r",* | ;. 3' :. f I [ 3' f I \;;,;";,) \o 0 b,d,) Our preferenceis for the secondfotm, whereR4 : R2 x {0} e {0} x R2 is thought of as R2 x R2. 2n(n) For any k with 0 < k < n the kth and (n +k)th columns of will be said to be partners. Thus in the examplethe columns (f) /0 \ .t0| uno arePartners. I ,, I \at I 2C(n)' Similar remarks apply to the algebra 2c Any algebra over c, 2R or may also be regarded as an algebra 2C over R, while any algebra over may be regardedas an algebra over C. In pa.rticular,as the corollary to the next proposition shows,the real algebrac is isomorphicto a subalgebraof the algebraR(2) of 2 x2 real matrices.

- 2 Real and complexalgebras 11

Proposition 2.1 Let C be identifiedwith R2 in the standardway. Then, for any c : a * ib e C, the resl linear map C - C; z a c z has matrix (a -b\ \b a)' -i Corollary2.2 Theset of - matrictt{ ( I ) : @,b)e n} ;, a subal- [\D a/ ) gebraof the algebraR(2), isomorphicto C. -b Note that the determinant of the real matrix ( : ) is equal to \D a/ a2+ b2: (a aTE)(a+ ib) : la + iblz.

Proposition2,3 Let X be a finite-dimensionalcomplex linear spaceand let Xn be the underlyingreal linear space.then, if t : X --+X is a complex Iinear map, detaf : ldetctl2, wheredetst is the determinantof t regardedas a complexlinear map and detpf is the determinantof t regardedas a real linear map.

Proof By Proposition 1.8 the matrix of r with respectto any basis for X is reducibleby elementarycolumn operations,all of determinant 1 in either the real or the complex sense,to a diagonal complex n x n matrix all of whoseentries except one are equal to 1, and for such a matrix the statementis clearly true, by the aboveremark. tr

Corollary 2.4 Let X be as in Proposition2.3, and let t : X--+X be a complexlinear map. Then detpt > 0.

Minimal idcals Let A: K(n),where K: R or C. A left ideal9 of A is a linear subspace 9 of Asuch that, for all x € I and all a e A, ax € 9. Right idealsarc similarly defined.

Proposition2.5 Let X be a K-linear space,and let t e EndX. Then the subset 9(t) : {at eEndX : a e EndX} is a left ideal of End,X. 12 2 Realand comPlexalgebtas proper subsetof A left ideal I of Ais said to be minimal if the only I which is a left ideal of ,a is {0}'

Theorem2.6LetXbeafinite-dimensionalK.linearspace'Theminimal Ieft idealsof EndX are thoseof theform 9(t):{atieeEndX}, wheret e EndX and rkt: l.

Then' for Proof Snpposefirst that g is a minimal left ideal of EndX' Since.-/ is any"te i,-.r61is a left ideal of EndX and a subsetof L minimal, it follows that I :9(t), for any non-zetot e '9' : l' 9(st) Supposethat tkt > 1. Then, for any s € EndX with rksr isapropersubsetof-q!),sincethereissuchans,itfollowsthat9(t)is not minimal. So 9(t) is minimal if and only if rk r : 1' n TheminimalleftidealsremainthesameevenifEndXisregardedas an algebraover any subfieldof the field K' integers wJshall generallyindex the columnsof anmxnmatrix by the O,I,2, 3,-.,fl - 1, a setwe denotebY n'

proposition2.7 The matricesinK(n\ all of whosecolumns except the zetoth K(n)' are zeroform a minimal left ideal of the algebra ForthisreasonitispossibletothinkofthelinearspaceKnonwhich algebra. the matrix algebra K(n) acts as a minimal left ideal of that Similar remarksmay be made about minimal right ideals'

Algebra maPs algebra- In practice one often wishesto construct an algebramap or an in so rev-ersingmap of one algebra,,4, to another, B, and such a map' linear far as it must be linear, will be determinedby its restrictionto any - free to basisfor /. However,the converseis no longer true we are not B'and assignarbitrarily the valuesin B of amap of somebasis for Ato general then to extendthis to an algebramap of the whole of '4 to B' In such an extensionwill not be possible' generates In fact what,onenormally Startswith is a subsetS of ,4 that AasaK-algebra,thesubsetSbeingsaidtogenerateAifeachelementof /isexpressible,possiblyinmorethanoneway,asalinearcombination

- 2 Real and complexalgebras 1,3 of a finite sequenceof elementsof A each of which is the product of a finite sequenceof elementsof S. For example, the set of matrices ((r o\ (01\) -i e"""'utesR(2)asanR-algebra'rhefollowing t(; /, ( i o )j is then true.

Proposition 2,8 Let A and B be algebrasouer a field K and let S be a subsetof A that generatesA. Then any algebraor algebra-reuersingmap t : A--+ B is uniquelydetermined by its restrictionto S.

There will be much interest later in certain automorphismsand anti- automorphismsof linear algebras. An automorphismof an algebra ,4 is an algebra isomorphism ,4. --+ .4, while an anti-automorphismof A is an algebra anti-isomorphismA --+A. An automorphism or anti- automorphism/ of ,4 such that f2 : 1,1is said to be, respectively,an inuolutionor an anti-inuolutionof A. In the sequelit will often be convenientto denotean automorphism{ of an algebra A by x F-+xO , rather than by x *+ $(x). The compositeErp of two automorphismsof Awill then be denotedby xr--+;du, and not by x r--+x'pf. In fact, however,in most of the usesof the notation that we have in mind, when we have two automorphismsof the samealgebra they will commute.

Proposition2,9 The only automorphismof R is the identity lp..

Proof Let f be an automorphismof R. Necessarily/ sends0 to 0 and 1 to 1, from which it follows by an easyargument that / sendseach rational number to itself. Also the order of the elementsof R is determinedby the field structure. So / also respectsorder and, in particular, limits of sequences.Since each real number is the limit of a convergentsequence of rational numbers and sincethe limit of a convergentsequence on R is unique,it follows that / sendseach element of R to itself. n

Proposition2.10 The only algebraautomorphisms of C, regardedas a real algebra,are the identity, lg and conjugation.

However,unlike R, thefield C has many automorphisms.(See Segre (1947) and also Fuchs (1963),page 122.) Conjugation is the only field automorphism other than the identity that sendseach real number to itself, but a field isomorphism of C need not send each real number to itself. It is true that each rqtional number must be sent to itself. algebtas 14 2 Realand comPlex real caseis no longer applicable' but the remainder of the proof in the of C sends to -J1' Indeed one of trre non-staidard automorphisms J2 the real subfield of C is not What is implied by these remarks is that of C alone' The field injection uniquely determinedUV ttte field structure piece of structure' In practice the or inclusion R --+ C is an additional granted' It is unusual to say so additional structureis usually taken for explicitly. 2R of the real algebra Proposition2.11 The only algebraautomorphisms -- 2R; (l,ti'Qt'L)' the analogywith are the identityond rwui o I'n temark that c:otniugationon C beingmade euident if we - -1)' (,1(1,1) * P(1,-\)" : ,l'(1'1) P(1' for swap'In Porteous(1969) There seemsto be no recognisednotation hyperbolic' notation hb, this being an abbreviation for I usedthe clumsy 2R set p)-€ : (L' P)o()" p) : and suggestedby the observationthat the {('t' l}' 1) is just the rectangularhyperbola {(1'p) :1p: of an algebtaA arc Two automorphismso, anti-uotomorphismsa,y a of A such that yu : uB' said to be similar if ;;* is an automorphism be dissimilar' If no such existsthen f andy are said to on C are dissimilar' Proposition2.12 The identity and coniugation are d'issimilar' Proposition2.13 The id'entityand swap on2R (L'ti'fit'L) andd : Proposition2.14 The inuolutions o :2C-t2C: 29 --+2C: Q,,trl --+@,7) arc similar.

Then :6Q; since'for Proof DefineQ :2C''zC;1)", tt)'(1'p)' Qo any()., p) e2C, : :6 6 o(1,tD : \Qt, 1) (4t,tr):a(L'F) 6Q' lt)' il

Itreducible automorphismsand anti'automotphisms a of A such that a2 : &' An idempotentof an algebra'4 is an element is the sameas the element (The word'meansthat uiy po*n' of the element of A that is not the itself.) A primitiue id"*pot)nt of '4 is an idempotent sum of two non-zeroidempotents of '4'

.a- 2 Real and complex algebras 15

Proposition2.15 Any automorphismor anti-automorphismof a real algebra A permutesthe primitiue idempotentsof A.

Proposition2.16 Let K be anyfield. Then,for any positiuen, the elements of the standqrd basisfor K" qre the primitiue idempotentsof the n-fold powerofK,"K.

A permutation n of a finite set S is said to be reducibleif there is a proper subset T of S such that TE(T): T, and an automorphism or anti-automorphism of an algebra A is said to be reducibleif the induced permutation of the primitive idempotentsof ,4 is reducible. A permutationor automorphismor anti-automorphismthat is not reducible is said to be irreducible.

Proposition2.17 Let n be a positiuenumber such that nK admitsan irre- ducibleinuolution. Then n: I or 2.

Proposition2.18 An automorphismof 2K is reducibleif and only if it is of theform 2K---rzK;(1, p)* (1x,pQ), 2K where74,0 :K--K are both automorphismsof K. It is an inuolutionof if and only if both y and Q are inuolutionsof K.

Such an automorphismis denotedby X x d. More interestingare the irreducibleautomorphisms of 2K.

Proposition2.19 An automorphismof 2K is irreducibleif and only if it is of theform 2K--2K; (A,p)- kP, Ax), wherey,6 :K--+K are both automorphismsof K. An inuolutionof 2K is irreducibleif and onty if it is of thatform with, moreouer,x: O-1.

Thus, for example,for any field K, swap is an irreducible involution of 2K. There is the following generalisationof Proposition 2.14.

Proposition2.20 For any automorphismsQ andy of afieldK the inuolutions 2K @ x 4-r)o and (x x N-\o of are similar. t6 2 ReaIand comPlexalgebras

Proof Let 6 : 7-r 6. Then x 1xx x-l;o1t x a): (1 x a)(@ Q-r)o' since,for anY()., P) e2K, (xxx-\o (xxx-1)o1r'"1(t l): (3 ,1) : (';" ,:-' ) xa)({ (1xa)(@'d-')'(t l):(1 ,i,) : ({ ^:r') n

Exercises said to be skew if 2.1 An elements of the matrix algebra R(n) is R(n) then' for any sr : -s. Show that if s is a skew elementof elements the linear x € Rn,x"sx :0. Deducethat for suchan and therefore map 1-s : R'--+Rn; x F x-s(x) is injective invertible. for all u'u eRn' 2.2 Let s be a skew elementof R(n)' Provethat' (1 - s)u: (1 + s)u+ u"u: ltxrt' of R(2) consisting Provethat )R i, iro-orphic to the subalgebra or), of the rorrn the map of all matrices ( i ur) a(1,1)* b(1,-1) ,- (i

being an isomorPhism. 2.4 Prove that -b (: 4 \ R(2)+ rtrz\:.rz);\b d.)-\_": )'- f d)

and (') R(2)-+n(2):( i :),- ( ".1,l,\bd) \b o/

are similar anti-involutionsof R(2)' T7 Prove that 'r) R(2)--+nrz); ( i ;),- (i ,

c\ / a-b\ R(2)--+R(-'/azr;(aa)-\_, a) and c\ ( d -") R(2)--+R(^'(azt;\r a)-\_a a/ are dissimilar anti-involutions of R(2). Hint:Track what happensto the matrices (r o\ (t o\ (o 1\ ^_^(o1\ \o 1/'\o -r/'\r o/'uno\-ro)' all of which have squaresin R : R 1n(z). 2.6 Prove that the primitive idempotentsof R(n) are those matrices similar to a, where u has a single non-zero entry on the main diagonal, all off-diagonal entries being zero, that is, is of the form aua-r, where a is any invertible elementof R(n). 3 Exact sequences

and group maps are The properties of exact sequencesof linear.maps are introduced' ,o.-urir.d and left-cosetspace representations

Exact sequencesof lineat maPs of s is also the source Let s and r be linear maps such that the target if ims : kerf' Note that of r. Such a pak of maps is said to be exact : which is equivalentto the this is strongerthan the assertionthat I s 0, infinite sequenceof linear condition im s c ker r only' A possiblydoubly with the sourceof its maps such that the targei of each map coincides maps is exact' 5o"-..rro, is said Io be exact if eachpair of adjacent

linear spaceX' Thenthe Proposition3.1 Let W be a linear subspaceof a sequenceof lineat maPs -.---+ -\ :' {0} w X X lW'----+{0}' wheret is the inclusionand n is the partition' is exact' An exact sequencsof linear maps of the form {o\--->w3^x3 Y-+{o} issaidtobeashortexactSequence.TheexactnessofthesequenceatWis equivalenttotheinjectivityofsandtheexactnessatYtothesurjectivity subspaceof X and of of t. Given such a r"qo.,t." one thinks of W asa the parallelsin X of Y as the quotient spiceX/W' The fibres of t are the linear subspacewhich is the image of s' techniqueknown The following proposition is a first exampleof the as diagram-chasing.

- 3 Exact sequences 19 Proposition3.2 Let

{0} ---+ w -5 x -\ y ------){o} la Lp 4l ----+ -5 l- yt ------+ {0} W' Xt {0}

be a diagramof linear mapssuch that the rows are exactand Bs: s,a, that is the squareformed by thesemaps is commutative.Then there exists -+Y' a uniquelinear map y : Y suchthat yt: ttB, and if a and B are isomorphismsthen y also is an isomorphism.

Proof Uniquenessof y: Supposethat 7 exists. By hypothesisy r(x) : /B@), for all x e X. Since for each y e Y there exists x in r-1{y}, V0) : t'B@) for any such x; that is, y is uniquely determined. Existenceof y: Let y e Y and let x, x1e r-1{y}. Then x1 - x € ker/ : ims and so x1 : x * s(w),for somew € W. Then

t'f(xi : t'B@ *s(w)) : t'f(x) + / B s(w) : {F@)+t'sta(w) : t'fl(x), since/s' :0.

The prescriptiony(y) : t'f(x), for any x in t-l{y), does therefore determinea map y : Y -+ I'such thatyt: t'f. Sincet'B islinear andt is a linear surjection,y is linear, by Exercise1.2. Now supposethat u and B areisomorphisms, and let q : Y --+Y, be the unique linear map such that 4 tt : t B-1. Then applying the uniqueness part of the proposition to the diagram

-----} ----+ {o} w -5 X_\ Y {0} L llw I'x 1,, ----+ -j- ----+ {0} W x-5 Y {0} yields4T : ly. Similarly 14 : Iy,. That is, q : y-t and so y is an isomorphism. n

Proposition 3.3 is an important specialcase.

Proposition3.3 Let W, X and Y befinite-dimensional linear spaces,and let {o}--+w3x3r--+{o} sequences ) 20 3 Exact dual sequence be a short exact sequence'Then the {0}-' Yt L x" \ wL'{0}

is exact.

Analoguesfor group muPS goesoverrrithout changeto sequences Tbedefinition of anexactsequence of a short exact sequenc'e'a group ;igt;;p maps, as doesthe definition group homomorphism'a map between map being rynony,ooot with a In work with multiplicative groups that respectt tltt gt"tn iroducts' - usedin placeof {0} to denotethe one- ;;ffi ir,, ,yrniot triit 'itt"riv of a group need not be normal' elementgroup. Of .o*'" u 'otg'oop *' shall encounter it will not be' and in many of tf," 'ltoutiont ihut definition' extendingthe notion of This possibilr,Vpr"-p"li"iollowing short exact sequence' and let Let F and G be groups,let -EIbe a set' r\eLn group injection and t is a surjection be a pair of maps such that s is a image of s in G' the set of suchleft whosefibres are the left cosetson the specificinjection s being understood. cosetsbeing o.not"ilv- c1r, rn" ThepairwillthenUesaiatabeleft-cosetexact'andtheinducedbijection spacerepresentutio'4 of the set 'E[' GlF '- H will be called a (teft-) coset Numerousexamples will be given later' be groups'let H'H.'M and N be Proposition3.4 Let F,G,F' and'G' sefi, and let F -5 G:'.H

+vl" lB j- F' e -\H' l,l, J'+ MN rowsand columnsate left-coset be a commutatiuediagtam of mapswhose u : M--+N such if thete is a (iecessatily unique) biiection exact. Then yt:t'P' is auniquebiiectionyi H'-+H' suchthat thatu1t:v{ there y if t and tt are group map, then If, morcouer,H and'nl *" i'o'pt and is a grouPisomorPhism'

.- Exercises 2I Group actions Let Gbe a group andX a set.Then amap G x X--+Xl g+ gxis said to be a (left) action of G on X il for all x € X, and g, g, € G, (g,g)x: g,(gx) with 11c;x : x.

ForanyaeX thesubset Go:{g€G: ga:a} is thenasubgroup of G called the isotropysubgroup ofthe action of G at a or the stabiliserof a in G.It is easyto verify that the relation - orrX definedby x-x, + forsomeg€Grxt:gx

is an equivalence. Each equivalenceset is said to be an orbit of the action' If there is only a single orbit, namely the whole of X, then the action of G on X is said to be trensitiue.In this case,for any a e X, the sequence t^ Ita ---+ (t :\X o r-) gA is left-cosetexact. similar remarks apply to group actions on the right. consider, in particular, the caseof a subgroup G of a group d acting on G/ on the right by the map d x G ---+d ; (g', g) * g, g.In this casethe set of orbits of the action coincideswith the set of left cosetson G in Gt, d /G. For this reasonthe set of orbits of a group G actingon a set x on the right may without confusionbe denotedby X /G.

Exercise 3.1 Let G x X--+f,; (g, x) *' gx be a left action of a group G on a set X. Provethat the subsetGo : {g e G : ga: a} is a subgroupofG and that the relation - onX definedby x - xt + for some g e G, x' : gx is an equivalenceon X. prove also that if the action is transitive then, for any e € X, the sequence -i G, e ]\ x is left-cosetexact, while, for any a, b e X with b : ha.,where h € G, and for any g € Go, then hgh-t € G6,the map Go--+ Gul g +, hgh-t being a group isomorphism. 4 Real quadratic sPaces

real linear spacewith an A real quadraticspace is a finite-dimensional assignedquadratic form'

Qaadraticforms is most convenientlyintroduced A quadraticform on a real linear spaceX on X' This' by definition' is a in terms of a symmetricscalar ptoduct bilinear maP X2 -' R; (a,b)v'+a'b ' : The map suchthat, for all a,b e X, b a a'b'

X -- R; qF--+a' a product' aQ)- a'a being called is called thequadratic form ofthe scalar (The notation a2 is reservedfor later thesquate or quadraticnorm of a' use.)Since, for eacha,b e X, 2q.b:o(z)1S@ -@-61Q), by its quadratic form' In the scalar product is uniquely determined if and only if its quadratic particular, the scalarproduct is the zero map form is the zero maP. products on R2: The following are examplesof scalar -xx' xy' + yx'' ((x,y), (x'y')) * O,xx', xx' * yy'' I yy' and

their respectivequadratic forms being (x,y) r- 0, x2,x2 * !2, -x2 * Y2 and2xY'

22 4 Real quadraticspaces 23 A finite-dimensionalreal linear spacewith symmetric scalar product will be called a (real) quadratic space,any linear subspaceW of a quadratic spacex being tacitly assignedthe restriction to w2 of the scalarproduct for X. A quadratic space X is said to be positiue-definiteif, for all non_ zero a e X, qQ) > 0, and to be negatiue-definiteif; for all non-zero a € X, aQ) < 0. An example of a positive-definitespace is the linear spaceR2 with the scalarproduct

((x,y),(x',y'))r+ xx' + yy'.

A quadratic spacewhose scalar product is the zero map is said to be null or isotropicand a quadratic spaceof dimension2n that is the direct sum of two null subspacesof dimensionn is said to be neutral. It is convenientto have short notations for the quadratic spacesthat most commonly occur in practice.The linear spaceRp+s with the scalar product (a,b)++ - t aibi * t aoaibp+j o

will therefore be denoted by Raa, while the linear spaceR2n with the scalarproduct

(a,b) v-+ an1;b;) \ {o,b,*, * O

may choose Proof since Q'a)' Q"$@ : TzoQ) and since a(2)I 0 we ,t: il1atzl1;;-t. n

proposition 4.2 If a and b are inuertibleelements of a quadraticspace X witi a@:bQ), thana*b anda-b are mutuallyorthogonal and either a*b or a-b is inuertible.

' - : Proof Since o(2) : bQ) it follows that (a + b) (a b) 0, while n @+-t1tzt*@-b)@:4s@ f 0. The elementsa*b and a-b neednot both be invertibleeven when (1, 1,1)(2) : a * -lb. Consider,for example,R1'2' In this casewe have (I,t,-Dpl : 1, and (0,0,2): (1,1,1)- (1,1,-1) is invertible,since (1,1,-1) is non- (0,0,2;tzl: 4 * 0. However,(2,2,0): (1,1,1)+ invertible,since (2,2,0)(2) : 0. For a sequelto Proposition4'2 seeProposition 5'L4'

Linear correlutions Let X be any finite-dimensionalreal linear spacewith dual spaceXL. Any - linear map ( : X --+XL; x t+ y( ((x) is said to be a linear correlation on X. An exampleof a linear correlation on Rn is transposition: r : R' -+ (R')t; xr--+x".

A correlation( is symmetricif, for all a,b e X,at(b) : b<(a)' A : symmetric correlation ( inducesa scalar product (a,b) r-> aES sa(b)' Conversely,any scalarproduct (a,b) = a'b is so inducedby a unique symmetric correlation, namely the map a t+ (a'), where, for all a,b e x,(a.)(b) : a.b. A real linear spacex with a correlation( will be called a real correlated(linear) space. By the above remarks any real quadratic spacemay be thought of as a symmetricreal correlatedspace, and conversely.

- 4 Real quailraticspaces 25 Non-degeneratespaces Let X be a real quadratic space,with correlation (. For such a space( is injective if and only if it is bijective,in which caseX, its scalarproduct, its quadratic form and its correlation are all said to be non-degenerate. The kernel of (,ker(, is also called the kernel of X and denoted by kerX. An element q € X belongs to kerX if and only if, for all x € X,a.x: a€x:0, that is if and only if a is orthogonalto each element of X. From this it at once follows that a positive-definite spaceis non-degenerate.The rank of ( is also called the rank of X and denoted by rkX. The spaceX is non-degenerateif and only if rkX: dimX.

Proposition 4.3 Let A be a finite set of mutually orthogonal inuertible elementsof a real quadraticspace X, with correlqtion(. Then the linear subspaceof X spannedby A is a non-ilegeneratesubspace of X.

Proof Let ), be any set of coefficientsfor .4 such that lo.o)*a t' 0. Then, by the orthogonality condition, : (\ t"a1t1\ Aoa?rtl D t|, o. oeA aeA aeA

Thereforethe kernel of the linear subspacespanned by ,a is {0}. tr

Corollary 4.4 For anyfinite p,q, the quadraticspace Rp,s is non-degenerate.

Proposition4.5 Let X be a real quadraticspace and X' a linear complement in X of ker X. ThenXt is a non-ilegeneratesubspace of X.

Orthogonal maps As always,there is interestin the maps preservinga given structure. Let X and Y be real quadratic spaces,with correlations ( and 4, respectively.A map f : X - Y is said to be an orthogonalmap if it is linear and, for all a,b e X,

f(a)af(b) : a(b or, informally, in terms of the dot notation,

f(a)'f(b) : a'b. 26 4 Realquadratic sqa'ces Thisconditionmaybere-expressedintermsofacommutativediagram involving the linear dual fL of /, as follows'

a linear map Proposition4.6 Let X,Y,C and'r7 be as-aboue' Then --+ and onty if : (' that is if and only t t X Y is orthogonalif fLllf if the diagram -tf X Y FI 'l ln tL Yt (' yt

commutes.

then any ottho- Corollary 4,1 If the quadraticspace X is non-degenerate --+ gonal map f : X Y is iniectiue'

: ( is injective and so / is Proof Let / be such a map. Then (fLlif tr injective.

Proposition4.sLetXandYberealquadraticSpqceswithcorrelations( all aeX' andrl,respectiuely. Alinear mepf :X'-+Y suchthat'for

f(a)qf(d : 6es

is orthogonal.

Proposition4.gLetW,XandYberealquadraticspacesandletf:X-+ g is y ind g i W --+X be orthogonalmaps' Then Iv is orthogonal'f orthogonaland, if f is inuertible,f-t is orthogonal'

'-+ an orthogonal An invertible orthogonal mapf : X Y will be called will be said isomorphism,and two quadratic spacesX and Y so related to be isomorPhic.

and Rf,\ are Proposition4.lO For any finite n the quadraticspaces R''n isomorphic. hy- Any two-dimensionalquadratic spaceisomorphic to the standard perbolic plane Rfl6will be called a hyperbolicplane' 4 Real quadraticspaces 27 Proposition 4.ll Let X be a real quadraticspace. Then any two linear complementsin X of ker X are isomorphicas quadraticspaces.

Orthogonal groups An invertible orthogonal map f : X --+X will be called an orthogonal automorphismof the quadratic spaceX. By Corollary 4.7 any orthogonal transformation of a non-degeneratequadratic spaceis an orthogonal automorphism. For quadratic spacesX and Y the set of orthogonal maps f : X --+Y will be denotedby O(X,Y) and the group of orthogonal automorphisms f : X --+X will be denotedby O(X). That this is a group is immediateby Proposition4.9. For any finite p, 4, h the groups O(nwl and O(ROr)will alsobe denoted,respectively, by O(p,q; R) and O(n; R) or, more briefly, by O(p,q)and O(n). An orthogonal transformation of a quadratic spaceX may or may not preservethe orientationsof X. An orientation-preservingorthogonal automorphismof X is said to be a specialorthogonal automorphism, or a rotation,of X. The subgroupof O(X) consistingof the specialorthogonal automorphismsof X is denotedby SO(X), the groups SO(Raa)and SO(RO,';also being denoted,respectively, by SO(p,q)and SO(n).These are the specialorthogonal groups. An orthogonal automorphismof X that reversesthe orientationsof X will be called an anti-rotstion of X.

Proposition4,12 For anyfinite p,q the groupsO(p,q) and O(q,p) are isomorphic,as are the groupsSO(p,q) and SO(q,p).

Adjoints Supposenow that f : X --+ Y is a linear map of a non-degenerate quadratic spaceX, with correlation ( : X --+XL, to a quadratic spaceY, with correlation4 : Y --+YL. Since( is bijective there is a unique linear map f- iY --+X such that (f. : fLq : Y --+XL, that is, such that, for anyxeX,yeY,

f'(y).x: y' f(x).

The map f. : 1-t 7r', : Y --+X is called the adjoint of / with respectto the correlations( and 4. 28 4 Realquadratic sqaces

Proposition4.|3LetXbeanon.degeneraterealquadraticspace.Thenthe map EndX--+EndX;f -f'

is an anti-inuolutionof the real algebraEndX'

anon-degeneratercnl Proposition4.14 Let f :X +Y bealinearmapof if and quairatic spaceX to a quadraticspace Y ' Then f is orthogonal onlYif f'f : 1t'

ProofLet(andtlbethecorrelationsonXandYrespectively.Since( is bijective, : : Ix. f-tqf : ( + f. f 7-tfL4f n

rcal quadratic Corollary 4.15A linear automorphismf of a non'degenerate : spaceX is ofthogonalif and only if f. f-r'

non-ilegeneratereal Proposition4.16 Let f be a linear endomorphismof a if : -f ' quahraticspace X. Ti"' *'f(x):0for all x ex if andonly f'

Proof x'l(x) : 0, for all x € X - - - b) e a' f(a) + b' f(b) (a b)' f(a X' : a' f(b) +b' f(a): 0' for alla'b e e X' o f (b\' a+ f'(b)' a:0' for alla'b kerX : 0' o (/ + /'Xb) : 0' for all x' e X' since + f +f':0' tr

of the spaceX' Then Corollary 4.17Let f be an orthogonalendomorphism - - -Lx' x' f(x) 0 for all x e X if and only if f2

ExarnPlesof adioints Thenextfewpropositionsshowwhattheadjointofalinearmaplooks likeinseveralimportantcases.Itisconvenientthroughouttheseexamples 4 ReaI quadraticspaces

to use the sameletter to denote not only a linear map t r Rp-+ Rq but also its q x p matrix over R. Elementsof Rp are identified with column matricesand elementsof (Rr)r with row matrices. For any linear map --+ t : Rp Rq, f? denotesboth the transposeof the matrix of t and also the linear map Ra -+ Rp representedby this matrix.

Proposition4.18 Let t : R01+R0,q be a linear map. Then t* : tt.

Proof For any x eRP, y € R4,

y. /(x) : !"tx : (t"y), x : f (y). x. Now R01 is non-degenerate,implying that the adjoint of / is unique. So t' : tx,

The case (p,q): (0,2) is worth considering in detail.

Proposition 4.19 Let t : R0,2+ R0'2 be a linear map, with matrix

(ac) \b d)'

( t u \ Then t* has matrix ana t is thereJbreorthogonal if and only if \ c a ), (au\(a,\_/10\ \c a)\u a/:\o t), that is, if and only tf o2+ b2 : c2 + d2 : I and ac * bd.: 0, from which -1 it follows that the matix is either of theform ( | ) ,, of theJbrm \, a/ ( i -a/-:\, ,un a2+ bz : t. Ihe map in thefirst caseis q rotationand \, in the secondcase an anti-rotation,as can be uerifiedby examinationof the sign of the determinant.

To simplify notations in the next two propositions Rp,aand Rfrf are identified, as linear spaces,with Rp x Rq and Rn x Rn, respectively.The entriesin the matricesare linear maps.

Proposition4.20 Let t :Rp,s--+ Rp,s be linear,and let , : ( ? ".\ . mm \o d/ / e" -b'\ t:(-."" d')' quadraticsqaces 30 4 Real Proof Fotall (x,Y),(x',Y') eRP'a' (x,Y)'(ax' I cY', bx'* d!') : -x"(ax' + cy')+ y"(bx'+ dy') : -(a"x)"x'-(c"x)"y' +(b"y)"x' +(d'y)"y' : -(d x - b"Y)" x' I (-c" x + fu)" Y' : (d x - b"Y,-c"x + trY)'(x''Y')' tr

t' dett. : dett' and' if t is orthogonal' Corollary4.21 For sucha linear map (de*)2:1. / a c\ wheret: rn"" proposition4.22 Len : Rfrfi-'Retbe linear' (; i )'

a\ , ld" c'\ r:\b' o")'

Exercises --Xbe transformatiol of a finite-dimensional 4.1 Lett i X a linear t2 is orthogonal' Discuss quadratic spaceX, and supposethat whetherornottisorthogonal'Discuss'inparticular'thecase whereX is Positive-definite' real linear space' Prove that the 4.2 Let X be a finite-dimensional map a(x)) (X x XL)2--+K;((x, t), (y, u))" t(y) + product on the linear space is a neutral non-degeneratescalar XxXL. form 4.3 Provethat R(2) with the quadratic R(2)-+R; t=> dett space the spaceR2'2' the is;;,.:jfi isomorphic as a real quadratic-?(? -i)yi\h (i i),),':'t: ?):G'uu,i'" ), ii invertiblewith un o*r'|,)i*ui v"'iri tnit t e n(z) only if it is invertible as an t.tn*, to the quadratic forrn if and elementof the algebraR(2)'

1"--'- Exercises 31 / d -c\ rorunv-\D( 1 1).nrzr.aenne(1 :)-, =\-a d/ \b d/ o),tn" spaceR(2) being assignedthe determinantquadratic form, and

let any,l e R be identifiedwitrr ( * 9 ) € R(2). verify that, \u,1,/ for any r e R(2), t-t: tQ) and that the subsetT : {t e R(2) : t I t- :0) is a quadraticsubspace of R(2) isomorphicto R2,1. 4.5 Let u e R(2) and 7ett e T, where 7 is as in Exercise4.4. Suppose also that / is orthogonalto u-tt. Show that tu € T. Hence prove that any elementof R(2) is expressibleas the product of two elementsof T. 4.6 With 7 as in Exercise4.4,prove that, for anyu e Sl,(2; R), the map T --+T; t+> -utu-r is reflectionin the plane(R{r})t. 4.7 Provethat, for any u € SL(2; R), the mapsR(2) --+ R(2); tF+ ut and t r-+tu arc rotationsof R(2). (It has to be shownnot only that the quadratic form is preservedbut also that orientations are preserved.) Find linearinjections a : R+R(2) and B: R1,1--R(2)such that, for all x e nt't, (B(r))' : -s1rc(z);. F f, The classificationof real quadraticspaces

and the Central to this chapter are the basis theorem (Theorem 5'9) are signature theorem (Theorem 5.22). Also important in the sequel hyper- the factorisationsof orthogonal automorphismsas compositesof plane reflections(Theorem 5.15) and a characterisationofneutral spaces (Theorem5.28).

Ort hog onal annihilators LetXbearealquadraticspacewithcorrelation(and|etWbealinear as the subspaceof X. The dual annihilator W@ of lll has been defined subspaceof XL annihilating W, namely : O}' {F e XL : for allw e W, B(w) : Moreoverdimw@: dimX - dim W. We now defineWL 1-rgrya\ That is, :0, Wr : {a e X : a' w for all w e W}' X. Its This linear subspaceis called the onhogonalannihilator of w in ( is dimensionis not lessthat dimX - dim W,being equal to this when bijective,that is, when X is non-degenerate' AlinearcomplementYofWinXthatisalsoalinearsubspaceof sum wr issaid to be an orthogonalcomplement of w in x. The direct decomposition w @Y of X is then said tobe an orthogonald.ecomposition of X.

proposition 5.1 Let w be a linear subspaceof a real quadraticspace X. ThenkerW: W nW'.

32 5 The classificationof real quailraticspaces 33 Proposition 5.2 Let w be a linear subspaceof a non-degeneratereal quadraticspace x. Thenx : w @wr if and only if w is non-ilegenerate, Wr, in this case,being the unique orthogonalcomplement of W1n X.

corollary 5.3 Let a be a non-zeroelement of a non-degeneratereal quailratic spaceX and let R{a} denote the line in X spanneitby a. Then X: R{a} @(R{a})a if anit only if a is inuertible.

Proposition 5.4 Let w be q linear subspaceof a non-ilegeneratereal quadraticspace X. Then (Wr)r : yry.

Proposition5.5 Let v and w be linear subspacesof areal quadraticspace X.ThenVcWt+Wc.Vr.

A first application ofthe orthogonal annihilator is to null subspaces.

Proposition5.6 Let w be a linear subspaceof a real quailratic spacex. Then W is null if and only if W c WL. corollary 5.7 Let w be a null subspaceof a non-degeneratereal quailratic space X. Then dimw < tdimX. By this corollary it is only just possiblefor a necessarilyeven-dimensional non-degeneratereal quadratic spaceto be neutral. As we noted earlier,Rfr[,, and thereforealso Rr,n,is such a space.

The basis theorem Let W be a linear subspaceof a real quadratic spaceX. Then, by Proposition 5.2, X : W @ Wr if and only if W is non_degenerate. Moreover, if W is non-degenerate,then, by proposition 5.4, IZa also is non-degenerate. Theseremarks lead to the basistheorem, which we take in two stages.

Theorem 5.8 An n-dimensionalnon-degenerqte real quadraticspace, with n > 0, is expressibleas the direct sum of n non-degeneratemutually or- thogonallines.

A linearly independentset s ofa real quadratic spacex such thar any two distinct elementsof s are mutually orthogonal, with the squareof any elementof s equal to 0,-1 or 1, is said to be an orthonormalsubset of X. If s also spansX then s is said to be an orthonormalbasis for x. i i i of teal quadraticspaces 34 5 The classification Any real quad'raticspace X has an Theorem 5.9 (The basis theorem) orthonotmalbasis'

X-of kerX' Then X' is a non- Let X' be alinear complementin Proof an orthonormal ;;, P'tposition.4-5' and so has degeneratesubspace 6 for n'oposition 4'l' Ler Abe any basis basis,B, say,by fheor-# S'a una basisfor X' tr kerX. Then ,4 U B is an orttto"ormal

As a corollarYwe have theorem'the secondpart Theorem5.10 (The first part of the classification Theorem 5'22') Any non'degenerate of which is the signaturetheorem' n Rp'q somefinite p'q' real quadraticspace X is isomorphic fot real quad' Let be an automorphismof a non-degenerate Proposition5.ll f : 1' (a"t : any rotation f of X' det / ratic spaceX' Then n' l' a'i' for

4'21' n Proof Apply Theorem 5'10 and Corcllury

Reflections

orthogonaldecomposition of a real Proposition 5.12 Let W @ Y be an quadraticsPace X. Then the maP X'-+X;w*Yt-+w-Y,

wherew e W and Y e Y, is orthogonal' of X inW' When Y :WL Sucha map is said to be areflection reflection of X \n a to be the reflecti'onof X in llz' A the map is said a a hypetplanereflection of X' Such linear hyper planeW is said to be hyperplanecan be chosento be an reflectionexists if dimx > 0, for the a is either an elementof kerX or orthogonal .o-pt"-*i of R(a)' where an invertible elementof X'

of a real quadraticspace X is an Proposition5.13 A hyperplanereflection anti-rotationof X. real quadratic sna1l X' Then the Let a be an invertible elementof a (R(a))r will be denotedby po' reflectionof X in the hyperplane 5 The classificationofreal quadraticspaces 35 Proposition 5.14 supposethat u qnd b are inuertibleelements of a real quadratic spaceX, such that oQ) :6Q). Then a may be mappedto b either by a single hyperplanereflection of X or by the compositeof two hyperplanereflections of X.

Proof By Proposition4.2either a-b or a*b is invertible,a-b and a*b being in any casemutually orthogonal.In the first case,pc_b exists and

p : - : - - o-u@) p o-u(|(a b) + t(a + b)) l{a b) + *(a + b) : u. In the secondc&st-, pa16 exists and

PuPa+u(a):p6Qb):b. n

Theorem 5.15 Any orthogonalautomorphism t of a non-degeneratereal quadratic spaceX is expressibleas the compositeof a finite numberof hyperplanereflections of X, the numberbeing not greater than 2dimX, or, if X is positiue-definite,dimX.

Proof This is a straightforwardinduction basedon Proposition5.14. The details are left as an exercise. !

By Proposition5.13 the number of reflectionscomposing t is even when / is a rotation and odd when / is an anti-rotation. The following corollariesare important. To simplify notations we write R2 for R0,2and R3 for R0,3.

Corollary5.16 Any anti-rotationof R2 is a refiectionin someline of Fi2.

Corollary 5.17Any rotation of R3 is the compositeof two planereflections.

Proposition5.18 The only rotation of R2 leauinga non-zeropoint of R2 fixed is the identity.

Proposition5,!9 Any rotation of R3, other than the identity, Ieauesfixed eachpoint of a uniqueline in Fi3.

The line left fixed is called the axis of the rotation. A common mistake is to assumethat every anti-rotation of R3 is a of real quadraticspaces 36 5 The classffication loosely to the orthogonal automor- reflection,for examplein referring iroiutio"s reflections' of R3' This is clearly phisms of R3 as tt" aid a linear irwolution of R3' wrong, any reflectionof R3 being Theorem5.l5hasanimportantpafttoplayinChapter16.

Signature of (J' Yt be orthogonaldecomp'ositions Proposition5.20 Let U @Y and @ d e (J" utQ)< 0' and' quadratr, 'po"" i 'uch that' fot all non-zero a real U with kernel V 0' Then the projectionof X on to nl, to""Z i,- oQ > maps(J' injectiuelYto U '

(J u.e V,' be any elementof U'' Then Proof Let 1tt: 1r* u, where u': € ' - g, '1t't: p(2)'implying that '7Q) o u,(2)- u(2)+ u(2),." ;tl;;;; "* thereforethat u' :0'

Corollarys.zllfalsou(2)<0'forallnon-zerou€u'andu'(2)>-0'for all u' e V' , then dim U : dim U/'

< dimU' By a similar argument Proof By Proposition5'20, dimUt tr dimU < dimU'. 5'20we then have the following' As a further corollary of Proposition

The real quadraticspacesRp'q and Theorem5.22 (Thesignature theorem') p : pt and q: qt' RP"e'are isomorphiciJ' and only if 5'20is the following proposition' Yet a further corollary of Proposition

Rp's--+ Rp'q be an otthogonalautomot Proposition5.23 Let (l i)t -+Rp withPJ xPlq'Then a : Rp phism ofP'q,w'q beinlia'itin"a as.usual and d :nq --+R'Iare linear isornorphisms' / a c\ of Raa.will be said to be The orthogonalautomorphism ;) li We una A are orientation-preserving' semi-orientation-presetuingif a that the set So+(p'4) of all semi shall prove futr, (propotj.nan 22'48) automorphismsof Rp's is a normal orientation-preservingorthogonal group isomorphicto t1' ;;;t*t oi so1p,q)]with quotient 5 The classificationofreal quadraticspaces 37

It follows from Theorem 5.10 that the quadratic form x r+ yQ) s1 ^ real quadratic spaceX may be representedas a sum of squares

x: t xi€it--+\t,*? 0

Proposition5.24 Let W be a null subspaceof the real quadraticspace R!'q. ThendimW < inf{p,q\.

Proof Clearly f,ac : X @Y, where X = Rp'o,and, Y 3 Ro,q.As in the proof of Proposition5.20, the restrictionsto W of the projectionsof Rp'q on to X and Y, with kernels Y and X respectively,are injective.Hence the result. n

The bound is attained,since there is a subspaceof Rp'qisomorphic to R"', wherer :inf {p,q\, andR''r is neutral.

l{itt decompositions Let X be a non-degeneratefinite-dimensional real quadratic space. A Witt decompositionof X is a direct sum decompositionof X of the form W @WI@(W @W')r,where W andW' arc null subspacesof X. (Some authors restrict the use of the term to the casewhere dim W : indexX.\

Proposition5.25 Let X be a non-degeneratefinite-ilimensional real quadratic spacewith a one-dimensionalnull subspaceW. Then there exists 38 5 The classificationof real quadraticspaces anotherone-dimensional null subspaceW' suchthat the plane spannedby W and W' is a hyperbolicplane.

Proof Let w be a non-zero elementof I7. SinceX is non-degenerate, thereexists x e X suchthat w'x + 0 and x may be so chosenso that w . x : l. Then, for any ). e R, (x * Aw)Q): *(z) 121, this being - - zero if 1 : -rx@. Let W' be the line spannedby *' x |x@w. The line is null since (w/;(2): 0, and the plane spannedby w and w' is isomorphicto Rl'l sincew 'w' : w' 'w : 1, and therefore,for any a,b eF., (aw -lbw')'w:b and(aw *bw'\'w' : a,bothbeing zero onlyif a: b :0. n

Corollary 5.26Let W be a null subspaceof a non-degeneratefinite-dimensional real quadraticspace X. Then there existsa null subspaceW' such that X : W @W' @(W @W)r (a Witt decompositionof X ).

Corollary 5,27 Any non-degeneratefinite-dimensional real quadraticspace may be expressedas the d.irectsum of a finite numberof hyperbolicplanes and a positiue-or negatiue-definitesubspace, a.ny two componentsof the decompositionbeing mutuallyortho gonal. The number of hyperbolic planes in the decompositionis of course equal to the (Witt) index of the space.

Neutral spaces By Proposition 5.24 a non-degeneratereal quadratic spaceis neutral if and only if its signatureis (n,n), for somefinite number n, that is, if and only if it is isomorphic to Rflf, or, equivalently,Rn'n,for somen. The following theoremsometimes provides a quick method of detecting whether or not a real quadratic spaceis neutral.

Theorem 5.28 A non-degeneratereal quadraticspace X is neutral if and only if there existsa linear map t i X -'+X suchthat t*t: -1.

Proof It is enough to remark that t*t: -l e for all x G X,((x);tz) - -xQ). n The next proposition has an important role to play in Chapter 15. 5 The classificationof real quadraticspaces 39 Proposition 5,29 Let w be a possibrydegenerate n-dimensional rear quadratic space.Then w is isomorphicto a subspaceof the neutrsl quadratic spaceR''n.

Euclidean spaces By Theorem 5.10any finite-dimensionalpositive-definite quadratic space is isomorphic to R0'' for some n. Throughout the remainder of this chapter quadratic this spacewill be denotedsimpry by Rr, and any such spaceis said to be a euclideanspace. Let X be a euclidean space. For all b e X the norm of a is, by definition : .1/@1, lal definedfor all a e X sinceae) : e. e> 0, and the distanceof a from b or the length of the line_segmentla,b] is, by definition,labl.In particular,for all A eR,le) : )?, and-lill: 41i1 is the usual absolutevalue.

Proposition5.30 Let a, b, c e R. Then the quadraticform (x, y) _ q.x2+ 2bxy : cyz is positiue-definiteif and only tf a > 0 anil ac _ b2-> O.

Proposition q 5.31Let X be euclideanspace. Then,for all a, b e X,i e R, (i) lal > 0, (ii) lal:0 tf andonly if a:0, (iii) la - bl : 0 if and only if a : b, (iv) p"al: lAllal, (v) a and b are collinearwith 0 if and only tf lbla: _flalb, (vi) a and b are collinear with 0 and not on opposite sidesof 0 if. and onlyif lbla : lalb, .bl (vii) la < lallbl, (viii)a.b

Inequality (vii) of Proposition 5.31 is known as the cauchy-schwarz inequality. It follows from this inequality that, for ail non-zeroa, b e x" -rs1rps1,, a.b 40 5 The classificationof teal quailraticspaces thequotientbeingequaltolifandonlyifbisapositivemultipleofa andequalto_lifandonlyifbisanegativemultipleofa.Theabsolute by angle betweenthe line segments[0, a] and [0, b] is defined

coso::*, o1o

S'>--+Rn;1u,r1- 1\, undefinedonly at the North pole, (0, l), is inuertible.

Proof Since(u, u) e 5", lul2*1t2 :1. So,if x: h, . ., 1.*u 2 1';-:1-r:1-r-l'

and u, and thereforea, is uniquely determinedby x. n The map defined in Proposition 5.34 is said to be the stereographic projection of S' from the North pole on to its equatorialplane Rn x {0}, identifiedwith R'. For, since(u, u) : (I -r)(h, 0) * o(0, l), the three points (0, !), (u, a) and G!-,0)'L-u' are collinear. Similarly the map Sn>--+Rn;(u, ,) * =!-, undefinedonly at the Southpole, is invertible.This is ,tereograpiliffiection from the South pole to the equatorialplane.

Exercises 5.1 ProveProposition 5.23 in the casethat p: e: L (Showfirst that any elementof SO+(1,1)may be written in the form ( sinha "?*' ) '.where,, € R. \ slnnu cosnu I Note that it fru, to be proved, that SO+(1,1) is a subset of so(1,1).) ' 5.2 Let t, u Rn+l + R'!+1be finear transformationsof Rr+l. prove that the following statementsare equivalent: (a) For eachx € S', (x, t(x), u(x)) is an orthonormal 3-frame; (b) r and a are orthogonal,tz : r.t2: -Lna1 andut: -tu. (UsePropositions 4.16 and 5.33and Corollarya.L5.) 'inverses' 5.3 Let i and j : Rn-> gn be the of the stereographic projection of Sn, the unit spherein R,+1 : Rn x R, from its North and South poles, respectively,on to its equatorial plane Rn : Rn x {0}. Prove that, for all x e R'\{0}, i-t i(*) : i-ri(x): x(-r). af,,'^*,' 5 The classirtcaion ::::;.tT::: :ffi 42 ;J:ryr;i *'.'' 5.4 ".Hr$*,Y#,$1t.ffi*** ffi,{:i #;fi

biiective' ?= o] is {hether":l*^':- :'-5:lo- tox* <5 #51il:;i:",-:T::'.::'_",,:Ji- :::'::''il':l"' i-. ", {x€R:i-8x?LL

5.6 (x'I'z)eR':x-r'I "-' 11 and x-z+1=0] ,"a-t -, t x'*-2:-r?+*+2y-42-o I y,y' z) e R'rr Pto.;?tove {(x,(r, --r^:nhtnklink in R3.R3' . -:-r vtuch disloint"ircloscirc B be mutually ' 4 T:"fi* 5:t *on - o)l\b- al I ;;;;-ro" (a,b)r--+ (b : n / B -s2 58'l**5'*:*-rlnjj"trhWj: ons3 definition'f-" .. :^r*r g'rea'icreat ci'rcrescircles

we saw in chapter 4 how a non-degeneratescalar product on R, induces an anti-automorphism of R(n), the adjoint R(n)-+R(1); ,,_* ,j, ,hi, being an anti-invorution in the particular casethat the scararproduct is symmetric. For example, for the standard scalar product (x,'y) r__+xr!, the adjoint of a matrix r is its transposer?. what can be said in the reversedirection? The answeris that any anti-invorution of R(n) is the adjoint of some symmetricor skew scalarproduct on Rn. An intermediaterole is played by reflexioescalar products.

Reflexiuescalar prodacts we are already familiar with symmetric scalar products. The scalar product induced by a correlation f on a finite-dimensionarreal linear spaceX is said to be skew il for all a,b e X,b(a: _otb. A scalar product (a, b)+> a€b on X is said to be reflexiueif, for all a, b e X. a(b:0+bta:0.

Not all correlationsare reflexive.For example,the R-bilinear product of R2,

-+R; R2 x R2 ((a,b),a',b,11 *, aa,+ ab,+ bb,, is induced by a correlation that is not reflexive. we prove first that any reflexiveproduct is either symmetricor skew and then show how any anti-involution of R(n) may be induced by a non-degeneratereflexive scalar product.

43 of R(n) 44 6 Anti-inuolutions ( onW' whete n > I' Proposition6.1 Any non-zetorefi'exite coruelation is either sYmrnetricor skew'

ProofSincebta:0. a(b:0foralla'b€Rn'itfollowsthatforany b r- bta and a,Ecoincide' non-zeroa € Rn the kernelsof the linear maps non-zeroreal number 'l'osuch Accordingly,by Proposition 1'5, there is a that, for all b e Rn, b(a : loaEb' a' For let a/ be any other Now .1odoes not depend on the vector vector c' independentboth of non-zerovector. Since n > 1 there existsa a and of a/ (separately!)'Then, since b(u+bEc: b1(a+c),

it follows that ).oa€b+ A"c(b: Ao+"kt+ c)Eb,

for all b e Rn. So loal ),"c: Lo+"(a*c)'

But a and c are linearly independent'So

)"o: ).a+s: ).2,

exists'/' € R' non-zero'such Similarly, La : A"' So 'la : )"o'Thatis' there that. foi all a, b eRn,b(a: )"atb' There are two cases. Then' since Supposefirst that a€a:0, for all a,b eB"' - - - Z1a€b + b(a) : ata + bEb (a Dc@ b)'

bEa: Aalb- -lb?a,

foralla,beF,:.Now,since(isnon-degenetate'thereexista'bsuch -1. correlation is skew' that bca f0. So ,t : That is, the 0, implying that the The alternative is that, for some a eRn, ata f tr correlation is sYmmetric. 4'13' We then have the following extensionof Proposition

or skewcorrelation on Proposition6.2 For any non-degeneratesymmetric n" ilrc ad'jointendomorphism of R(n) is an anti-inuolution' il lr 6 Anti-inuolutionsof R(n) 45 I ; The following proposition is requiredearly in the proof of the converse i to Proposition 6.2

Proposition6.3 Let a be an anti-automorphismof R(n) anil let t e R(n), with rkt: l. Thenrktd : l.

Proof By Theorem2.6 the map I generatesa minimal left ideal of R(n). Since s is an anti-automorphismof R(n) the image of this ideal by a is a minimal right ideal of R(n). This ideal is generatedby fo; so, by Theorem2.6 again,or, rather, its analoguefor right ideals,rkto : I. n

Now for the converseto Proposition 6.2.

Theorem6.4 Any anti-inuolutiona ofR(n-fl) is representableas the adjoint anti-inuolutioninduced by a non-degenera.terefiexiue correlation on Rn*l .

Proof Throughout the proof we think of R'+1 as Rn x R. - / 0 0\o Letw: rnen w2:w, while,byProposition6.3, w has (; ;). rank 1. Accordinglyw:l)lt, wherez : Rn--+Rand a :R-+Rn arelinear maps,u surjectiveand a injective. Now w2 : w, so uuuu: uu, implying thatuu:1: uwu,while,forall(: ) aRn xR, \d / *(3;)":(3?).(',;)":(3;)"

The map 'o)" rP: R--+R; ^--r(oo , is the identity, sinceuwu : l. Now define

( :R' x R+(Rn x R)t; (;)*,(3;)"

Clearly ( is linear. Moreover it is injective, forif u( : 0, for 3 ; )" 46 6 Anti-inuolutionsof R(n)

xR,then(S :,u( "", (l ).*" ; ). S ; )':o,imprvingthat ( : )- :0, sinceu2: l.So ( is anon-degeneratecorrelationon R'xR. \d /

This correlationis reflexive,since, for all (;), (;,) € R' x R, ((;)'(;))' :;[?!1,,{[jl; 'r) ;0" =.(3 ; )" (Z (:)', : '(3;)"(i)t")(:)' =(i)' (i),,

that ( is reflexive' wherepr:(10)(:) u € R, from which it follows

(',, teR(n*1), Finally, for any (;) , I €Rn xR, andany \a /

rr( u r z,rt( r ('o',,))'(,) : (;,)'(i),

:t.rhatis, eachsidebeingequal., r (; tod rnis ( z';)' ),since the adjointof r with respecito the.correlatron(. n

Real symplecticsPaces

A real symplecticspace is a finite-dimensionalreal linear spacewith a skew correlation (. 6 Anti-inuolutionsof R(n) Let X and Y be real symplectic spaces,with correlations ( and 4 respectively.A linear map f : X --+Y is said to be symplecticif, for all a,beX, f(a)nf(b) : a(b.

Proposition6.5 The real linear spaceR2 with the bilinear product o\ o' R2xR2 -^'\\.-p. ( ( ( \\,- -ab, u)'\u')t ab, is a non-degenerateneutral symplecticspace.

That the spaceis neutral follows at once from the remark that any one-dimensionalsubspace of a symplecticspace is'null. The symplectic space of Proposition 6.5 is called the stqndardreal symplecticplane and denoted ty n,*. Any symplecticspace isomorphic to this one is called a real symplecticplane.

Proposition6.6 Let x be a non-degeneratereal symplecticspace and let W be any one-dimensionalsubspace of X, automaticallynull. Then there exists a one-dimensionalsubspace W' , necessarilydistinct from W, such thst the plane spannedby W and W' is a real symplecticplane.

Theorem 6.7 (The classificationtheorem for real symplecticspaces.) Zer X be a non-degeneratereal symplecticspace. Then X is isomorphicto (R$) : (R3n)k,where 2k: dimX, dimX necessarilybeing euen.

/ o c \ Proposition6.8 Let t : u" an endomorphismof the symplectic ( ; ; ) -t: spaceR{. Then the adjoint oy t r, t. : ( !'. ) . \-b' a') The group of symplecticautomorphisms / : X + f, will be denotedby Sp(X;R). For any finite n :2k the groupsSp(nff) will also be denoted eitherby Sp(2k;R) or by Sp(n;R).These are the ieal symplecticgroups.

Proposition 6.9 An endomorphismt of a real symplecticspace X with correlation( is symplecticif and only if t*t : l.

Corollary 6.10 Any columnof the matrix of an automorphismof the symplecticspace R![ is orthogonalin the ordinarysense to euerycolumn of the matrix exceptone, its partner. 48 6 Anti-inuolutionsof R(n)

Proposition6.ll The determinantof a sympleci:ticautomorphism of a non- ilegeneratereal symplecticspace X is equal to I.

Proof Use Proposition 6.8, Corollary 6.10 and Exercise1.4, considering first the casethat dimX :4. tr 7 Anti-involutionsof C(n)

The classificationof the anti-involutions of c(n), regarded as a real algebr4 parallels the work of the last chapter,but also extendsit with the introduction of hermitianproducts andunitary groups.

Complex quadratic spaces Much of the last three chaptersextends at once to comprex quadratic spaces,or indeed quadratic to spacesover any commutativefield K (of characteristicnot equal to 2), R being replacedsimply by C, or by K, in the definitions,propositions and theorems.An exceptionis the sijnature theorem,Theorem 5.22,which is false. The main classification theorem for complex quadratic spaces-r---- is.the following.

Theorem 7.1 Let x be a non-degeneraten-dimensionar comprex quadratic space.Then x is isomorphicto cn with its standardcomplex'scalar product --Cn C" x Cn : (a,b)++ D otu,' 0

Proposition7.2 Let x be any non-iregeneratecomprex quadratic space of euendimension 2n. Then x is neutrar,being isomorphic not onry to c2n but also to Cn'nand tu 2Cn. For any complex quadratic spaceX the group of orthogonal auto_ morphismsof X will be denotedby O(X;C), the group O(i,;C) being normally denotedby o(n;c). Theseare the comprexortho'goni,at'groups.

49 50 7 Anti-inuolutionsof C(n) As in the real case,the determinant of any elementof O(n;C) is equal to *1 or -1, the subgroup of elementsof determinant 1 being denoted by SO(n;C).Such groups are the specialcomplex orthogonal groups.

Complex symplecticspaces A complexsymplectic space is a finite-dimensionalcomplex linear space with a skew correlation (. Let X and Y be complex symplecticspaces, with correlations( and 4 respectively.A linear map -f : X --+Y is said to be symplecticif, for all a,beX, f(a)af(b) : a:b.

Proposition7.3 The complexlinear spaceC2 with the bilinear product

cz xcz._,c,'"' (/ g \. (:l )) t+ ab,_ab, \\ b )'\b' ) ) is a non-degenerateneutral symplectic space.

That the spaceis neutral follows at once from the remark that any one-dimensionalsubspace of a symplecticspace is null' The symplecticspace of Proposition7.3 is called the standsrdcomplex symplecticplane and denotedbV C3n.Any symplecticspace isomorphic to this one is called a complexsymplectic plane.

Theorem7.4 (Classificationtheorem for complex symplecticspaces.) Zer X be q non-degeneratecomplex symplectic space. Then X is isomorphicto (C3f): (C3o)u,where 2k: dimX, dimX necessarilybeing euen'

'O) Proposition7.5 Let t : (i o" an endomorphismof the svmplectic -.: : spaceC$. Thenthe adjointof t is t* (. _r. ) a, / The group of symplecticautomorphisms / : X --+X will be denotedby Sp(X;C). For any finite n:2k the groupsSp(Cff) will alsobe denoted eitherby Sp(zk;C)or by Sp(n;C).

Proposition7.6 An endomorphismt of a complexsymplectic space X with correlation( is symplecticif and.only if t*t: L 7 Anti-inuolutionsof C(n) 51 Proposition7.7 The determinantof a symplecticautomorphism of a non- degeneratecomplex symplectic space X is equal to l.

The proof follows the sameroute as the proof of proposition 6.11

Heymitian spaces complex quadratic spacesare not to be confusedwith hermitian spaces, finite-dimensionalcomplex linear spaces that carry a hermitian form. These arise when the field c is regarded as a real algebra, assigned complex conjugation as an anti-involution, e then denoting c with this assignment. Let X and Y be complex linear spaces. Then an R_linear map --+Y f : X is said to be semi-linearover C if for all x e X,)" e C eitherfQ,x): ),f(x) or f(Ax):7f@\ In the former casethe map is cJinear. In the latter caseit will be said to be e-finear. A semi-linear map t uniquely determinesthe automorphism rp of c with respect to which it is semi-linear,unless / : 0. on occasionit is convenientto refer directly to rp, the map t then being said to be c,p-Iinear (not ,cv-semi- linear' since,when g : lc, C,pis usually abbreviated 'C-semi-linear' to C and the term could thereforebe ambiguous.l. The composite of any pair of composablesemi-linear maps is semi- linear and the inverseof an invertible semilinear map is semi-linear.An invertible semi-linearmap is said to be a semi-Iinearisomorphism.

Proposition7.8 Let f :x--+y be a semi-linearmap ouer c. Theniml ls a C-Iinear subspaceof Y and ker/ is a C-Iinear subspaceof X.

Rank and kernel rank are defined for semi-rinearmaps as for linear maps.

Proposition7.9 Let f : x --+y be a cv-lineqr map, wherex and,y are finite-dimensionalc-Iinear sp&ces,and p is complexconjugation. Then, for anyyeYL,pyfeXL.

Proof The map vyf is certainly R-linear. it remains to consider its interaction with conjugation.However, for eachx €.X, ). e C, rpyf(tx) :rt@ : rtf @):il|@) : hiflu;r: Arpyf(x). ! 52 7 Anti-inuolutionsof C(n)

The map f" : Y"-XL, defined,for all y e YL, by the formula f"(y):Vyf ,is calledthe dual of f . -'+Y Proposition 7.10 The dual f" of a e-hnear map f : X is e-hnear. Many propertiesof the duals of Rlinear mapscarry over to semi-linear maps over C. A correlationon a finite-dimensionalClinear spaceX is a semiJinear mapC : X -- XL; vt--+x(.The map X x X + (a,b),- a€b: 46(b)is the product induced by the correlation, and the map X - C; a t+ aEa the form inducedby the correlation. Such a product is R-bilinear, but not, in general,C-bilinear, for although the map X --+C; xr-- a(x is Clinear, for any a e X, the maP X -+C; yr-+x(b. for any b e X, is, in general,not linear but only semi-linearover C. Such a product is said to be sesqui-linear,the prefix being derived from 'one a Latin word meaning and a half times'. Let rp denote either of the involutions of C. Then a Cp-correlation ( : X --+XL and the induced product on the Clinear spaceX are said to be, respectively,symmetric or skew with respect to rp or ouet Cv accordingas, for eacha, b e X, 61s: 1s

Proposition7,ll Any skewe-correlationon a C'linear spaceX is equiualent to a symmetrice-correlationon X, and conuersely. 7 Anti-inuolutionsof C(n) 53 Proof Let ( be a skew e-correlation on X. Then i( is a e-correlation on X since,for all x € X, A e C,

(i3(xl) : i(((xl)) : ih((x): z(i(Xx). Moreover,for all a, b e X, bi€a:ib(a:(_le&D:F6. That is, i( is symmetricover e. Similarly, if ( is symmetricover e then i( is skew over e. tr

Equivalent semi-linearcorrelations ( and 4 on cn induce the same adjoint anti-automorphismof C(n). Supposenow that f : X --+ y is a linear map of a non-degenerate hermitian spaceX, with correlation (, to a hermitian spacey, with correlation 4. Since( is bijectivethere is a unique Clinear map t* : y --+X such that (t* : 1-rtt4, that is, such that, for any x € X,y € y, t.(y).x : y't(x). The map t' : 1-rtr, is calledthe adjoint of r with respectto ( and 4. The adjoint of a linear map u:x --+x with respectto a non-degenerate correlation ( on a complex linear spaceX will be denoted by u(. The map u is said tobe self-adjointif ut : u &r:tdskew-adjoint if ut : -u. The real linearsubspaces {ueEndX :u( :u} and{u e EndX :ut : -u} of : the real linear spaceEndX L(X, X) will be denotedby Endl(X, o and End-(X, O, respectively.

Proposition7.12 Let x be a non-degeneratehermitian space. rhen the map --+ End X End X; f - f. is an anti-inuolutionof End,X, regardedas a real algebra.

Proposition7.13 Any non-zeroreflexiue correlation ( on a hermitianspace X of dimensiongreater than one is either symmetricor skew.

Proof This is just a re-run of the proof of proposition 6.1, but taking account of conjugation. For all a,b e X,b{o:0 + glo:0 e atb:0. Thatis, for any non-zeroa e X, the kernelsof the surjectiveC-Iinear maps b ,-Fd and a6 coincide. Therefore,by Proposition 1.5, there is a non-zerocomplex number,l,osuch that, for allb e X,

6{a: loa(b, 56 7 Anti-inuolutionsof C(n)

Proposition7,20 For any finite n, and p * q : n, with a rather obuious definitionof O(p,q;C), O(n) O(n;C)n Gr(n;R) : O(n;C)oU(n), SP(2n;R): SP(2n;C)nGL(2n;R), O(p,q) O(p,q;C) nU(p, q). ClearlyU(1) : S1,the circlegroup of complexnumbers of modulus1. MoreoverU(l) = SO(2). 8 Quaternions

Sincefor &r! z : x*iy e C,lzl2 :Zz : xz + y2, the field of complex numbersc, when identifiedwith the linear spaceR2, can alsoin a natural way be identified with the real quadratic spaceR0,2. Moreover, since, for anyz eC,

lzl:Lex2+y2:1,

the subgroupof the group of invertible complex numbersc*, consisting of all complex numbers of absolutevalue 1, may be identified with the unit circle51 in R2. In what follows the identification of C with p2 - po,z is taken for granted. We have akeady remarked in Proposition 2.I that for any complex numberc: a*ib, the map C---rC; z -> cz may be regardedas the -' reallinear map R2+ R2 with ^ut i* ( ? ). torrover. lcl2: Ec : \b a/ . / a -b\ :1+ detl I lcl:1, while,for allc e C, with lcl:1, andall \D a/ z e C, lczl : lzl. Thus,(cf. Proposition 4.19) multiplication by a complex number of modulus 1 inducesa rotation of R2, and any rotation of R2 may be so representedby a point of the unit circle,S1. The following statementsums this all up.

Proposition8.1 SO(2)= St.

The group Sl is called the circle group. Anti-rotations of R2 also can be handled by C, for conjugation is an anti-rotation, and any other anti-rotation can be regarded as the composite of conjugation with a rotation.

57 58 8 Quaternions The algebraof quaternions,H, the subjectof this chapter,is analogous in many ways to the algebraof complexnumbers C. For example,it has application to the description of the groups O(3) and O(4). The letter H is the initial letter of the surnameof Sir William Hamilton, who first studied quaternionsand gave them their name (1844).

The algebra H Let 1, i, j and k denote the elementsof the standard basis for R4. The quaternionproduct on Ra is then the R-bilinearproduct Ra x Ra --Ra; (a,b)++ab with unit element1, definedby the formulae i2: j2: k2 : -l

andij: k: -ji, jk: i: -kj, ki: j: -ik.

Proposition8.2 The quaternionproduct is associatiue. On the other hand the quaternion product is not commutative. For example,ji + ij.Moreover it doesnot necessarilyfollow that, if a2: b2, thena: *b. For example,i2 : j2, but i I !. The linear spaceRa, with the quaternion product, is a real algebra H known as the algebraof quaternions.In working with H it is usual to identify R with R{1} and R3 with R{i,j,k}, the first identification having been anticipatedby our use of the symbol 1 to denote the unit elementof the algebra.The subspaceR{i,j,k} is known as the subspace of pure quaternions.Each quaternionq is uniquelyexpressible in the form rcq+puq, wherereq e R and puq €R3,req beingcalled the real part of q and pu4 the purepart of q.

Proposition8.3 A quaternionis real if and only if it commuteswith euery quaternion.That is,R is the centreof H.

Proof > : Clear. <: Let q: q+bi-fcj*dk, wherea,b,c,d arcrcal,be a quaternion commuting with i and j. Sinceq commuteswith i, ai- b * ck- dj: iq: qi: ai- b - ck-tdj, implying that 2(ck- di :0. So c : d.:0. Similarly,since q commutes with j, b : 0. So e : a, and is real. ! 8 Quaternions 59 Proposition8.4 The ring structureof H inducesthe reql linear structure, and any ring automorphismor anti-automorphismof H is a real linear automorphismof H, and thereforealso a real algebra automorphismor anti- automor phi sm of H.

Proof By Proposition 8.3 the injection of R in H, and hence the real scalarmultiplication R x H --+H; ()",q)- ).q, is determinedby the ring stfucture. Also, again by Proposition 8.3, any automorphism or anti-automorphism r of H maps R to R, this restriction being an automorphism of R and therefore the identity, by Proposition 2.9. Therefore f not only respectsaddition and respectsor reversesmultiplication but also,for any ,t e R and q e}J.,{),q) : tQ,)t(q): Lt(q). n

This result is to be contrastedwith the more invorved situation for the field of complex numbersdescribed following proposition 2.10. The automorphisms and anti-automorphismsof H are discussedin more detail below.

Proposition 8.5 A quaternionis pure if and only if its squareis a non- positiuereal number.

Proof + : Consider the pure quaternion e : bi * + dk, where "j b, c, d e R. Then q2 : -(b2 + c2+ dz),whichis real and non-positive. < : Considerq: a+bi+cj+dk, wherea, b, c, d e R. Then

q2 : a2- 6z - c2- d2* 2a(bi+ cj + dk).

Ifq2 is real,either a:0 and4 is pure,ot b: c: d:0 anda< 0. in which caseq2 is positive. So, if q2 is rcal and non-positive,q is pure. n

Proposition8.6 The direct sum decompositionof H, with componentsthe qnd real pure subspaces,is inducedby the ring structurefor H.

The conjugateQ of a quaternion q is defined to be the quaternion req - puq.

Proposition8.7 Conjugation: H-+H; q e q is a real algebraanti- inuolution.Thet is,for all a,b eH andall,le R, aTE :a +6, TA: )"a,a: aundA| :6d. 60 8 Quaternions -a, Moreouer,a € R e a: a anda € R3+ a: whilergq: l1a+a) : -a). andpua ){o Now let H be assignedthe standardpositive-definite scalar product on Ra,denoted as usualby ..

Proposition8.8 For all a, b eH, a'b : re(db): j6a+E d. In particular, for anyq eIJ, aa: a. a, soda is non-negatiue. In pafticularalso,for all a, b eRi3,a.b : -l1aU+b q) : -rc(ab), with aQ)- a' &: -a2 andwith a.b :0 if andonly if a andb qnti-commute.

The non-negativenumber lal: $a is called the norm ot modulusor absoluteualue of the quaternion a.

Proposition 8.9 Let x be a non-reql elementof H. Then R{1, x} is a subalgebraof H isomorphicto C. In particular. R{1, x} is commutatiue.

Proof lt is enoughto remark that

xz+bx*c:0, whereb : -(x * x), and c :ix, b and c both beingreal. n

Proposition8.10 Each non-zero a eH is inuertible,withq-r :lal-2a and withla-rl: lal-r. Note that the quaternion inverse of a is the conjugate of the scalar productinverse of a, seD : @l-2a. By Proposition 8.10,H may be regardedas a non-commutativefield. The group of non-zeroquaternions will be denotedby H-.

Proposition8.ll For all a, b eH, labl: lallbl. A quaternionis said to be a unit quaternion

Proposition8.12 The set of unit quaternionscoincides with the unit sphere 53 in R4 and is a subgroupof the group H-.

Proposition8.13 Let q eH be suchthat q2 : -1. Thenq e 52,the unit spherein Fi3.

Proof Sinceq2 is real and non-positive,q e R3 while, sinceq2 - -1, lql: l. Soq e s2. n 8 Quaternions 6L The crossproduct axb of apair (a,b) of pure quaternionsis defined by the formula

axb:plu(ab).

Proposition8.14 For all a, b e R3

ab : -a, b a x : _ + b, anda x b L@U b a) : _(b x a), q while x e : a. (ax b): b. (ax b): 0. If a and b are mutuqily orthogonarelements of R3 then a and b anti- commute,that is, b a: -ab, and a x b : ab. In particular,

i. fi x k) :i. 0k):i.i : _i2: l.

Proposition 8.15 Let q be any quaternion. Then there existsa non_zero pure quaternionb suchthat qb also is a pure quaternion.

Proof Let b be any non-zero pure quaternion orthogonal to the pure part of 4. Then qb: (req)b+ (puq) x b, beingthe sum of two pure quaternions,also is pure. !

Corollary 8.16Any quaternion q is expressibleas the proiluct of a pair oJ. pure quaternions.

Proof Let b be any non-zero pure quaternion orthogonal to the pure part of q. Then qb: (req)b+(puq) x b, beingthe sum of two pure quaternions,also is pure. rr

Proposition8,17 For any a,b, c € H,

re(abc) : re(bc a) : re(cubl.

Moreouer,for any a, b, c € Fr3,

re(abc) : -a ' (b x c) : - det(a,b, c), where (a, b, c) denotesthe matrix with columnsa, b and c. We call the real number re(abc): ct.(bc) the scalartriple product of the quaternions a, b, c, in that order. (In the case that a, b,, ur" pure it is more usual to take the negative of this as the scalar triple product.) 62 8 Quaternions Automorphismsand anti-automorphismsof H By Proposition 8.4 the field automorphismsand anti-automorphismsof H coincide with the R-algebraautomorphisms and anti-automorphisms of H. In this section we show that they are also closely related to the orthogonal automorphismsof R3. The relationshipone way is given by the next proposition.

Proposition8.18 Any automorphismor anti-sutomorphismu of H is of the formH--+H; a+ re&-lt(pua), wheret is an orthogonalautomorphism of R.3.

Proof By Proposition 8.4, Proposition 2.9 and Proposition 8.5, u is a linear map leaving eachquaternion fixed and mapping R3 to itself. Also, for eachx € R3, @@))2: u(x2): x2, sincex2 € R, while lxl2 : -x2. So

I : R3--+ R3: x r.1 u{xt is linear and respectsnorm. Therefore, by Proposition 5.32, it is an orthogonalautomorphism of R3. n

In the reversedirection we have the following fundamentalresult.

Proposition8.19 Let q be an inuertiblepure quaternion.Then, for any pure quaternionx, qxq-r is a purequeternion, and the map

-pn , R' -- R3; x t--+-q x q-l is reflectionin the plane(R{q})t.

Proof Since(q*q-t)t: x2, which is real and non-positive,qxq-1 is pure,by Proposition8.5. Also -pq is linear, and-pn@): -q, while,for anyr e (R{q})I, Pq(r): -qrq-l :rqe-r: r. Hencetheresult. n

Proposition8.19 is usedtwice in the proof of Proposition8.20.

Proposition8.20 Each rctation o/ R3 ls of theform pn for somenon zero quaternionq, and euerysuch map is a rotationof R3.

Proof Since,by Corollary 5.1,7,any rotation of R3 is the compositeof two plane reflectionsit follows, by Proposition8.L9, that the rotation can 8 Quaternions 63 be expressed in the given form. The converseis by Corollary g.16and Proposition8.19. n In fact, eachrotation ofR3 can be so representedby a unit quaterniron, unique up to sign. This follows from proposition g.21.

Proposition --+SO(3); 8.21 The mapp: H* e+ pq is a groupsurjection, with kernelR. , the restrictionof p to s3 alsobeinf surieitiue,wrti kerner 5o: {*1}.

Proof proposition The map p is surjective,by g.20,and is a group map since,for all q, r e H., and all x e R3,

pq,(x): qr x(qr)-r : pqpr@).

Moreover, q € kerp if and only if qxq-l : x, for all x e R3, that is if and only if qx: xq, for all x e R3. Therefore,by propositiong.3, kerp-RnH*:R*. The restriction of p to 53 also is surjectivesimply because,for any ,t e R- and for any q eH*, pt"q: pq, aLfid). may be so chosenthat Itrql:1. Finally,ker(p l53) : kerpO 53 : R. n 53: S0. n Proposition8.22 will be usedin propositiong.26.

Proposition 8.22Any unit quaternionq is expressiblein the form ab a-L b-r. wherea and b are non-zeroquaternions.

Proof By Proposition 8.15 there is, for any unit quaternion q, a non_ zeropure quaternionb suchthat qb is a pure quaternion.Since I ql: l, I q b | : I b l. Thereis therefore,by proposition g.20,a non-zeroquaternion o suchthat qb: abe-r, thatis, such that e: abq-t b-r. n

Proposition8.20 also leadsto propositiong.23, converse to prooosi- tion 8.18.

Proposition8.23 For eacht e O(3), the map

u:H-_+H;a>reA*t(pta) is an automorphismor anti-qutomorphismof H, u being an automorphism qn if t is a rotation and anti-automorphismif t is an anti-rotationof R3. 64 8 Quaternions Proof For eacht € SO(3),the map u can,by Proposition8.20, be put in the form H --+H; a,-- q a q-t : rea * q (Pua)q-r,

where q c H*, and such a map is an automorphismof H. Also, -1*, is an anti-rotationof R3,and if ,: -1ns, u is conjugation, which is an anti-automorphismof H. The remainderof the proposition follows at once,since any anti-automorphismof H can be expressedas the compositeof any particular anti-automorphism,for exampleconjugation, with someautomorphism. !

Corollary 8.24 An inuolutionof H either is the identity or correspondsto the rotation of R3 tlrough n about some axis, that is, reflectionin some line through0. Any qnti-inuolutionof H is conjugationcomposed with such an inuolution,and correspondseither to the reflectionof R3 in the origin or to the reflectionof R3 in someplane through 0.

It is convenientto single out one of the non-trivial involutions of H to be typical of the class.For technicalreasons we choosethe involution H--+H; a+ iaj-l, correspondingto the reflectionof R3 in the line R{j}. This will be called the main inuolutionof H and, for each a e H, 6: iuj-r will be calledthe inuoluteof a. The main involutioncommutes with conjugation. The compositewill be called reuersionand, for each a e H. d. : G: ? will be called the reuerseof a. A reasonfor this is that H may be regardedas being generatedas a real algebraby i and k, and reversionsends i to i and k to k but sendsik to ki, reversingthe multiplication. Reversionin a more generalsetting will be definedlater, in Chapter15. Proposition 8.25is required in the proof of Proposition 10.9.

Proposition 8.25 The map }I--+H; x ++ ix hqs as image the three- dimensionalreal linearsubspace {y e ff :T : y}: R{1, i, k}.

Proof It is enoughto provethat the map 53-- 53; x t--+i x: i-1 x has as imagethe unit spherein R{1, i, k}. So let y e H be suchthat T y : I andy :|. Then I-ly:TyIl:(i+1)1.

So,if y # -1, y :i-i x,wherex : (1+yxll +yl)-l. Finally,-1 8 Quaternions 65 Rotations of Ra Quaternions may be used to represent rotations of Ra.

Proposition 8.26 Let q be a unit quaternion, with H identified with F(4. Then qy : Ra--+f,.a; t--+ x qx is a rotation of R4, as also is the man 4n :Ra--R4; x t--+xq.

Proof The map er is linear, and preservesnorm by proposition g.11; so it is orthogonal, proposition by 5.32, That it is a rotation foilows fiom Proposition 8.22, which states that there exist non-zero quaternions a andb suchthat q: aba-r b-r,for thenq7: a1b1(a7)-r(U")-r,implying that : detp(ar.) 1. Similarly for 4p. r-1

Proposition8.27 The map

p : ^S3x 53+ SO(4);(q,r) r+ qTTp is a group surjectionwith kernel{(1, 1), (_1, _1)}.

Proof For any e, 4,, r, r, g ,S3and any x € H, p(q'q, : r'r)(x) (q'dL(rrr)nx : e,4xll : p(q,,r,)p(q,r)(x). Therefore,for any (q, r), (q,,r,) e 53 x 53,

p((q', r')(q, r)) : p(q,,r,)p(q, r): that is, p is a group map' That it has the statedkernel forlows from the observationthat q if and,r are unit quaternionssuch that qx\ : x, for all x e H, then, by choosingx: l, qF : l, from which it follows that qxq-r: x for all x e If, or, equivalently,Ihat qx: xq for all x e H. This implies,by Propositiong.3, that q e {t, _tf : R n i3. To prove that p is surjective, 7et t be any rotation of Ra and let s : (1). Then : p+ --+Rai lsl 1 and the map x r_+S(r(x)) is a rotation of Ra leaving 1 and thereforeeach point orn fixed. so, by proposition g.20, there existsa unit quaternion r such that, for all x € Ra,

3(/(x)) : r xr-l or, equivalently, t(x) : e xF, where 4 : 51.. n Anti-rotations of Ra also are easily representedby quaternions,since conjugation is an anti-rotation and since any anti-rotation is the com_ li posite of any given anti-rotation and a rotation. ll l' irl 66 8 Ouaternions Exercises 8.1 Let q and r be non-zeroquaternions such that lql: lrl, and : -r). let a: i@+r) and B itq Provethat eitheru or B is non-zero (cf. Proposition 4.2) and that if a is invertible then B a-1 is a pure quaternion,while if B is invertible then a B-r is a pure quaternion. 8.2 Let a and B be non-zeroquaternions such that B d is pure. Prove that, as vectorsof R4, a and B aremutually orthogonal. 8.3 For any unit quaternion rc and any 0 e R let eoadenote the quaternioncos0 * rcsin0.Prove that, for any s, / € R, : ent, whererc: i cos2s- j sin2s. "ksritn-ks (This identity is the starting point of a recent paper by V.I. Arnol'd (1995), that applies the algebra of quaternionsto the geometryof curveson the sphereS2.) 9 Quaternioniclinear spaces

Much of the theory of rinear spacesand linear maps over a commutative field, summarised in chapter 1, extends over -[I. Becauseof the non- commutativity of H it is, howevet,necessary to distinguishtwo types of linear space over H, namery right rinear spaces and reft rinear spaces. There are therefore restrictionson possibretypes of rinear maps over H.

Right and left linear spaces A right linear spaceover H consistsof an additive group X and,a map X x H --+X; (x, A)+>x) such that the usual distributivity and unity axioms hold and such that, for all x e X,), 1, eH., (x ),)1': x(A).'). A left linear spaceover H consistsof an additive group x anda map H x X -- X; (p,x)r- px such that the usuar distributivity and unity axioms hold and such that, for all x e X, p, lt' e H, lt'(px) : (t' tix. The additive group Hn, for any finite n, and in particular H itself, can be assigned either a right or a left H-linear structure in an obvious way. unless there is explicit mention to the contrary, it will normally be assumed that the fight H-rinear structure has been chosen. (As we shall seebelow, a naturarnotation for H, with the obvious left H-rinear structureis either(H,)r or (Ht)r.)

67 68 9 Quatetnioniclinear sqaces of such spaces Subspacesof right or left H-linear spacesand products hold' except are definedin the obviousway. Likewisethe basistheorems thatcaremustbetakentoputscalarmultipliersonthecorrectside. number n of Where there is a finite basis for an H-linear spacethe by definition the elementsof the basisis independentof the basisand is quaternionicdimension,AimnX,ofX'ltwiLtbetacitlyassumedinallthat is Tottow that any H-linear sp&ceencountered finite-dimensional' spaces'may be Lineat maps t ; { --+Y, where X and Y are Hlinear a right linear space defined,prouia"a that eachof the spacesX and Y is and Y are both or that each is a left linear space' For example' if X linear if it respects right linear spaces,then t is said to be linear or right (t(x)Y" an analogous uJditlon and if, for all x e X,)" eH,t(xl): definition holding in the left case' Thesetoflinearmaps':X.-+Ybetweeneitherrightorleftlinear caseby L(X'Y)' spacesX and Y over H will be denotedin either fails' For Iio*"u"r, the usual recipe for L(X,Y) to be a linear space )' e H' a map supposethat we define, for any t e L(X, Y) and any right Hlinear t i i X'-+ Y by the formula (t ),)x : t(x) )",X and Y being spaces.Then, for anyt e L(X,Y) and any x e X' -l(x)k' l(x)k: (rijXx): (tixxj): r(xj)i: possible' Normally leading at once to a contradiction if t + 0, as is H' namely R' In L(X, i) is regardedas a linear spaceover the centreof L(X'X) is pur,i"uiur, foi any right H-linear spaceX,rhe set EndX: normally regardedas a teal algebta' left Hlinear On the other hand, for any right linear spaceX over H' a p(t(x))' for all ,t.o"tu." can be assignedto L(x,H), by setting(pt)(x): teL(X,H),xeHandpe['Thisleftlinearspaceiscalledthelinear a left H-linear dual of x andis also denotedby xL. The linear dual of spaceis analogouslydefined. It is a right HJinear space' For any finite-dimensionalHlinear spaceX, dimnXL: dimnX' --+Y map tL : Any right H-linear map t : X induces a left linear YL -- XL by the formula rt(T): yt, for eachYeYL, Y all being right and if t e L(X,Y) and u e L(W,X), W,X and HJinear sPaces,then (tu\L : YL 7L' 9 Quaternionic linear spaces 69

Quaternionicmatrices Any right Hlinear mlp t tHn --+Hmmay be representedin the obvious way by an m x n matrix with entries in H. In particular, any elements of the right Hlinear spacesH' and H may be representedby column matrices.Scalar multipliers have,however, to be written on the right and not on the left as has beenthe customhitherto. In fact this is really more logicalanyway, as the equationt(x,)'\: t(xY" may then be regardedas exemplifyingthe associativelaw for compositionof matrices. For example,suppose that t e EndH2, x eH2 and A e H' Then the statementthat t(x 1) : t(xY" becomes,in matrix notation, (nfl)(;:i): ((';ll )(;l)) " The left H-linear space (H')t dual to the right H-linear spaceHn may be identified with the additive group Hn assignedits left H-linear structure. Elementsof this spacemay be representedby row matrices. A left Hlinear map u : (H )L -*(H')' is then representedby an m x n matrix (sic) that multiplies the row vector elementsof (H')t on the right. H(n) will denote the real algebra of n x n matricesover H. Any quaternionic linear spaceX may be regarded as a real linear space,with dimn X : 4 dimsX. Such a spacemay also be regardedas a complex linear space,once some representativeof C as a subalgebraof H has beenchosen, with dim6 X : 2 dimn X. In the following discussion g2n : gn C is identified.with R{1, 4 in H, and, for each frnite n, x Cn is identified with H' by the (right) complexlinear isomorphism

Cn x C" --H' ; (u, u) ++ u I ju.

Proposition9.1 Let a -t jb e H(n), whqle a and b e C(n). Then the / a -h\ correspondingelementofC(2n) u (; d )

Proof For any tr,u,Lt',u' € C'and any a,b c C(n), the equation u'+ju' :(a*ib)(u+lb) is equivalentto thepair ofequations

1,r': au -Eu, I)' : bu*du. tr

In particular, when n : l, this becomesan explicit representationof 70 9 Quaternioniclinear sqeces H as either a complex or a real subalgebraof C(2), analogousto the representationof C as a subalgebraof R(2) given in Corcllary 2'2' Noticethat,foranyq-a*ib e H,witha,beC, -2) lql':4 q: qa*1, : o* (i

This remark is a detail in the proof of Proposition9.3 below' The lack of commutativity in H is most strongly felt when one tries to introduce products,as the following proposition shows.

Proposition 9,2 Let X, Y and z be fight linear spacesouer H and let t:X xY --+Z;(x, y)- x'y bearight bilinear map. Then t:0.

Proof For any (x, y) e X x Y,

(x'y)k: (x'y)ij : (x'yi)j - xj'Yi: (xj'Y)i : (x'v)ii: -(x'v)k.

Sincek * 0, r'.1 : 0. So r : 0. n

It follows from this, a fortiori, that there is no non-trivial n-linearmap Xn --+H,for a right Hlinear space,for any n> 1. In particular, there is no direct analogueof the determinantfor the algebraof endomorphisms of a quaternioniclinear space,in particular the right H-linear spaceHn. However, any n x n matrix over H is reducibleby column operationsto a diagonal matrix all of whose entries except one are equal to 1, just as in the real or complex case as describedin chapter 2, though this number is no longer independentof the route taken, as Exercise9.2 will show.However, any two such numbershave the sameabsolute value, as follows from the next proposition.

Proposition9.3 Let X be a right H-linear spaceand let t : X'--+X be an H-linear map. Then det6 t is a non-negatiueteql number,

Proof Mimic the proof of Proposition2.3. tr

For any Hlinear endomorphismt : X --+X, the (positive) squareroot of det6r is definedto be the (absolute)determinant of t,dett. Clearlyt is invertible if. and only if det I + 0. The group of all invertible right linear maps t : H'-> Hn over H or, equivalently,the group of all invertible n x n matrices over H will be Exercises 71 denotedby GL(n;H) and the subgroupof all suchmaps or matricesof determinant1 by Sf(n;H). clearly earlier remarks about direct sum decompositionsand the 2R(n) 2c(n) matrix algebras and extend to direct decompositionsof quaternionicspaces and the (real) matrix algebrastll(r).

Exercises 9.r Verifythat tr,. -atri* ( 1 is invertible in H(2), but that i ) thematrix(1j),rno,. \1K/ 9.2 verifyrhat ttre matrix ( 1 may be reducedby a pair of i ) -2k elementarycolumn operations 0 \ ' not only" to f \ 0 r)butalso 'oIo/t 0\ 2k) 9.3 Does a real algebrainvolution of H(2) necessarilymap a matrix o of theform(: ),*h.r, aeH,tooneof thesameform? \u 4,/ 9.4 Verifythat themap a: C2-'H; x" xo*jx1 is a right C-linear isomorphism,and compute

1afu)),for anyx, y e C2. "-11/[ Let Q : {@, y) € (C\2 : x0.y0I xtyr: 1}. Proverhat, for any (a, b) e H- x C,

(o-t@),a-l@-t(r + jb)))e e and that the map

H' x C - e; (a,b)+> (u-r@),u-r1a-r1t + j b))) is iniective.

l 10 Anti-involutionsof H(tz)

Proposition 9.2 has shown us that there are no non-trivial quadratic forms on a quaternionic linear space. There are, however, analogues of hermitian forms and these induce adjoint anti-involutions of real quaternionicmatrix algebras.

Correlated quaternionic speces LeI X and Y be right or left quaternionic linear spaces.Then an H- linear map f : X --+Y is said to be semi-linearover H if there is an automorphism or anti-automorphismrp of H such that, for all x e X, A e H, f(xl) : f(x)lv, f(il,) : )vf(x), f(lx) : ['pf(x) or f().x) : f (*) )"e, as the casemay be, rp being an automorphismof H if H operates on X and Y on the sameside and an anti-automorphismif H operates on X and Y on opposite sides. The terms right, right-to-left, left and left-to-right semi-linearmaps over H have the obvious meanings. The semilinear map r uniquely determinesthe automorphismor anti- automorphismrp of H with respectto which it is semi-linear,unless / : 0. On occasionit is convenientto refer directly to rp, the map t then being saidto be Elv-linear. The following maps are invertible right-semilinear maps over H:

H--+H; xF+,x x> ax, for any non-zeroa c I{, xe, xb-t,,for any non-zerob g H. xe axb-r, for any non-zefoa, b eII, , x r-- i (: jrj-r), and H2 ---rH2; (x, y) - (ax, b y), for any non-zeroa, b eFI, (x, y),--t (by, ax), for any non-zeroa, b eH,

72 10 Anti-inuolutionsof H(n) 73 the correspondingautomorphisms of H being,respectively,

111,111, Ap-+ b Lb-r, ),r-+b ).b-1, Ar-j and 111,111. By contrast,the map --rI* H2 ; (x, y) - (x a, y b), with a, b € H,

is not |:ght semi-linearover H, unless,l,a : pb, with A, p eR. The maps

112--H2i (x, y)r--+(X,y), (x, Y)++(y-,7) are invertibleright-to-left E-linear maps. The compositeof any two composablesemi-rinear maps is semi-linear and the inverse of an invertible semllinear map is semi-linear. An invertible semilinear map is said to be a semi-Iineir isomorphism.

Propositionl0.l Let f :X-+y be a semi-linearmap ouerlH.Thenimf is y unH-Iineqr subspaceof and ker/ ls qnH_linearsubspace of X. Rqnk and kernel rank are defined for semi-linearmaps as for linear maps.

Proposition 10,2Let f : X--+Y be anH'p-linearmap, where X anil y are finite-dimensionalH-linear spaces,and p is an automorphismor anti- automorphismof H. Then,for any y e yL,g:yf e XL.

Proof The map Vyf is certainly R-linear. It. remains to consider its interaction with H-multiplication. There are four cases,of which we consider only one, namely the casein which X and I are each right H-linear. In this case,for eachx e X, A eIJ,

rp-lyt(x 1) : p-ly(t(x) A*) : ,p-1((yt(x))Av) : (p-1yt(x)fi.. The proofs in the other three casesare similar. n yt-XL, -The map f" , defined,for all y e yL, by the formula ft(y): Vyf , is calledthe dual of f .

Proposition 10.3 The duat fL of an Elv-linear map f : X__+y is H'P-t-linear. '14 10 Anti-inuolutionsof H(n) Many propertiesof the duals of R-linear mapscarry overto semi-linear maps over H. A correlation on a finite-dimensionalHlinear spaceX is an H-semi- linearmap ( :X --+XL;x,--, xt. The map X x X--+ (a,b),--ta(b: qE(b) is the producr induced by the correlation,and the map X --+C; aF+ sEq the form induced by the correlation. Such a product is R-bilinear, but not, in general,H-bilinear, for although the map

X --+H; xr--ta(x is H-linear, for any a € X, the map

X --+H; xr--,xcb, for any b e X, is, in general,not linear but only (right-toJeft) semi-linear over H. As in the parallele casesuch a product is said to be sesqui- lineur. An He-correlation ( : X --+XL and the induced product on the right H-linear spaceX are said to be, respectively,symmetric ot skew with respectto 1por ouer FItpaccording as, for all a, b e X,

bca : (at61vor -(acb)v.

A correlatedquaternionic space is a finite-dimensionalHJinear space assignedan H-correlation. As alwaysan invertible correlation is said to be non-degenerete. Semi-linearcorrelations (,4 : X -- XL on a right Hlinear spaceX are saidto be equiualenrif, for someinvertible ,1 e H,4 :,,1(. This is clearly an equivalenceon any set of semi-linearcorrelations on X. Thereare two quaternionicanalogues of Proposition7.11.

Proposition10.4 Let p be any anti-inuolutionof H other than conjugation. Then any skewH'p-correlation on a right H-linear spaceX is equiualent to a symmetricH-correlationon X, and conuersely.

Proof We give the proof for the caseHe : fr. Let ( be a skewfr-correlation on X. Then j ( is an H-correlationon X, since,for all.x e X, )" eIJ'

AO@1):j),((x): 1AO@\ 10 Anti-inuolutionsof H(n) IJ Moreover,for all a, b e X,

Ata: jb(a: -jtqb :m j: a{6. That is, j ( is symmetricover H. n Proposition 10,5 Let p be as in proposition 10.4. Then any symmetric H'p-correlation on a right H-linear spaceX is equiualentto a skewE- correlationon X, and conuersely.

Supposenow that f : X --+y is a linear map of a right quaternionic spacex, with non-degeneratecorrelation (, to a right quaternionicspace Y, with correlation 11. Since ( is bijective there is a unique Hlinear map t' y --+ : X such that t" : 1-rrrr, rhat is, such that, for any x y € X, e Y,f (y)ex: y,rt(x).The map t* - C-1tLryis called tiheadjoint of r with respectto ( and n. The adjoint of a linear endomorphisma of a right quaternionicrinear space X with respectto a non-degeneratecorrelation ( will be denoted by u4' The map u is said to be serf-adjointif L$ : r,t and skew-adjoint if uc : -u. The real linear subspaces{z e End X i u( : u} and, e EndX : -u} {u : ut of the real linear spaceEndX : L(X,X) will be denoted by End..(X, O and,End_(X, O, respectively.

Proposition10.6 Let ( be a non-degeneratecorrelation on a right quaternionic linear spaceX. Then the mao

EndX+EndX;f-f.

is an anti-inuolutionofEnd X, regardedas a real algebra.

Equivalent semi-linearcorrelations ( and rlon the right quaternionic spaceHn induce the sameadjoint anti-automorphismof ffln). Symmetric and skew correrationsare particular casesof reflexivecorrelations, a correlation ( on a quaternionicrinear spacex being said to be reflexiueif, for all a, b e X.

bta:0oatb:0. It is almost the casethat any reflexivecorrelation is equivalenteither to a symmetricor to a skew one,a counter_examplein the one_dimensional casebeing provided by the correlation on H with product

Y12--+H2 i @, b) ++ d(t + j)b. 76 10 Anti-inuolutionsof H(n) -correlation( on a right H- Proposition 10.7Any non'zeroreflexiue W or skew' spaie X of d.imensiongreater than one is eithet symmetric

6'1' but one has Proof Thisis naturally a further re-run-of Proposition H' and that it need not to remember that p is an anti-automotphismof be an anti-involution. e atb:0' That is' For all u,b e X,(bta)v-t:0 + b€a: O H-linear maps for any non-zero a € X, the kernels of the surjective quaternionicanalogue of T,--d';;; u"a o'coincide' Therefore'by the suchthat' for allb e X' Proposition 1.5there is a non-zeroquaternion '1o (b(61v-':;oa(b' c e X linearly To prove that )"ois independentof a one then considers since independentof a, c existingsince dimX > 1' Then' bta+btc:bE(a*c),

it follows that ),oa|b + lrctb : Ao+,fu* c)(b,

for all b e X. So a1[-' : l|i'": 1'!-''

But a and c arelinearly independent'So 1Y-': [Y*i:1Y-', suchthat' for all implying that)"o.:l'"' So, as before'there exists IeH a, b e X,(bEs)w-' : )"acb' Again there are apparentlytwo cases' (b(a)v-\: -lbt a for However,if aea: O,for ail a e X,it follows that' : that )' : -1 and all a, b € X. But, for suitablea andb, b(a l, implying casedoes not arise' that p: 1n. But p cannot be the identity' so this implying that' for The alternative is that, for some x € X'xtx * 0' : 7v' someinvertible p e H, (l'-')'^ : llt-r'or' equivalently'lt-r Q;r)v Then, for all a, b e X, : pgneb)vtrt-t- bpta- 1t1Aa(b)v: plpalb)v(1t-t)r)J rlou|blvp-t' The conelationp(' Moreovet,for all ). €H,(bl'1uco: (P1'pP-r)bP(a' where' for any equivalentto'(, is thereforea symmetricHp'-correlation' v g H. vrp,: ltyv!t_r. 10 Anti-inuolutionsof H(n) 77

That is, the correlationp(, equivalentto (, is symmetric. !

There is a converseof Proposition 10.6,analogous to the conversesof Propositions6.2 and 7.L2.

Theorem 10.8Any anti-inuolutiona of the real algebraH(n * I) is representableas the adjoint anti-inuolutioninduced by a non-degeneraterefiexiue correlationon Hnrr.

Proof This follows the proof of Theorem 6.4 or 7.I4. At the end of the first stage of the argument one proves that the map 1pis an anti- automorphismof H, not necessarilyan anti-involution. The correlation ( is defined as before and proved to be H'p-linear,and injective,and so is a non-degenerateHpJinear correlation on H' x H. The remainderof the proof then goesthrough without change. n

The next proposition, analogousto Proposition 4.1, is required in the proofofTheorem10.10.

Proposition l0.g (i) Let X be a right H-lineqr spaceand ( a symmetric H-correlationand supposethat x e X is suchthat xtx f 0. Thenthere exists7 eH suchthat (x l)a(x1) : 1.or -L. (ii) Let X be s right H-linear spaceand ( a symmetricfi-correlation and supposethat x e X is such thqt xex * 0. Then there exists). e H suchthqt (x \c@ 1) : L.

Proof Let rp be either of the anti-involutions of H. Then, since, for all ), eIJ, (xl)e(xl1 : )v(xexfi., it is enoughto prove that, for some ). €IJ, (1-r1v1-t : (1p)-t1-r : lxvx. Now, when r,ois conjugation, 7;: x(x,by the symmetryof €, and xlx is thereforereal. So in this casewe may take l-1 to be the squareroot of lx(xl, proving (i). On the other hand,if HP : fr, 7;: x(x, andso, by Proposition8.25, xf x belongsto the image of the map H-+H; p> TL2L.Then we may take A : Lr-r,proving (ii). n

Theorem10.10 (The basistheorem for symmetriccorrelated quaternionic spaces.)Each finite-dimensional symmetric rt- or fi-correlotedspace has an orthonormalbasis.

By Theorem 10.8 this holds also for skew correlated quaternionic spacessince any skew correlation is equivalentto a symmetricone. 78 10 Anti-inuolutionsof H(n) Theorem 10.11 (The classification theorem for symmetric correlated quaternionic spaces.)(i) Let X be a non-degeneratesymmetric H-correlatedspace of finite dimensionn. Thenthere existsa uniquepair of numbers (p,q),with p-lq: n, suchthat X is isomorphicto-#'n , this beingthe right H-linear spaceHp+s, with the sesqui-Iinearproduct --+H; -D (Hv+ay (a,f) * 6 bi * | a* Ar+1. 0

(ii) Let X be a non-ilegeneratesymmetric fi-correlq.ted space of finite dimension n. Then X is isomorphic to Hn, this being the right H-lineat spaceIJn, with the sesqui-linearproduct ''- dibt. (Hn)2- H; (a,U) | 0

Quaternionicgroups

'r\ Proposition 10.12(i) t"t t : ( O, an enilomorphismof the cor- | d/ -f,'n \D related quaternionic tpace . Then the adjoint of t is r:(1, -#)

(ii) Let t be an endomorphismof the correlatedquaternionic space H',. Thenthe adiointof t is t- :V. There are quaternionicgroups analogousto the orthogonal and symplectic groups over R or C. The classicalnotations make no mention of quaternionsand so are quite inappropriatefor our purposes.It turns out that the natural definitionsare as follows: to definelhe orthogonalquater' group of automorphismsof nionicgroup of degreen, O(n;H), lo be the the correlatedquaternionic space Hn, and to define the symplecticquater- q;H), to be nionicgroup of degreen: P*q andsignature (p,q), Sp(p, -#'n. the group of automorphismsof the correlatedquaternionic space of course,by Propositions10.4 and 10.5either may be regardedas pre- servingeither a symmetric or a skew form. As in the unitary case'but by contrast to the real or complex case,there is no requirementin the Exercises 79 symplecticquaternionic case for the degreeof the group, the dimension of the correlatedspace, to be even.The group Sp(O,n;H)is frequently denoted simply by Sp(n), and referred to as the symplectic group of degreen, without evenmention of the quaternionsH. Note that sp(l) : s3, the group of quaternionsof absolutevarue 1. The 'symplectic' rather varied uses of the word tend to be a bit confusing at first. The word was first used to describe the groups sp(2n;R) and sp(2;c) to indicatetheir connectionwith the set or nuil planes in the correlated spacesRff and clf;. such a set of null planes, regarded as a set of projective lines in the associatedprojective space, is known to projective geometersas a rine complex. The groups were thereforeoriginally called,comprex groups. This was leading to hoperess confusion when Hermann weyl (1939) coined the word 'symprectic,, ,complr"i. derivedfrom the Greek equivalentof the Latin word wh.th., the situation is any lesscomplicated now is a matter of dispute! The next propositioncontinues proposition 7.20.

Proposition 10.13For anyfinite n, anrl p * q : n, with a rather obuious definitionof Sp(2p, 2q; C), sp(n) Sp(2n;C)n GL@;H) : Sp(2n;C)o (r(n), O(n;H) O(2n;C)o GL(n;H), sp(p,q) Sp(2p, 2q ; C) n U(2p, 2q).

Proof As an example,the equationSp(n): Sp(2n;C)au(2n) followsat oncefrom the observationthat, for all z I jw, z, * jw, e H, - 1i +fiy12' * iw,) (zzt tfr wt)- j(, ,, _ z w,) - 0 if andonly if zz'+'Ww':0 andw z,-zw,:0. fl corolfary 10.14Each element,either of the group o(n;H) or of the group Sp(p,q),has determinant l.

Exercises 10.1 Let t € SU(3). Showthat

fro lrr | - /oo ror I too tot l" lrl:- I . &OCl fnr : tzo tn I' tzo tzt l' trc ttt 80 10 Anti-inuolutionsof H(n) t0.2 Show that the diagram of maPs SP(1): SU(z) il SP(2) ---+ sU(4) :' T Ji 5z -i- 5z ' -T whereany z*jw insp(l)is identifiedwita' ( .t,, ) t" su(2)' \vu z / is commutative,the top row and the two columnsbeing special and casesof the left-cosetexact pairs definedin Theorem13'13' the map n beingthe surjection,with imageT a subsetof SU(4)' defined,for all t e SU(4),by the formula n(t): tfl whete -tot L3l 'ii) l6s -130 ':l'1,'i,i-toz 133 toz -132 --+ Hence, by Proposition 3.4, construct a bijection 53 T that makesthe square SU(3) ------)5: I I su(4) :. T

commuteandshowthatthisbijectionistherestrictiontoSJ -+ with target T of an injective real linear map 1r : C3 C(4)' (Exercise 10.1 is relevant at one point of the argument' We in ,hull .rr.oonter this example again in Proposition 17'3 and Diagram 24.5.) tl Tensorproducts of algebras

The tensor product of algebras is a special case and generalisationof the tensor product of rinear spaces thai canbe defineddirectly. w. t uu. chosennot to develop the theory of tensor products in general,as we have no need of the more generalconcept.

Tensorprodacts of real algebras certain algebrasover a commutatiue fierdK admit adecompositionsome- what analogousto the direct sum decompositionsof a rinear space,but involving the multipricative structurerather than the additive strucrure. suppose that B and c are subalgebrasof a finite-dimensionaralgebra A overK, the algebrabeing associativeand with unit erement,such that (i) for anyb e B, c e C, cb : bc, (ii) ,4 is generatedas an algebraby B andC, (iii) dim,a: dimBdimC.

Then we say that .4 is the tensorproduct B @xC over K, the abbreviation B C @ being usedwhen the field K is not in doubt.

Propositionll'l Let B qnd c be subargebrasof afinite-dimensionalargebra A ouerK, suchthat A: B x C, the ilgebra A beingassociatiue and with unit element.Then B a C :K (the field K beingidentffied with the set of scalar multiplesof the unit elementl1a1).

It_istempting to supposethat the conditionB OC :K can be usedas an alternativeto condition (iii) in the definition. That this is not the case is shown by the following example.

81 82 11 Tensorproducts of algebras

Examprerr.2Let^:{( e R(3): e,b, c. -), [ ;3)

Condition (iii) is essentialto the proof of the following proposition.

Propositionll.3 Let A be a finite-dimensionalassociatiue algebra with unit elementouer a commutatiuefield K qnd let B and C be subalgebrasof A suchthat A: B @C. Also let

{ei:0

{",fi t0 < i < dimB,0 < i < dimC} is a basisfor the linear spaceA.

This may be usedin the proof of the next proposition.

Proposition ll,4 Let A and A' be finite'dimensionalassociatiue algebras with unit elementsouer a field K and let B and C be subalgebrasof A, and B' and C' subalgebrasof A' suchthat A : B @C andA' : B' @C'. Then, if S =- B' and if C = C', it follows that A = A'.

Proposition 11.4 encouragesvarious extensionsand abuses of the notation I. In particular, if A, B, C, B' and C' are associativealgebras with a unit elementover a commutativefield K such that

A:B@C,8,=BandC'=_C, then one frequently writes ,4 'r B/ @C', even though there is no unique construction of B/ I C'. The precisemeaning of such a statementwill alwaysbe clear from the context. The following propositions involving the tensor product of algebras will be of usein determiningthe table of Clifford algebrasin Chapter 15. 11 Tensorproducts of algebras g3

Propositionll.5 Let A be an ussociatiuealgebra with unit elementouer e commutqtiuefield K and let B, C and D be subalgebrasof A. Then A:B@C + A:C8B andA:BO(C8D) + A:(BSC)OD.

In the latter caseit is usual to write, simply, A : B A C A D.

Propositionll.6 For any commutatiuefield K, andfor anyfinite p, q, K(pq) = K(p)sr K(q).

Proof LeI Kpq be identified as a linear space with Kpxa, the linear spaceof p x q matricesover K. Then the maps K(p) - K(p q); cre eL and K(q) -K(pq); U ,* bn are algebrainjections, whose imagesin K(p q) satisfyconditions (|-(iii) for @,aa and b* being defined,for each a eK(p) and b e K(4), and for eachc €KPrQ,by the formulae

aa(c): qc andb"p(c) : 96".

For example,the commutativity condition (i) follows directly from the associativityof matrix multiplication. n In particular, for any finite p, q,

R(pq) = R(p)en R(c). In this casewe can say slightly more.

Proposition ll.7 For any finite p, q, let Rp, R4 and Fipq be regarded as positiue-definite quadratic spacesin the standard way, and let Rpxq be identified with Rpq. Then the algebra injections

R(p)-R:(pq); ee eL andR(q)-R(pq); b++b,^ send the orthogonal elementsof R(p) andR(q), respectiuely,to orthogonal elementsof R(p q). corollary ll.8 The product of any finite ordered set of elements beronging either to the copy of O(p) or to the copy of O(q) in R(pa) ls an element of o(p q).

In what follows, C and H will both be regardedas real algebras,of dimensions2 and 4, respectively,and g : gr. 11.Tensor products of algebras

Proposition11.9 R8R:R, C8R:C, H8R:H, c@c =2c, Idac = c(2) H8H=R(4).

Proof The first three of these statementsare obvious. To prove that C 8 C : 2C it is enoughto remark thatzC is generatedas a real algebra (/r 0\ I (/- n\ \ bythe subalgebras {{ i i \ t' e Cf and1('": ):zeCf, each t\uz/ ) t\uz/ ) isomorphic to C, conditions (i) to (iii) being easilyverified. To prove that H a C = C(2) let C2 be identified with H as a right complexlinear spaceby the map C2--.rH; (z,w)r-+ z *jw, as before. Then, for any q e H and arryc e C, the maPs qy:H--+H; xe qx andcx : H-+H; xa xc are complexlinear, and the maps

H--+C(2);e+ er andC "+C(2);c> cR arc algebrainjections. Conditions (i) and (ii) are obviously satisfiedby the imagesof theseinjections. To prove (iii) it is enough to remark that the matrices (t o\ (i o\ (o-1\ro-i\ \o 1/'\o-il'\r o/'\-i o)' -l (; i),( I),(: ; ),(?I), representlng it, kr, j1 ip, k1.ip, respectively,span C(2) linearly. The proof that H @H = R(a) is similar, the maps

qy:F{--+H;x+ qx andVp:H-+H; x*t xV being real linear, for any q, r € H, and the maps

H--+R(4);q+ qL andH--+R(4); rr-+Tp being algebra injectionswhose images satisfy conditions (i) - (iii), condition (i), for example,following by the associativityof H. ! r

11 Tensorproducts of algebras g5

In this last caseit is worth recallingproposition g.26,which statesthat the image,by either of theseinjections, of a quaternionof absolutevalue 1 is an orthogonal elementof R(4). In fact we have the following analogueto propositiong.2T.

Propositionll.l0 For any q e 53 and any c e. Sr the map

gz--C2| xe qxc

is unitary, c2 being identifiedwith H in the usual way. Moreo,er, any elementof U(2) can be so represented,two distinct elements(q, c) and (q', c') e 53 x S1 representingthe sameunitary map if and only if (q,, c,) : -(q, c).

Proof The map x + qxc is complexlinear, for any (q,c) e 53 x Sr, sinceit clearlyrespects addition, while, for any). e C, q(x))c: (qxc))", since,1c : c )". To prove that it is unitary it is then enough to show that it respectsthe hermitian form

az--+Rixv-+x'x.

However,since, for all (x6, x1) e C2, -xriXxo xoxo* x1x1: (ro *jxr) : lro+j*r12, it is enoughto verify insteadthat the map, regardedas a map from H to H, preservesthe norm on H, and this is obvious. Conversely,letf e U(2) andletr: (1). Then lrl : 1 and the map gz --+C2;xt--+Tt(x)

is an element of u(2) leaving 1., and therefore^every -C point of c, fixed, and mapping the orthogonal complementin of C, the complex line jC : {j z : z € C}, to itself. It follows that thereis an elementc of 51, defineduniquely up to sign, such that, for all x e C2, T{x\:Zxc or. equivalently,t(x): qxc, wheree: rE. Finally, sincec is definedup to sign, the pair (q, c) also is definedup to sign. An alternativeproof of the conversegoes as follows. Let t e U(2). Then t: cpu,where c2 : det/ andu e SU(2\. Now the matrix of u can

which follows that u: qy, whercQ: a *jb. The resultfoilows at once. 86 11 Tensorproducts of algebras

corollary ll.ll Thefollowing is a comnrutatiuediagram of exactsequences of group maps:

{1} {1} J J -----) -! -> {1} 5o 5o {1} J J, J' I proJ. ------) € ------t -, {1} 5: 53xSl Sl {1} J J' j* ------) I det ------' {1} su(2) - u(2) - SI {1} l j I {1} {1} {1} the mapf beingdefined by theformula

f (q,c):4LcR, for all (q,c) e 53 x 51, and the map sq. beingjust the squaringmap z r- 72. In Particular,Sp(l) : 53 rz SU(2\.

Now, by Proposition 11.9,C(2) = H I C, the representativeof any q e H being q1 and the representativeof any c e C being ca. It follows,by Proposition11.10, that the product of any finite orderedset of elementsbelonging either to the copy of Sp(1): 53 or to the copy of U(1) : 51 in C(2) is an elementof U(2). This result is to be comparedwith Proposition11.7 and the remark following the proof of Proposition11.9. Note also that in the standard inclusionof C in R(2) the elementsrepresenting the elementsof U(i) : 51 are all orthogonal. For an important application of theseremarks seeproposition 15.2g.

Complexification Roughly speaking,complexification is the operation of tensoring by C. Thus c(2) is the complexificationof H and 2c is the complexificationof C, but in the applicationsthat we have in mind the object complexified will frequentlybe a real superalgebra,that is a real algebrafurnished with an anti-involution, in which casethere aretwo kinds of complexification, tensoring by C, that is by C with the identity as (anti-)involution, and

-t 11 Tensorproducts of algebras g7 tensoring by e that is by c with conjugation as (anti-)invorution. For H I C there are four diflerent outcomes,as follows.

Proposition ll.l2 We considerthe isomorphismC(2) = H @C, wherethe quaternione: w * zj is representedby the matrix

lw -z\ (, w) and the complexnumber c is representedbv the matrix /c 0\ \'0,) (i) Let H be assignedconjugation and c the identity. Thenthe induced anti-inuolution of C(2)(; = ;)-H I C ls(-x -:) (ii) LetHbe assignedconjugation andc conjugqtion.Thenthe induced anti-inuolutionof C(2) = H A C is conjugatetransposition. (iii) LetHbe assignedthe anti,inuolution e:w lz j+>d:w _Z j, wherew, z e C, and let C be assignedthe identity. Thei the induced anti-inuolutionof C(2) = H I C ls

( a c\, . ( o b\ \a a/-(, d)' (iv) Let H be assigned the anti-inuolution q : w * z j r_->4:w- j, 7 where w, z e C, and let C be assigned conjugation. Then the inducedanti-inuolution of C(2)= H I C is (;;)-(-t-:)

Proof rn eachcase all one hasto verify is that the assertedanti-involution is correct both on the copy of H in C(2) and on the copy of C in C(2). In every casethis is immediate n Likewisefor c x c =2c there are three different outcomes,as folrows.

Proposition 2c ll.l3 we considerthe isomorphism ^i c g c, where the complexnumber w in thefirst factor is representedby the matrix (w o\ \ 0 tr/ 88 LL Tensorproducts of algebras

and the complexnumber c in the secondfactor i.srepresented by the matrix /'o\ \o c)'

(i) Let both copiesof C be assignedthe identity. Then2C 3 C x C is assignedthe iilentity. (ii) Let the first copy of C be assignedthe identity qnd the second copyconjugation. Thenzc = C x C is assignedthe swap(anti-) inuolution _.(a0\ /d0\ "'\o d)* \o r/'

(iii) Let both copiesof C be assignedconjugation. Then2C 3 C x C is assignedconjugation.

The recognition of subalgebras It is an advantageto be able to detectquickly whetheror not a subalgebra of a given associativealgebra ,4 is isomorphic to one of the algebras

R, c, H, 'R,'c,2H, R121,c121, or H(2),

or whether a subalgebraof a given complexassociative algebra,A is isomorphic to one of the algebrasC, 2C or C(2). The following proposition is useful in this context.

Propositionll.l4 Let A be a real associatiuealgebra with unit elementL. Then (i) I generatesR, (ii) any two-dimensionalsubalgebra generated by an elementes of A suchthat e3: -f is isomorPhicto C, (iii) any two-dimensionalsubalgebra generated by an elementes of A suchthat eE: I is isomorPhicto 2p., (iv) anyfour-dimensional subalgebra generated by a set {no, nt} of mutually anti-commutingelements of A such that ef, : e? : -l is isomorphicto H, (v) anyfour-dimensional subalgebra generated by a set {ro, ,t} of mutually anti-commutingelements of A suchthat ef - e?: f. is isomorphicto R(2),

j 11 Tensorproducts of algebras (vi) any eight-dimensional subalgebragenerated by a set {es,e1, e2}of mutuallyanti-comnuting elements of A suchthat ef,- €2r: e7: _I is isomorphicto 2fI, (vii) any eight-dimensional subalgebragenerated by a set{eg, e1, e2}of mutually anti-commutingelements of A suchthat e/ - e?: e|,: L is isomorphicto C(2\.

Setsof elementsmeeting the required conditionsinclude /o r\) [(01\ forR(2), l.\ 1 o/'( , , )]

{(L\0: _r/'\0e),fi _j/,\0e),ft _k))e)} (/io\ r j o\ (k o\l *t(o ,/,(o-j/,\.ok)], (/io\ ou'notllo /j0\ (k o\I r/'(."0,),(o u,/i (/i o\ (19),fl -t(o -r/'\o j/,\o -u)t9)I.ro,,s. "'d{(? I ),(l_? ),(i ; )},",.r,, severalof those listed in the following proposition we have had before.

Proposition ll.l5 The subsetof matricesof the real algebraK(2) of the form

Ol (i 1) * asubalgebraisomorphicto2R,2Cor2H, \D a/ -1 Otl ( i ) * a subatgebraisomorphic to C,2C or C(2), \D a/ ( a b'\ (ur) is a subalgebraisomorphic to 2F.,R(2) or C(2), ( O ;, ) -:', 2r., (i") ( i ) * , subalgebraisomorphic to c, H or \, a/ : accordingas K R, C or H, respectiuely,where, for any a e K, er : e, a or d, respectiuely.

Each of the algebraslisted in proposition11.15 is inducedby a (non- unique) real linear injection. For example,those of the form (iii) may be regarded as being the injections of the appropriate endomorphism 90 11 Tensorproducts of algebras algebrasinduced by the real linear injections 2R--R2; (*, y)*' (x I y, x - y), R2 --+C2; (x, y) r- (x * iy, x - iy), and C2--H2; (2,w)++ (z Ijw,Z +jW). Real algebras,A, B, C and D, say,frequently occur in a commutative squareof algebrainjections of the form A ------+C

I t, i------'i) the algebra D being generatedby the images of B and C. Examples of such squares,which may easily be constructedusing the material of Proposition 11.15,include R ----+ zp zp ----) R(2) ll JJ I l, C ------) R(2) R(2) ----- 2n1Z)

C ------' R(2) C __+ 2C zg -____, c(2) ttl +JT tl I H -----+ C(2) zg --+ C(2) C(2) -----+ 'c(z)

H -----+ C(2) 211 -----+ H(2) and JJ' I I 211 -----+ H(2) H(2) -----+ 2H(2)

Exercise 11.1 Showthat thereis a groupmap SU(2)--+ 53 x 53 makingthe dia- 53x53 gram V J commute,but that there is no group SU(2) __) so(4) 53 x 53 map U(2)--S3 x 53 that makes z J commute, U(2) --+ so(4) the vertical map in either casebeing the group map definedin Proposition 8.26 and the horizontal maps being the standard groutriinjections induced by the usual identification of C2 with R4. 12 Anti-involutionsof ,X(r)

In this chapter we learn to think of the generallinear groups GL(n;K) For examplean ?:i::1? :ro.ups! etementof GL(n;ci.uv, accordimg to the context,appear as a matrix of the form (t ) - (t ,J,-), where a is an invertibre n x n matrix over c. The first of thesereaves invariant the symmetricsesqui_linear form zgn r2gn--r", o)).-1r', o (( ; \ ;) Y// \ 0 *'y)' with swapplaying the part of conjugation,and the secondthe symmetric sesquiJinearform

2gn r2gn--'", (( , (; * ; ;) ;)) (u;:,5, with swap composed with conjugationtaking the part of conjugation.

2K linear specesand maps The double 2R and 2C fields were introduced at the very beginning of chapter 2 and2H at the end of chapter 9. Theseur. ooifi"tir, ior it is not true that every non-zero elementhas an inverse.Nevertheless much of the standardtheory of rinear spacesand -K-trnear maps over a field holds also Ior spaces, : for K R, C or H, as doesthe theory of semi_linear correlations. The part of a 2K linear spaceis played by a K_linear spaceA with a prescribeddirect sum decomposition,40 el A1, whereAs and A1 are 92 12 Anti-inuolutionsof2X1n) isomorphic linear subspacesof ,4 (so in particular in the case that ,4. is finite-dimensionaldim,4s : dimlr) with scalarmultiplication defined by ((1, ti,xo * xr) H (Axo+ Px)' which mav also be written as ((3 :),(r :, ))-' (';' ;, ) 2K-linearmaps are definedin the obvious way' / an 0 \ Lett:X-+Y be a 2K-lineatmap. Then t is of thef*t" (; ;, ), whereao € L(Xo, Ys) and a1 €L(X1, Y1). Conversely,any map of this 2K-linear form may be regardedas a map. one has to take care with extending the basis theorems of linear algebraovera field to double fields. We say that an elementx of a2K- linear spacex is linearlyfree of a subsetA of x if, and only if, (1, 0)x is linearly independentof (1, 0),4 in the Klinear space(1, 0)X and (0' l)x is linearly independentof (0, 1),4in the Klinear space(1, 0)X' Thus, in 2K itself,(1, 1) is not freeof {(1,0)}. Then abasisfor X is a free subset of X that spansX. 2K-linear The following is the basistheorem for spaces' 2K-linear : Theorem l2.l Let X be a spacewith a basisA' Then Xs (1, 0)X andX1: (0, 1)X are isomorphicas K-Iinearspaces, the set (1, 0)'4 beinga basisof the K-linear spaceXs and the set (0, l)A beinga basisof the K-linear spaceX1. Moreorserany twofinite basesfor X hauethe same numbetof elements. 2K-linear Any 2KJinear spacewith a finite basisis isomorphic to the space2Kn : (2K), n being the number of elementsin the basis' 2Klinear It should be noted that not everypoint x of a spaceX spans a 2KJine. For this to happen,both (1, 0)x and (0, 1)x must be non-zero. A point that spans a line will be called a regular point of X. Similar 2Klinear considerationsshow that the image and kernel of a map need not be 2K-linear spaces.

. Anti'imtolutionsof2K 2K-linear For the theory of semi-linearcorrelations on spaceswe needto 2K. know all possibleanti-involutions of we begin by recalling Proposi- 2X1n) 12 Anti-inuolutionsof 93 tions 2.18and 2.r9,extending these to apply not only to the fieldsR and c but also to the non-commutativefield H, in which casewhatever was said earlier about automorphismsapplies also to anti-automorphisms.

Proposition 12.2 Let K : R, c or H. Then any automorphismor anti- 2K automorphismof is of theform either (i)2K-.+,*, (3 ;)-, (f lr), *

(ji)2K.+2K;( * 3 ; ) ( { :,), wherey, @ : K -+ K are both automorphismsor both anti-automorphisms of K, any irreducibleautomorphism or anti-automorphismnecessarily being of type (ii).

In case (i) the map is an involution or anti-involution of 2K if and only if y and Q are both involutions or both anti-involutions of K, case (ii) is of greater interest. such an automorphism or anti-auto- O\ /-t o\--li o\ morphismr.nor(1--- andisan^-r \o -r/'o\ o t):-(.o -i)' 2K involution or anti-involution of if and only if it is of the form 2K-+2K;(ti) -({ ^l-), where { is an automorphism or anti-automorphism(not necessarilyan involution or anti-involution) of K. As in Chapter 2 the swapinvolution 2K--2K:(lo),.--(ry 0\ '\o tt/ \0 1) will be denotedby o, theinvolution or anti-involution 2K--2K:(10),-/ut "'\0tt)''\o 9 \ Aa') then being the compositeinvolution or anti-involution (6xd-\o:o(Q-rxd).

Proposition12.3 For eny two automorphismsor two anti-automorphismse and y of K the inuolutions or anti-inuolutions(6 x Q-\o and (y x X-\o of 2K are similar. 94 12 Anti-inuolutionsof zX(n) Proof The proof is just as for Proposition2.20. n

SemiJinear maps ooer 2K Let X and Y be right or left linear spacesover the double field 2K, whereK: R, C or H. Then a2K-Tinearmap f : X--+Y is said to be semi-lineqrover 2K if there is an automorphismor anti-automorphismr;r 2K 2x, of such that, for all x € X,)" e 71x11: f(x)lv, f(xl) : )"vf(x), f( x) : L'pf(x) or f(Ax): f(x)Ae, as the casemay be, rp being an automorphismof 2K if 2K operateson X and Y on the sameside and an anti-automorphismif 2K operateson X and Y on oppositesides. The terms right, right-to-left, left and left-to-right semi-linearmaps over 2K havethe obviousmeanings. The semi-linearmap r uniquely determinesthe automorphismor anti- automorphism rp of 2K with respect to which it is semi-linear,unless f : 0. On occasionit is convenientto refer directly to g, the map t then being said to be 2K'P.linear. The first of the maps H2 --H2; (x, y)r+ (x, y), (x, Y) r+ (!, f) is a right-to-left 2Hlinear map and the seconda right-toJeft zHo-linear map. Semi-linearmaps over 2K areclassified by the following proposition.

Proposition 12.4 Let X and Y both be 2K-linear spaces. Then any 2Kx'Q-linear map X --+I is of theform Xo @Xt --+Is O Yi (xo,x1) r-+ (/(xs), g(x1)), wheref :Xs--+Ys isKx-lineqrand g : Xt --+Y1ls KQ-linear,while any 2Ko(x"il-Iinear mapX -+Y is of theform Xo @Xr --+Ys @ Yr; (xo,x1) r-r (/(xs), g(xl)), wheref : Xs --+Y1 is Kx-linearand g : X1--+Yg is K6-Iinear.

Proof We indicatethe proof for u2yo(xx$-linearmap X --Y, assuming, for the sakeof definiteness,that X is a right 2KJinear spaceand Y aleft 2Klinear space.Then, for all a € Xs, b e Xy t(a,0): r((a,0)(1,0)) : (0,1)t(c, 0), t(O,b): ((0, bX0,1)) : (1,0X0,b). 2IK(fi 12 Anti-inuolutionsof 95 So mapsf : Xo--+Y1 and g : X1--+ys arc definedby (0, (al) : f t(a, 0) and (g(b),0) : r(0,b), for all (a, b\ e X. It is then a straightforward matter to checkthat thesemaps / and g have the requiredproperties. The proofs in the other casesare similar, ! An --+y Aplinear map t : X is said to be irreducibleif p is irreducible. otherwise it is said to be reducibte.rf r is irreducible, and if rp is an involution or proposition2.LT,A: anti-involutionthen, by K or 2K. Of the two proposition maps in r2.4 the first is reducible,while the second is irreducible. As before, the compositeof any two composabresemi-linear maps is semi-linear and the inverseof an invertible semi-linear mapis semi-linear. An invertible semi-linearmap is said to be a semi-rinearisomorphism. Duals of 2KJinear spacesand semi-rinearmaps are then definedjust as for HJinear spacesand semi-rinearmaps. In particular, correrationsand their induced sesqui-linearforms are similarly defined,as aresymmetric and skew correlations.For example,the product 2R 2R --2R; x ((o, b), (a,,b,)) * (bat, ab,) is a symmetricsesqui-linear product over 2R. A co*elated 2K space is a finite-dimensionarright 2K-linear space assigned 2K'p-correlation, a whereqo is some anti-involution of 2K. Semi-linear -t 2KJinear correlations (,4 : X XL on a right spaceX are said to be equiualentif, for someinvertible ). e2K,rt : A(.iiri, i* clearly an equivalenceon any set of semi-linearcorrelations on x. There propositions followsan analogueof 7.11, 10.4and 10.5. 2K,p-correlqtion Proposition12.5 Any irreducibreskew on a right 2K-rinear spaceis equiualentto a symmetriccorrelation on X, and conuersely.

Proof Let ( 2K'p-correlation be an irreducible skew on x. Then, for alr a,beX,

6o'-t)(o : (1,-1) bta : -@€b1v(1, -1) : (acb)v(I, -1)e, sincerp is a swapanti_involution. : 1a0,-r)€61v. That is, the correlation(1, -1)( is symmetric. Similarly, if ( is symmetric,then (L, _DC is skew. tr 96 12 Anti-inuolutionsof 2X1n1

The adjoint of a 2Klinear map X -, Y , with respectto correlations( on X and 4 on Y, is then definedjust as in the quaternioniccase. Equivalent semi-linearcorrelations ( and 4 on the right quaternionic spaceHn induce the sameadjoint anti-automorphismof H(n).

Proposition 12.6Let ( be a irreduciblesymmetric or skewcorrelation on a right 2K-linearspace X. Then the map

EndX +EndX; *-+ tc

is a real algebraanti-inuolution.

As always, symmetric and skew correlations are particular casesof reflexivecorrelations, a correlation ( on a 2K-linear spaceX being said to be refiexiueif, for all a, b e X,

bta:0 o a(b-- 0.

Proposition12.7 Any non-zeroirreducible reflexiue 2K'P'correlation ( on a right zK-spaceX of dimensiongrea.ter than one is equiualenteither to a symmetricor to a skewone.

(A counter-examplein the one-dimensionalcase is the correlation on H with product H2 --H'i (a, b) ++ a(I + j)b.)

Proof This is our final re-run of Proposition 6.1, but one has again to rememberthat p is an anti-qutomorphismof 2K, and that it neednot be an anti-involution. The proof is basically as before, but care has to be taken, since a non-zero elementof 2K or of X is not necessarilyregular. One proves that there existsan invertible I e2K such that, for all regular a e X and aTlbeX, (b(q1v-1: )"a€b.

It is then easyto deducethat this formula also holds for all a e X. The remainderof the proof is as before. tr

Next we state an analogueof Proposition 6.3.

Proposition12.8 Let X be a finite-dimensional2K-linearspace, let a be an anti-automorphismof the real algebraEndX and let t e EndX, with rkt : l. Thenrktd : l. 12 Anti-inuolutionsof 2X1n1 97

Proof This is almost as before.The only differenceis that the left ideal generatedby an elementr of EndX, with rkt: 1, is not minimal, but has exactly two minimal left ideals as proper subideals.However, this can only occur if rkt : L The details are left to the reader. n

What then follows is our final generalisationof Theorem 6.4.

Theorem129 Let X be afinite-dimensional2K-Iineqrspace, with dimX > L Thenany anti-inuolutiona ofthe real algebraEndX is representableas the adjoint anti-inuolutioninduced by a non-degener&tereflexiue correlqtion on X.

Proof Entirely as for Theorem 10.8. !

Proposition 12.10Let ( be a non-zeroirreducible reflexiue correlqtion on 2K-Iinear a right spaceX . Thenfor somex e X, x( x is inuertible.

Proof Since( is an irreducible correlatiorr, ( : o(rt x O, where ( : X1-'+X! and 4 : Xs---+X! are semi-linear.Since ( is reflexive, (0, aab) : (a, 0)<(a, b) : 0 + (b(a,0) : (0, b)<(a,0) : 0, thatisa'tb*0+b(a*0. Since ( is non-zero,either 4 or ( is non-zero. Supposethat 4 is non- zero. Then there exists(a, b) e Xs x Xl such that aqb + 0 and b(a * 0, that is suchthat (a,b)t@, b) is invertible. !

If ( is symmetric we can say more, this case being required in the proof of Theorem 12.14below.

Propositionl2.ll Let t be an irreduciblecorrelation on a right zK-lineqr 2K, spaceX, symmetricwith respectto the anti-inuolution(Q x Q-1)o of and supposethat, for somex e X, x

Proof As in the proof of Proposition 12.10,( : o(q x O. Now ( is symmetricwith respectto ({ x d-\o, so, for all (a,b) e X,

((a, b)( (a, b))@x6-')o: (a, b)<(a, b), that is,

((aab)6,(b{a16-'1: (b(a, aab). fir zX(n) Anti-irwolutionsof 98 12 if x : (a' b) is and only b( : !' Now' it In particular, a4l :1 if if il 'rli (i-;;4 ' l)' f\ invertible,b( a * l'ct'""* is said @ be non-d'egenetate' lil An invertible correlation i1 Il cortelstionon a finite'd'imen- I' Let ( be a non-d'eg-enerate Proposition12'12 elementof X' ltl i' and'-let,x b"eany non-zero sional right K-Iinear ;,;;' "filr- x!x' :l' tntr" existsx' e-X suchthat irreduciblecorrelafion on a 12'13Let ( be a non'degenuate Proposition let x be a regularelement 'ight"'i-t'nu'no 'u.o1X' and finite-dimensional x(x' :1 (: (1' 1))' the'" existsJ t x thot of X. Then "ih itreduciblesym- Eachfinite-dimensional Theorem 12.14 (Basis theorem') hnsan orthonormalbasis' metric 2Ko-cotrelated space

ByTheoreml2.gthisholdsalsoforirreducibleskew2Ko-correlatedone' is equivalentto a symmetric spacessince any tkt;;;;;htion synx- Let X be a non-degenerate 1'2.15(Classification theorem') Theorem n' whereK : R' C or H' 2T -correlatedspace- of dimension rnetric finite 2K-linearspace 2Kn' this b"ing ii" 'i'gl't Then X is isomorph:ic-i" enf"' with the sesqui'IinearProduct v, Ur, ,zgn12-+2Kt (a,b) I W 0Si

Generullineat grouPs

the cowelated' : * an endomorphismof Proposition12.16 Let' ( 3 2) t is : C or H' Thenthe adjointof space(2T )', whereK R' .1-(t 0 \ ":\o d )' the endomori automorphismsof (K'1" are Corollary 12'17 The correlated of wherea is any automorphism ,w of the phisrnsof Jbnn \ o (a)-, ), Kr. I '' Exercise 99 Corollary 12.18 The group of correlatedautomorphisms of (2Ko) is isomorphicto the generallinear group GL(n;K).

We definethe determinuntof sucha correlatedautomorphism to be the determinantof a e K(n). Accordingly the subgroupof all suchcorrelated automorphismsof determinant equal to 1 is isomorphic to the special lineargroup SL(n;K).

Exercise t2.1 Let twK denote the twistedsquare of K, that is, K x K with the --+ product (K x K)2 K x K; ((1,p), ():, p'))- Q,X, p,p). Show that, for any finite-dimensionalKlinear spaceX, the Klinear spaceX xXL may be regardedas a (twK)Jinear spaceby defining scalarmultiplication by the formula (x, a)()",p): (x ),,pa), for any (x, a) e X x XL, (,1,p) e twK. Develop the theory of twK-correlated spaces.Show, in particular, that, for any finite-dimensionalKlinear spaceX, the map --+twK; (X x XL)2 ((a, a), (b, P))++ (a(b),p(a)) is the product of a non-degeneratesymmetric (twK)o-correlation on X x XL, o being an anti-involution of twK. 13 The classicalgroups

In this chapter we tie together the results of the last few chapters.The main result, Theorem 13.8,states how any real algebra anti-involution of A(n) may be regarded as the adjoint involution induced by some appropriateproduct on the right AJinear spaceAn, where A is equal to K or 2K, and K : R, C or H, n being finite' Theorem 13.8 and The- orem 13.7together classifythe anti-involutions of A(n) into ten classes, and associatedwith theseare the ten families of classicalgroups The left-coset exact pairs of the last section will play a role in the discussionof Lie groupsin Chapter 22.

Equiaalent correlated spaces Let A : K or 2K, where K : R, C or H. Then a correlatedA-space is a finite-dimensionalright A-linear spaceassigned an Ae-correlation,where rp is some anti-involution of 2K. One speaksof the A'p-correlatedspace (X, C),where ( is the assignedAp-correlation. Let (X, O and (Y, q) be correlatedA-spaces' Then a correlatedmap t : (X, O -(Y , q) is a (right) Aolinear map, wherea is an automorphism of A, such that, for all a, b e X, t(a)at(b): (a€b)",

an invertible map of this type being a correlatedisomorphism.If such an isomorphismexists then the correlatedspaces are said to be equivalent, (X, €) = (Y, q). Recall also that two anti-automorphismsof A are said to be similar if there existsan automorphisma of A such that atp : yu' The next proposition generalisesseveral propositions of previouschapters.

100

,ti 13 The classicalgroups 101

Proposition l3.l Let ( be an Alp-correlation on a right rt-rinear spqce X and let y be any anti-qutomorphismof A similar to p. Then there exist a right a-linear spaceY and qn Ax-correlationq on y such that (Y, rD= (X, C).

Proof Sincey and p are similar, there exists an automorphisma of A such that cxlp: Xd,. Let Y : Xo, where Xd consistsof the set X with addition definedas before,but with a new scalarmultiplication, namely,

Xo x A-- Xo; (x, 1) r-+x 1"-t, and let q : Y -+YL be defined,for all a,b eY, by the formula on6: (a

The image of 4 is genuinelyin Yr since,for any p e A,

an16 ,a-t 7: 1aEb p"-')o : 1atb1op. Moreover,for any 1e A, - (a)d-\y6 : ((a 1o-'1161o 17vo-'a€b)" : 1xo4'6, since1 : dtpa-[. That is, 4 is ArJinear. Finally, the set identity (X,O--+(Y,4) is a correlatedisomorphism, sinceit is a semi-linearisomorphism and, from its very definition, alb : (a(b)6 for all a, b e X. n

The adjoint t* of a linear map f betweencorrelated spaces (X, O and (Y,4) isthemap(-rtLq : Y --+X,andissuchthat,forallaeX,b ey, bat(a): t.(b)

the adjoint of a linear mapu: X+X with respectto ( being denoted by u€. The map u is said to be self-adjoint if u( : u and skew-adjoint : -u. if u€ The real linear subspaces{u e EndX i uE : u} and {u e EndX : u( : -u} of the real linear spaceEndX : L(X, X) will be denotedby Endl(X, O and End-(X, O, respectively.

Proposition 13.2 Let ( be an irreduciblereflexiue correlation on a finite- dimensionalright A-linear spaceX of dimension> 1.. Then the map EndX --+EndX; t r-+7t is a real algebraanti-inuolution, equiualent cor- relationsinducing the sameanti-inuolution of EndX. lO2 13 The classicalgroups

Proposition 13.3 Let (X, O anrl (Y, fi be non-degeneratefinite-dimen- -correlated sional iP spaces.Then an L-linear map t : (X, O -->(y, q) is correlutedif and only if t. t : ly, wheret* is the adjoint of t with respect to ( and 4.

Corollary 13.4Let (X, O and (y , q) be as in proposition 13.3.Then any correlatedmap t : (X, O --+(Y, 4\ is injectiue.

Corollary 13.5Let (X, O be as in propositionlj.j and let f € EndX. Then t is a correlatedautomorphism of {X, O if and only if ttt:7y.

The final proposition in this sectionwill be usedin the constructionof charts on quadric Grassmanniansin Chapter 14.

Proposition13.6 Let (X,O and (Y,q) be as in proposition 13.3,and suppose,further, that ( and 4 are each symmetricor skew. Then,for - eny t e L(X,Y), (t"t1e tf t, the I sign applyingif ( and 4 are both symmetricor both skew,and the - sign if one is symmetricand the other skew.

The ten product types The following theoremcombines many earlier results.

Theorem13.7 Let ( be an irreduciblecorrelqtion on a right A-rinearspace of finite dirnension> 7, and thereforeequiualent to a symmetricor skew correlation. Then ( is equiualentto one of the Jbllowing ten types,these beingmutually exclusiae.

0 a symmetric R-corr elation ; 1 a symmetric,or equiualentlya skew,zVo-correlation: z a skew R-coruelqtion: 3 a skew C-conelation; 4 a skew tI- or equiualently a symmetricE-conelation; 5 a skew, or equiualentlya symmetric,2Ho-correlation; 6 a symmetricFl-, or equiualentlyq skew,fr-correlation; 7 a symmetric C-correlation ; 8 a symmetric,or equiualentlya skew,e-conelation: 9 a symmetric,or equiualentlya skew,2eo-correlation. 13 The classicalgroups 103

Explicitlythis combinesPropositions 7.1L, I0.4,10.5, 12.5 and 13.1and Propositions6.I. 7.I3, 10.7and 12.7.The logic behindthe codingof the ten types listed in the theoremwill be explainedlater. Theorem 13.8follows at oncefrom earlier work.

zK-linear Theorem 13.8 Let X be a finite-dimensionalright K- or space. Then any irreducibleqnti-inuolution u of the real algebra EndX is rep- resentableqs the adjoint anti-inuolutioninduced by a symmetricor skew correlationon X.

By Theorem 13.7there are ten cases.In eachof thesethere is a family of groups of correlated automorphisms analogous to the orthogonal groups. These are known as the classicalgroups. In a later chapter (Chapter 22) we shall show that they are all Lie groups,groups that arc also differentiablemanifolds, the group product being smooth. As such each has a real dimension.What we then prove is that for the group G : {t e A(n) : t( t : 1} the dimensionis equal to the dimensionof G : {t e A(n) : $ + t: 0} as a real vectorspace. First we summarisehere the explicit anti-involutions for each type. The notations are all as before, with the additional convention that, if, for some number n, a eK(n) and if rp is any anti-involution of K, then a'pdenotes the elementof K(n) whosematrix is obtainedfrom the matrix of a by applying the anti-involution rp to each term of the matrix. The r\dp a1PIis then the transposeof a'p.

Table 13.9 X (X,C) t eEndX

(K'PT Kn tlqx (K,P: C orfi) K"', W xKq -5) (R: R,e orn1 (; ;) (: KnxKn (K")i[ (; ;) (!i",:i") ( a c\ KnxK' (Ke)# -:;'") \'a d) (_1i", 2K 12gv61n (s:)(t-2.") We are now in a position to list the classicalgroups. 104 13 The classicalgroups

Table 13.10The tenfamilies of classicalgroups, with their real dimensions, are asfollows, where n: p { e: Code Group Dimension : - 0 O(p,q;R)or O(p,q),with O(n) O(0,n) ln(n l) I GL(n;R) n2 2 Sp(2n;R) n(2n* l\ a J Sp(2n;C) 2n(2n-t I) 4 Sp(p,q;H)or Sp(p,q),with Sp(n): Sp(0,n) n(2n* 1) 5 GL(n;H) 4n2 6 O(n;H) n(2n- l) O(n;C) n(n- l) 8 (l(p,q),with U(n): U(O,n) n2 9 GL(n;CJ 2n2, the subgroupconsisting of all elementsof the group of determinantI in eachcase being as follows: Code Group Dimension - 0 SO(p,4;R)or SO(p,q),with SO(n): SO(0,n) tn(n L) I SL(n;R) n2- 1 2 Sp(2n;R) n(2n* l) J Sp(2n;C) 2n(2n* I) 4 Sp(p,q;H)or Sp(p,q),with Sp(n): Sp(O,n) n(2n-t I) 5 SL(n;H) 4n2- | 6 SO(n;H) n(2n- l) 7 SO(n;C) n(n- L) 8 SU(p,q),with SU(n): ^lU(On) n2- 1 9 SL(n;C) 2n2-2.

Note that the three families of generallinear groups in this context all 'unitary' turn up as groups.In the simplest case,that of a symmetfic 2R form, the elementsof the group preservingthe form actually occur as matricesof the form (a o \ \ o (o")-') where a is any invertible m x m matix and d denotesits transpose,the group in this way being isomorphicto the generallinear group GL(rr;R). In the skew real caseor in any of the complex or quaternioniccases the automorphismin the bottom right-hand slot may be different, but that is irrelevant to the nature of the group. 13 The classicalgroups 105 Complexification There are two distinct complexificationsof real algebrasfurnished with an anti-involution, accordingto whetherthe algebraof complexnumbers is assignedthe identity or conjugation as (anti-)involution. In the tables of the following theorem we use the code numbersfor the various types that were introduced in Theorem 13.7.

2K-Iinear Theorem l3.ll Let X be a finite-dimensionalright K- or space, whereK: R, C or H, and let a be an anti-inuolutionof the real algebra EndX of type m, where0 < m < 9. Then the inducedanti-inuolution of the real algebraEndX I C restricting to a on EndX and to the identity on C is of the type giuenby thefollowing table: 0123456q 89 793\3e7 g29, while the inducedanti-inuolution of the real algebraEndX I C restricting to a on End X and to conjtgation on C is of the typegiuen by thefollowing tqble: 01234567 8 9 89898989 z3z9

Proo/ None of theseare difficult to verify, if one has the canonicalforms of Table 13.9in mind. Indeed the casesin which K : R are immediate. For the casesin which K : H the argument is a mild generalisationof Proposition 11.12,while in the casesin which K : C the argument is a mild generalisationof Proposition11.13. As an exampleconsider the correlated spaceH' of type 4, inducing the anti-involution t t+ f of H(n). If each entry_q : Z *j w is rep- -D \ resentedby the 2 x 2 uurrrPrE^_r_,,complexmatrix ( .z *rrn q : z-Jw \ r, ), representedby the matrix ( .z I ) , tnrn the matrix of I may be \-r.v z / representedin block form as an element of C(n)(2) = C(2n), namely ( z -7\ e C(n),mappedbytheanti-involutiontothe ( t Z ),whereZ,W matrix (7w') , in accordancewith both the anti-involutions of -Iry. Zr t' \ -i".) C(2n),(;;) '-( -{" **oe3 and (; ;) - ( T #) 106 13 The classicqlgrouqs of type 8, the first being appropriateif the (anti)-involution on C is the identity and the secondif it is conjugation. By contrast, the anti-involution H(n) -+ H(n) ; t - 7, of type 6, z -F\'i ( z'tl'\'T transforms. / ,- accordwith both ( ; ) \ _tr ),in /- -\ /a" b.\ theanti-involutionsof c(2n), type7 and l;;)'- (; ;" )* (a c\ ( t -c\ '* tvpe8' the latterbeing a differentversion ( -; \; ;) ; ) " of type 8 from that above,but type 8 nevertheless,the first again being appropriateif the (anti)-involution on C is the identity and the secondif it is conjugation. n

For completeness'sake and for future referencewe also give the 2R. followinganalogue of Theorem13.11 involving tensoring by

2K'linear Theorem13.12 Let X be a finite-dimensionalright K- or space, whereK: R, C or H, and let a be an qnti-inuolutionof the real algebta EndX of type m, where0 < m < 9. Then the inducedanti-inuolution of 2R the real algebraEndX @2Rrestricting to u on EndX and to swapon is of the type giuenby thefollowing table: 012345 67 8 9 L2rlg52559 929

The reasonfor separatingoff cases8 and 9 from the others in these two theoremswill becomeapparent when we come to applicationsin a later chapter (Chapter 17).

Quasi-spheres There are analoguesof the unit sphereSn in R'+l for all non-degenerate finite-dimensionalcorrelated spaces. Suppose first that (X, O is a symmetriccorrelated space Then

g(X, 0: {x € X : xtx: !}

is defined.tobe the :uirritquasi-sphere in (X, O, with g(R"+r) : Sn,while g(e'*t) andv(#+r) are identifiablein an obviousway with 52n+141d S4n+3,respectively, for any n. 13 The classicqlgroups 107

Note alsothat, for (X, O:rft"*t,with fr : R, C or fr, g(X, g+r O : {x e2 : x(x: l} : {x e2K+r : (4x0,'4*): (1,1)} : {x e2K"+t : rcxy: l\, sinceilxs : 1 if and only if 76xr:1,. A slightly different definition is necessaryin the essentiallyskew cases. The appropriatedefinition is s(K?p: {(x,y) e 1rff),: x.y : L}, . where denotesthe standardproduct on KS, or, equivalently,

s(K?;): "o). 61,"2: a. d-b.c: 1}, {(; ' where denotesthe standard scalarproduct on Kn, K being either R or C, The verification of the following theoremis a straightforwardcheck!

1] p, q, _I*lt"y rl3^Forgny n-letthe correlatedspaces Rp,s+1, gn+r,-gP'a+l , H"-',11n,a+t,2Rrnl, 2gn*t,2l1n+t,R!6+2, gz"+z -C'q be identifie:dwiti Rp,qx R, C'xC, xe, H-nxH,llr,nxll,zp,"zp, 2Cnx2c,rfr,xrg,n3f,xnrrn C?l x C?e,respectiuely, in the obuiousways. Then the pairs of mapi

o(p, q) ----+ O(p, q + l) ------)g(Rr'u+r1, SO(p,q) -----) SO(p,q*l) -----+9(Rn'a+1, p+q>0, o(n; C) -----+ O(n* I, C) -----+g(C"+\, -----+ So(n; C) SO(n*1.,C -> g(Rin,t+r1,fl)0, u(p, q) -+ u(p, q + r) ----+ g(d'n*'), SU(p, q) ----) su(p,q+t) ---+ g(d'n*t), p+q>0, o@;fi) -----'O@+l;fi) ----+ g(fin+r), sp(p,q) ------' Sp(p,q -t l) ------)g(fl'n*'), GL(n;R) ----+ GL(n* 1; R) ----+ 9(2R+r1, Sf(te; R) -----) SI,(ru* 1; R) -----+,q(2R"+\, n) 0, GL(n; C) -> GL(n* l; C) -----' 9(2C+t; SL(n; C) -----> SL(n* I; C) -----+9(2C+r1, GL(n;H) -----+ GL(n * I; H) ------)g('Hn*t), Sr(n;H) -----+Sr(n * 1; H) -----' g(fl*t), Sp(2n;R) -----) Sp(2n* 2; R) -----+ s(R?i+\, Sp(2n; C) -----+ Sp(2n* 2; C) -----),e(C?X+2) 108 13 The classicalgroups are each left-coset exact (Chapter 3), the first map in each casebeing the . / s 0\ iniectiont'-- and the secondbeing, in all but the last two cases, ( ; ; ) the map r -r t(0, l), the last columnof t, and, in the last two cases,the m&pt,-, ((0, (1,0)), (0, (0, 1))),the last two columnsof t.

Note, in particular, the left-cosetexact pairs of maps

O(n) ----> O(n* L) ----+ S', SO(n) ----->SO(n * 1) ---+ S', n) O, U(n) -----> U(n* I) -----) 52n*1, SU(n) -----+SU(n*1) ____)S2n+1, n)0, Sp(n) ----+ Sp(n+ 1) ___-_+54n+3,

For applications of Theorem 13.13 see Corollary 22.38 and Corol- Iary 22.40.

lVitt decompositions The Witt constructionof Proposition 5.25generalises to eachof the ten classesof correlatedspace as follows.

Proposition13.14 Let (X, O be a non-degenerateirreducible finite-dirnen- sionalsymmetric or skewA'p-correlated space, and suppose that W is q one- dimensionalnull subspaceof X. Then there existsanother one-dimensional null subspaceW' distinctfrom W suchthat the plane spannedby W and W' is, rcspectiuely,a hyperbolicor symplecticA1p-plene, that is, isomorphic to (Av)fr' or to (Ae)2ro.

Proof In the argumentwhich follows, the upper of two alternativesigns refersto the symmetriccase and the lower to the skew case. Let w be a regular element of W. SinceX is non-degeneratethere exists,by Proposition 12.12or 12.13,an elementx € X such that w( x : L Then,for anyAe A,

(x +w )")€(x*w ),): xtx* ),'P+ ),, thisbeing 0 if ,t:|lx€x, sincex(x: +1x(rs;lp.Letwt: xT \wxax. Then w(.w,: l, w,(w : *1 and w,(w, :0. Now let W' : A{w'} be a 2KJine in the 2K case,since w'6w : t1(: +(1, 1)). Then the plane spannedby W and W' is a hyperbolic or symplecti"Av-plane. fI 13 The classicalgroups 109 corollary 13.15Let w be q null subspaceof a non-degenerateirreducible finite-dimensionalsymmetric or skew Nt -correlatedspace X. Then there existsanull subspaceW' of X suchthatX:W @W'@(W @W)L. Such a decompositionof X will be called a Witt decompositionof X with respectto the null subspaceI,I/.

Corollary 13.16Let (X, O be a non-degenerateirreducible symmetric or skewfinite-dimensional |t'p-correlated space. Then there is a uniquenumber k suchthat X is isomorphiceither to (A'rXf x Y in the symmetriccase, or to (1$)!! x Y in the skewcsse, where in either caseY is a subspaceof X admittingno non-zeronull subspace.

The numberk is the (Wifi) index of the correlatedspace (X, O. It is the dimensionof the null spaceof greatestdimension in (X, O.

Proposition 13.17 The index of a non-degeneratefinite-dimensional 1{p- correlatedspace (X, () is at most half the dimensionof X. -(n+k, Proposition 13.18 The correlatedspaces pn,n*k, R?f, Crr, g2n+r, -('"*o C?;, fr", llzn+t, , (2Kv1znand (2Kv12n+rall haueindex n for uny non-negatiueintegers n and k. t4 QuadricGrassmannlans

The central objectsof study in this chapter are the quadric Grassman- nians of finite-dimensionalcorrelated spaces.Particular topics include parabolic charts on a quadric Grassmannianand various coset space representationsof quadric Grassmannians. There is no attempt to be exhaustive.The purposeof the chapteris to provide a fund of illustrative examples. All linear spaceswill be finite-dimensionallinear spacesover A : K or 2K, where K : R, C or H. On a first reading one should assume that A: R or C and that the anti-involution rp is the identity, ignoring referencesto the more complicatedcases. We start by consideringordinary Grassmanniansof linear spaces.

Grassmannisns Let X be a right Alinear spaceof dimensionn. Then, for any k, the set 9p(X) of linear subspacesof X of dimension/c over A is the Grqssmannian of linear k-planesin X. In the casethat A is the field of real numbersR there are also the Grassmannians9[(X) of orientedlinear /c-planesin X. An important example is the Grassmannian9{X) of lines in X through 0, also called the projectiuespace of the linear spaceX. The projective space G1(K'+1) is also denoted by KP", and said to be n- dimensional.A zero-dimensionalprojective spaceis called a projectiue point, a one-dimensionalprojective space is called a projectiueline and a two-dimensionalprojective space is called a projectiueplane. Each point of the projective space9{X) of a linear spaceX is a line through 0 in X, this line being determinedby any one of its points x other than 0. The line, or projective point, K{x} will also be denoted by [x]. For example,[x, y] denotesa point of KP2, namely the line through 0 and

110 (

14 QuadricGrqssmannians 111

(x, y). (confusion here with the closedintervals of R, which are similarly denoted,is most unlikely in practice.) Let X be a finite-dimensionalKlinear spaceand let x e X\{0}. Then we denoteby K{x} the linear subspaceof x dimension 1 spannedby x. The map x\{0} --+.9{X); xr-.-+[x] is called the Hopf map over the projective spaceg{X). various subsetsof the Grassmanniangp(x) may be regarded in a natural way as linear spaces.

Proposition l4.l For any linear subspacew of x of dimensionk, any complementarylinear subspaceY of x of codimensionk and any linear map t : W --+Y , grapht is an elementof gx(X).

The injectivemap L(W,Y)---+gp(X); r r-+ graphr will be calleda standqrdchart on 9*(X). A standardatlas on 9p(X) is a set of standard charts on g1r(X) such that every point of 9p(x) is in the image of at least one chart, the dimensionof 9p(x) being defined to be the dimensionover A of anv of the linear spacesL(W , Y ), namelyk(n - k), wheren : dim X.

Example 14.2 The maps --+ K KPI ; x r--+[x, I] and y ,- U, y] form a stqndardatlas for KPl. That is KPl may be thought of simply as the union of two copiesof the field K glued togetherby the map K >+ K; x r- x-1. Only one point of the secondcopy of K fails to correspondto a point of the first. This point [1, 0] is often denotedby oo and called the point at infinity on the I projectiveline. Every other point [x, y] of Kp is representedby a unique point xy-1 in the first copy of K. when we are using this representation of KP1 we shall simplywrite K u {oo} in placeof Kp1.

Example 14.3 The maps y2 --KP2; (x, y)- fx,y,lf, (x, z)-> lx, L,zl and (y, z)r-+ ll, y, zl form a stqndardatlas for KP2. As was the casewith KPI it is often convenientin working with Kp2 to regard the first of thesecharts as standardand to regard all the points tI2 14 QuadricGrassmqnnians not lying in its image as lying at infinity. Observethat the set of points then lying at infinity is a projectiveline, namely the projective spaceof the plane {(x, y, t) € K3 : z : 0} in K3.

Quafuic Grassmannians Now let ( be an irreducible symmetricor skew correlation on the right AJinear space X. The kth quadric Grassmannianof the correlated spaceis, by definition, the subsetlf n(X,Q of 9xV) consistingof the k-dimensionalnull subspacesof (X, O.

Proposition14.4 Let ( be sucha correlationon X and let 11be any correlation equiualentto (. Then,for eachk, ..tr*(X,q) : ,ff*(X,O. ln particular, tf n(X,-O : [n(X,€), for eachk.

The inverse image of ,[x(X, O by any one of the standard charts of 9x(X) will be called an ffine form of ,ffp(X, O or simply an ffine quadric Grassmannian.We shall mainly be concernedwith the casewhen ( is non-degenerate.In this casethe dimensionof a null subspaceis at most half the dimensionof X. When (X,6) is a non-degenerateneutral space,necessarily of even dimension,null subspacesof half the dimensionof X exist.Such subspaces will be termed semi-neutral(or, when (X,4) =- Rlff or C!1, Lagrangian (Arnol'd (1974)) subspaces)and the set of semi-neutralsubspaces of (X,0 will be called a semi-neunalquadnc Grassmannian. There are null lines in (X, O unless (X, O is positive- or negative- definite.The subset.,{{X,O of the projective space9{X) is called the projectiuequadric of (X,O. A line W in X is null with respect to the correlation ( on X, or, equivalently,is a point of the projectivequadric, if and only if, for every x € W, xcx:0. This equationis frequentlyreferred to as the equation of the quadric .,f {X,O. Just as the elementsof 9p(X) may, when k > I, be interpreted as (/c- l)-dimensional projective subspacesof the projective space9{X) rather than as k-dimensional linear subspacesof X, so the elements of ..(p(X,€) may, when k > L, be interpretedas (k - l)-dimensional projective spaceslying on the projective quadric ,.{{X,O rather than as k-dimensionalnull subspacesof (X, O. We shall refer to this as the projectiveinterpretation of the quadric Grassmannians. 14 Quailric Grassmqnnians 113

When (X, O is isomorphiceither to RS or to C!{, everyline in (X, O is null. In thesecases, therefore, the first interestingquadric Grassmannian isnot,,[{X,O but .tfz(X,O,the set of null planesin (X,O. This set is usually called the (projectiue) line complexof (X,O, the terminology reflectingthe projectiverather than the linear interpretation of -(2(X,O. Seealso the remarks precedingProposition 10.13.

Grassmsnniansas coset sp&ces

pk pn-/<. Proposition 14.5 Let R be identifiedwith Then the map f : --+ " O(R') 9*(R'); / '. r(Rk x {0}) is surjectiue,its fibres being the left cosetsin O(R) of the subgroupO(Rk) x O(n,-r.;.

Proof With Rn identifiedwith Rk x R,-k, any linear map t i Rn-+ Rn is of /ac\-, theform I I l, wherea € t(Rk,Rk),b €.L(Re,Rn*o),r€ Z(R,-k,Rk) \b d /' and,d € L(R'-k,11n-r'1. Since the first k columns of the matrix span t(Rk x {0}), (Rk x {0}) : Rft x {0} if and only if b : 0. However, if / is orthogonal with b : 0, then c also is zero, since any two columns of the matrix are mutually orthogonal. The subgroup O(Rft)x O(R,-k), / n O\ consistingofall ( ; ; ) € O(RkrRn-o), is thereforethe fibre of/ over \v u / Rk x {0}. The map / is surjective by Proposition 5.2. This follows since any element of 9p(R") is a non-degeneratesubspace of R, and so has an orthonormal basisthat extendsto an orthonormal basisfor the whole of Rr.

Finally, if r and u e O(R) are such that r(Rk x {0}) : a(Rk x {0}), then (a-lt)(Rk x {0}) : Rk x {0}, from which it follows that the fibres of the map / are left cosetsin O(R,) of the subgroupO(Rk) x O(n,-r,;. n

Accordingly the pair of maps O(k) x O(n - k) -\ O(n) J-, go(n\ i, left-cosetexact, and the induced bijection

o (n)/ (o (k) x o (n - k)) --+e p(Rn) is a cosetspace representation of 9p(R\ Ir4 14 QuadricGrassmannians

Proposition 14,6 For eachfinite n, k, with k < n, there are cosetspace representations u(n)/(u(k)x u(n-k)) --r 9n(C"), Sp(tc)/(sp(k)x Sp(n - /r)) -----' 9n(H"), So(n)I $o(k) x So(n- /c))-----,EI(R"), analogous to the coset space representation o(n)/(o(k)x o(n-k)) -----+ 9*(R") constructedin Proposition 14.5.

Churts on qaadric Grussmannians Let (X, O be any non-degenerateirreducible symmetric or skew A'p- correlatedspace, and considerthe quadric Grassmanniantfp(X, O. By Corollary 13.15there is, for any W e .,fp(X, O, a Witt decomposi- tionW@W'@Z of X, whereW' e.,fp(X, O and7 : (W@W')t.There are,moreover, linear isomorphismsAft-- W and N --rW'such that the product on X inducedby ( is given with respectto theseisomorphisms by the formula (a, b, c)c(a',b', c')- b'td + 4t 6t a c( ct, where4 is the (symmetric)correlation on (Av)k and ( is the correlation induced on Z by (, and whereX has been identified with W x Wt x Z to simplify notations. Both here and in the subsequentdiscussion, where there is a choiceof sign the upper sign applies when ( (and therefore 0 is symmetric and the lower sign when ( (and O is skew Nowlet Y :W'@Z =W xZ and considerthe standardchart on 9*(X), L(W , y ) --+9p(X); (s, z)r-' graph(s,t). The inverseimage by this chart of the quadric Grassmannian-(y(X, o is given by the following proposition.

Proposition14.7 Let (s, r) e L(W, Y), the notationsand sign conuentions being thosejust introduced.Then

graph(s,t) e ltrk(X, O o s t s4+ t* t : 0, wheret* is the adjoint of T with respectto the correlationsq on Ak and ( on Z. 14 Quadric Grassmannians 115

In particular,when Z : {0},that is,when tf p(X, () is semi-neutal,the counterimageof .ffp(X, () by the chart

L(W , Y) --+9n(X); (s, r) r-+graph(s, r) is a real linear subspaceof its source.

Proof For all a, b €. W,

(a, s(a),t(a))((b, s(b), t(b)) : s(a)ab-t aas(b)* t(a)(t(b) : (s(c)t sa(a)-f t. t(a\nS by the definition of the adjoint precedingProposition 13.2.Therefore

graph(s,t) e -,fp(X, O : s * sa* t' t : 0. The second part of the proposition follows from the remark that k), Enda(Ak,q) and End-(Ak, 4) are real linear subspacesof End(, while /:0, whenZ : {0}. !

Proposition14.8 Let the notationsbe as aboue.Then the map

f : Enda(Ao,q)xL(*,4--L(*, Y); (s,r)r--+(s -lt.t,t) is injectiue, with image the ffine form of ,ffn(X,() in L(Ak,Y) (: L(W, Y)).

Proof That the map / is injective is obvious. That its image is as statedfollows from the fact that, for any t e L(Ak, Z), (t- t1n: !t* t, by Proposition 13.6.Therefore, for any (s, t),

6 - it. r) * (s- lt. t1'I t. t : 0' That is, the image of / is a subset of .,{p(X, O, by Proposition 14.7. Conversely,if graph(s',t') e tf n(X, O, let s: s/ * f,t'"t' andlet t: tt. Thens*s'r :0;so s € Endl(At,4) andt e L(Ak,Zl,while { : s-lt. t and t' : t. That is, the affine form of .,{n(X, O is a subsetof im/. The image of / and the affine quadric Grassmanniancoincide. n

The compositeof the map f with the inclusion of im/ in .,f p(X, o will be called a parabolic chart on -,[n(X, O at W. A set of parabolic charts on,,fp(X, (), one at each point of .,{n(X,(), will be called a parabolicatlas for ff*(X, O. r1.6 14 Quadric Grassmannians

Qaadric Grsssmanniansas coset spaces Coset spacerepresentations analogous to those of Proposition 14.6exist for each of the quadric Grassmannians. We begin by consideringa particular case,the semi-neutralGrassman- nian -,(,(Czof,)of the neutral C-correlatedspace Cfl[,.

Proposition14.9 There existsa bijection O(2n) / U (n) --- .,tr*(Cfr), whereO(2n)/U(n) denotesthe set of left cosetsin O(2n) of the standard imageof U(n) in O(2n).

Proof The bijection is constructedas follows. The linear space underlying the correlated spaceCf,i is C' x Cn, and this samelinear spacealso underliesthe positive-definitecorrelated -C space xen. Any linear map t: Cn xCn-+Cn x Cn that respects '-\, both correlationsis of the form ( : with db*ba:0 and \D a/ da:tb: 1, for, by Table 13.9,the respectiveadjoints of any such (a c\ /d c'\ ,/d rrt'Pt: {),unotheseareequal \., a )ut"\a o")uno\t d / if and only if d:d. andc :6. By Proposition11.15 such a map may be identified with an elementof O(2n) or, when b :0, with an elementof U(n),the injectionU(n)'-+O(2n\being the standardone. Suppose that W is any null subspaceof Cfrfl of dimension n. A positive-definiteorthonormal basismay be chosenfor W as a subspace -C. of en x Supposethis is done, and the basis .t"?._t1 arranged in someorder to form the columnsof a 2n x n matrix Then I{z is l; ) / 7\ the image of the null subspaceC' x {0} by the map^ ( i : \. Moreover, \D a/ db + ba : 0, since I7 is null for the hyperbolic correlation, while da+Tb: 1, sincethe basischosen for W is orthonormalwith respect to the positive-definitecorrelation. Now let / be the map 6\ / a\ o(2n)--+.r-rc?1\t__>",v,\clg))( : l, u )+ffi\r/ The map is clearly surjective; so none of the fibres is null. Secondly, f-l(Cn x {0}) : U(n). Finally, by an argumentsimilar to that usedin the F

14 Quadric Grassmannians II7

proof of Proposition 14.5,the remainingfibres of f are the left cosetsin O(2n)of the subgroupU(n). n

Theorem 14.10 Let (X,O : (A\|t or (Av)!{, for somen, where p is irreducible.Then in eachof the ten standardcases there is a cosetspace representationon the semi-neutralGrassmqnnian ffn(X,(), asfollows: (o(n)x o(n))/o(n)-----+ ,rr"(R?L), -----+ u(n)/o(n) "r"(R?l), o(2n)/u(n)- ,.{"(c?t), (u(n)x u(n))/u(n)----) r1/ _)h : _)h "(C;:;) "(C:,i), Sp(n)/u(n)-----+ x^(c([), "f -: -1, u(2n)/Sp(n)+ Jr,(H!hL): ,(Hi), "f -1'- (Sp(n)x Sp(n))/Sp(n)-----+ rr"(IJif: ,X^(H;:il, o(2n)/(o(n)x -----+ o(n)) "r,(R#), U(zn)/(U(n)x U(n)) - Jl,(e;), Sp(2n)/(Sp(n)x Sp(n)) - rt/"('H;:i).

Proof The third of theseis the caseconsidered in Proposition 14.9.The detailsin eachof the other casesfollow the details of this case,but using the appropriatepart of Proposition11.15. n

Cayley charts The first of the caseslisted in Theorem 14.10merits further discussionin view of the following remark.

Propositionl4.ll Let f be the map

O(n) x O(n)-----+ O(n); (a, b) ++ ab-l .

Then the inuerseimage of the identity elementof O(n) is the imageof O(n) by the injectiuegroup map

O(n) ------>O(n) x O(n); a+> (a, a)

and the induced map

(o(n) x o(n))/ o (n)- o (n)

is bijectiue. 118 14 QuadricGrassmannians That is O(n) may be representedas the semi-neutralGrassmannian .rf ,8?t).The chartson O(n) correspondingto the paraboliccharts on lf will be called the Cayley charts on O(n). The following is an "B?U independentaccount of this case. LeI (X, O = Rn'' = Rig, and consider the quadric -,f{X,t\ Its equationmay be taken to be either

x"x: !"!, where(x, y) € R, x R,, or 7fD :0, where(u, u) € R' x Rn, accordingto the isomorphismchosen, the two modelsbeing related,for example,by the equations

u:xI!, 0:-x*!.

Now any n-dimensionalsubspace of Rn x Rn may be representedas the image of an injectivelinear map

@,D:(;) : Rn r-.- Rn x Rn.

Proposition14.12 The linearspace fr*(q,b), where(a,b) is an injectiue elementof L(R,R'xR'), is anull subspaceof R''" if andonly if a and b are bijectiueand b a-r e O(n).

Proof +: Let im(a,b) e .,{r(Rn'n), let w € Rn be such that x : a(w) :0 and let y : 61r;. Since(x, y) belongsto an null subspace of R''n, x"x : y"y, buI x : 0, so that Jl : 0' Since (4, b) is injective, it follows that w : 0 and therefore that a is injective. So a is bijective. Similarly, & is bijective. Since a is bijective, a-l exists; so, fot any (x,r1 : @(w)b(w)),y : ba-r(x)' Bnt yxy : x"x. So b a-r e O(n). €: Suppose that a and b are bijective; then, as above, for any (x, y) e il^(a,b), y : ba-r(x). If also ba-| e o(n), then!"y : x"x. n

Corollary 14.13Any n-dimensionalnull subspaceof Rn'" hasan equationof theform y : t(x), wheret e O(n),and any n-planewith suchan equation is null.

Proposition 14.14 Any elementof ,.{"(R'") may be representedas the imageof a linear map (a, b) : Rn--+ R' xR" with a and b eachorthogonal' 14 QuadricGrassmannians 1.19

This leadsat onceto the cosetspace representation for -,f ,(Rn'"\ whose existenceis assertedin Theorem14.10. Note that X"(R'"): {graphr : t e O(n)\ dividesinto two disjoint classes,according as t preservesor reversesorientation. So far we have consideredthe projectivequadric ff{R'").We now considerthe quadric lf {R?,U.Let s e End(R') be suchthat graphs e -{"(R?U. Then, for al7u, ut ePi',

s(r/)"ul u'"s(u):9, implying that s*s" : 0, that is, that s e End_(Rn),this being a particular caseof Proposition14.7. Now graphs : m(1, s). We can transfer to .,{{R'") by the map | / | -t \ ( ,, ' nffi --+R','. Then the image of graphs in Rn,', namely + ; ',/) VZ\' -l '-#(1)(l):'-(l;:), is an elementof .,{r(R"'"). So, by PropositionL4.12, or by Exercise2.1, 1 - s is invertible. By Proposition 14.12 again, or by Exercise2.2, the product(1 +sX1 -s) : 1 -s2 : (1-s)(1 *s). Moreover,since

(1-s):1f s": (1+s)', (1+sxt -s)-1 e SO(n).

The following proposition sumsthis all up.

Proposition14.15 For any s € End-(Rn), the endomorphism| - s is inuertible,qnd (1 + s)(1- s)-1 e SO(n).Moreouer, the map

End-(R') --+SO(n); s r-+(1 * sXl - s)-1 is injectiue.

The map of Proposition14.15 is the Cayley chart on SO(n)(or O(n)) at nl.. For n > 2 it is not surjectiveeven on SO(n). For example,when n :2 the rotation -21 doesnot lie in its image. The direct analogueof Proposition 14.15,with Rp,qin place of Rn and SO(p,q)in placeof SO(n),is not true when both p and q are non-zero; lo 1\ / r -1-l \ for(l I )e nna-1nr'r;,Uut( j \r v/ \-r t/)irnotinvertible.Thereis, however,the following partial analogue. r20 14 QuadricGrussmannians Proposition 14.16 For any s € End-(Rae; for which 1 - s rs inuertible, (1 + sX1- s)-1 e SO(p,q). Moreouer,the map End-(nae;>;SO(p,q); s'-+(1*sXl -s)-t is injectiue. The map given in Proposition 14.16is, by definition,the Cayleychart on SO(p,q) or O(p,q)) at"L An entirely analogousdiscussion to that givenabove for the orthogonal group O(n) can be given also for both the unitary group U(n) and the symplecticgroup Sp(n). It was remarked above that the semi-neutralGrassmannian .,|'"(R?i) divides into two parts, the parts correspondingto the orientations of Rn. The semi-neutralGrassmannian;f divides similarly into two "Qf,i) parts, the parts corresponding,in the cosetspace representation --+ O(2n) I U(n) ",tr,(Cfr), to the two orientations of R". (By Corollary 2.4, any elementof U(n) preservesthe orientation of R2'.)

Further coset spacerepresentations Cosetspace representations analogous to thoselisted abovefor the semi- neutral quadric Grassmanniansexist for all the quadric Grassmannians. The resultsare summarisedin the following theorem.

Theorem 14.17 Let (X, O be a non-degenerateirreducible symmetric or skew A'p-correlatedspace of dimensionn. Then,for each k, in each of the ten standardcases, there is a cosetspace decomposition of the quadric Grassmannianffp(X () asfollows: ----+ (o(p)x o(q))/(o(k) x o(p- k) x o(q- k)) lf k(Re's), u(n)/(o(k)x u(n-k)) ------)lf k(R?l), o(n)/(u(k)x o(n- 2k))-----+ "{*(C'), (u(p)x u (q))/ (u(k) x u(p- k) x u (q- k)) -----+'trn7'\, sp(n)/(u(k)x Sp(n-k)) ------)'f kG?$, u(n)/(sp(k)x u(n-2k)) ----+ lf x(E\, (sp(p)x Sp(q))/(sp(k)x Sp(p -k) x sp(q-k)) -----+,rtrt(fl'\, o(n)/ @(k) x o(k)x o (n- 2k)) ----+ ,ttrilzR"), u (n)/ (u(k) x u (k)x u (n- 2k)) ------)..fr,(U), Sp(n)/ (Sp(k) x Sp(k) x Sp(n - 2k)) ------),'f*(fl). F

14 Quadric Grassmannians l2I The resourcefulreader will be able to supply the proof! Certain of the caseswhere k : I are of especialinterest, and we concludeby consideringseveral of these. Consider first the real projective quadric ,ff{Rt't1, where p > 1 and q>-L

Proposition14.18 The map

Sp-r x Sq-1______),.{{no,ay (x, y)_ R{(", y)}

is surjectiue,the fibre ouerR{(x, y)} beingthe set {(x, y), (-x, -y)}. That is, there is a bijection

(Sr-1 x 5e-t;750--.,f {R!'q), wherethe action of S0 on Sp-1 x Sq-1 is definedbv the formula

(x, y)(-1) : (-x, -y),

for all (x, y) e 5r1 a 5t-1. This result is in accord with the representationof rll {Rp,q) given in Theorem 14.17,in view of the familiar cosetspace representations

O(p)/o(p- 1)- SP-' and O(q)/o(q- 1) ------+$e-1

of Theorem13.13. The complexprojective quadric tf {C") handlesrather diflerently.

Lemma 14.19For anyfinite n let z: x * i/ € C, wherex, y eRn. Then zQ): O if and only if xQ): y(2)a.nd x. y :0.

Now let R(x, y) denotethe orientedplane spannedby any orthonormal pair (x, y) of elementsof Rn.

Proposition14.20 For any orthonormalpair (x, y) of elementsof Rn, C(x* iy) e.,{(C) andthe map

gI (R') -> .,{ {C"); R(x, y) r--+C{x * iy} is well-definedand bijectiue.

The cosetspace representation So(n)/(So(2)x So(n -2))-st(R") t22 14 Quadric Grassmannians given in Proposition 1.4.6is in accordwith the cosetspace representation

o(n)/(U(l) x o(n -2))--+ -,f {C") givenin Theorem14.17, since SO(2) = St = U(1). Now consider lf {R!p. In this case every line is null; so lf {n?#) coincides with 9{Fi2"), for which we already have a coset spacerep- resentationO(2n)/(O(l) x O(2n- 1)), equivaTent,by Theorem 13.L3,to 52n-tf 50,where the action of -1 on 52'r-1ir the antipodal map. By The- oremI4.I'7there is alsoa representationU(n)/(O(I)xu(n-l\). This also is equivalent,o5z'rt/$0 by the standardrepresentation (Theorem 13.13 again) U(n)/U(n- 1)---+52r-1.

Finally, the same holds for lf tG?il, which coincideswith gilCz"), for which we alreadyhave a representationU(n)/(U(l) x U(2n - l)), equivalent,o5+n-t/gt. Herethe actionof Sl is right multiplication,Sa'-l being identified with the quasi-spttercf1t\ in C2". Theorem 14.17 provides the altercative representationSp(n) /(U(l) x Sp(n - 1)), also equivalent ,o 5+'r-t/$t via the standard representation(Theorem 13.13 Yetagain) sp@)/Sp(n- 1)-- s4n-1'

Exercises 14.1 Show that the fibres of the of the Hopf map

C2-, CPr ; (zo, zr)r+ fzs,z1f

to the sphere53 : {(to,r) e Cz : Tozo*4zt:1} are circles, any two of which link. (Cf. Exercise5.8 and Hopf (1931) and (1e35)). I4.2 Show that the fibres of the restriction of the Hopf map

H2 (qo,q),--, fqo,qil

to the sphereS7 : {(go,qi eFIz i !a.eo*6qt: I} are 3- spheres,any two of which link. 14.3 Prove that the map RP3 ---'SO(3), induced by the map p : H.+SO(3); 4+ pq of PropositionS.2l,is bijective.In this representationof SO(3) by RP3 how are the rotations of R3 about a specifiedaxis through 0 represented?

{ ui r

15 Clifford algebras

We saw in Chapter 8 how well-adaptedthe algebra of quaternionsis to the study of the groupsSO(3) and SO(4). In either casethe centre of interest is a real quadratic spaceX, in the one caseR3 and in the other caseR4, and in either casethe real associativealgebra H contains both R and X as linear subspaces,there being an anti-involution,namely conjugation,of the algebra,such that, for all x e X, Xx: *(2)' In the former case, when R3 is identified with the subspaceof pure quaternions,this formula can also be written in the simpler form

y2 : _*{Z).

In an analogous,but more elementaryway, the real algebra of complex numbersC may be usedin the study of the group SO(2). Our aim is to put theserather specialcases into a wider context. To keep the algebra simple,the emphasisis laid at first on generalisingthe secondof the two displayed formulae. It is shown that, for any finite- dimensionalreal quadratic spaceX, there is a real associativealgebra, A say,with unit element1, containing isomorphic copiesof R and X as linear subspacesin such a way that, for all x e X, y2 - -*{z). If the algebn A is also generatedas a ring by the copies of R and X or, equivalently,as a real algebra bV {t} and X, then A is said to be a (real) CIffird algebrafor X (Clifford's term (1876) was geometic algebra).It is shown that such an algebracan be chosenso that there is also on A an alsebraanti-involution

A--+A; a= e- such that, for all x € X, x- 124 15 Clffird algebras Tosimplifynotationsintheabovedefinitions,RandXhavebeen identified with their copiesin ,4. More strictly, there are linear injections a : R-+,4 and t : X --+Asuch that, for all x € X,

(r(x))2: -a(x@1, the unit elementin,4 beinga(1). - -xQ) The minus sign in the formula x2 can be a bit of a nuisance at times. One could get rid of it at the outset, simply by replacing in the quadratic spaceX by its negative.However, it turns up anyway applications,and so we keePit in. proposition l5.l Let A be a ctffird algebrafor a teal quadraticspace X and let W be a linear subspaceof X. Then the subalgebraof A genetated by W is a Clffird algebrafot W.

ByProposition5'2gandPropositionl5'ltheexistenceofaClifford algehrafor an arbitrary n-dimensionalquadratic spaceX is implied by the existenceof a Clifford algebra for the neutral non-degeneratespace (An R,,,. Such an a|gebra is constructedbelow in Corollary 15.18. alternativeconstruction of a Clifford algebra for a real quadratic space X dependson the prior construction of the (infinite-dimensional)tensor algebraofX,regardedasalinearspace'TheCliffordalgebraisthen defined as a quotient algebra of the tensor algebra,this definition being that adopted by Chevalley(1954). For details See,for example,Atiyah, Bott and Shapiro (1964).

Clffird algebrasfor low'dimensionalspaces Examplesof clifford algebrasare easily given for low-dimensionalnon- degeneratequadratic spaces' For example,R itseli is a Clifford algebraboth for R0'0and for R1'0'C' regardedas a real algebta,isa Clifford algebrafor R0'1,and H, regarded as a real algebra,is a Clifford algebra for both R0'2and R0'3,it being usual, in the former case,to identify R0,2with the linear image in H of has necessarilyto be identified with {i, k}, while, in the latter case,R0,3 quaternions'Moreover' it th.lit"ut imageof {i, j, k}, the spaceof pure follows easily from Exercises4.4 and 4.8 that R(2) is a clifford algebra for each of the spacetp2'0, ft1'l and R2'1. 15 Clffird algebras L2S It is provocativeto arrangethese examples in a table as follows.

Table 15.2 Clffiril algebrasfor ga'a,for low ualuesof p and q -p+q -4 -3 -2 -l 0 1 2 3 p+q 0 R 1 RC 2 R(2) R(2) H a J ?R(2)?H 4???2

A completetable of cliflord algebrasfor the non-degeneratequadratic spacesRp's is given later in this chapter as Table 15.27. As can be seenfrom that table, one can alwayschoose as clifford algebrafor such a spacethe spaceof endomorphismsof some finite-dimensionallinear 2R 2H, spaceover R, C, H, or the endomorphismspace being regarded as a real algebra. In chapter 16 we examinein some detail how a clifford algebraA for a real quadratic spaceX may be used in the study of the group of orthogonal automorphismsof X. Here we only make two preliminary remarks.

Proposition15.3 Let a, b € X. Ihen, in A, 1 a.b:-)(aU*ba).

particular, I h a and b are mutuallyorthogonal if and onty if a and b anti- I commute. I I I Proof I 2a.b : a.a*b.b-(a-b).(a-b) | -a2-b2+1a-b12 | : I - -ab-ba. ln I I Proposition I 15.4 Let a € x. Then u is inuertiblein A if and only if it is inuertible - I with respectto the scalarproduct, when a-7 -aeD. I I I I I I \ 126 15 Clffird algebras

Proof + :Letb: a-7,in,4. ThenaQ)b - -a2b: _a,implyingthat a@+0 andthat b: aet. e :Let b- a'eD:-(a?\-r a.Thenba:-fu(z)\-ral:1. Similarlv. (rb:1,. Thatis, b: a-r. n Notice that the inversein A of an elementof x is also an elementof X.

Orthonormal sabsets one of the characteristicproperties of a clifford algebra may be re- expressedin terms of an orthonormal basisas follows.

Proposition 15.5 Let x be a finite-dimensionarreal quadraticspace with an ofthonormalbasis {e; :0 < i < n}, wheren - dimX, and let A be a qssociatiue real algebrawith unit elementr containingR qnd x as linear - subspaces.Then x2 -xQ), for all x e X, if and only if

t? : -r(,'), for all i,

e;ei * €j er:0 for all distincti andj. This prompts the following definition. An orthonormalsubset of a real associativealgebra,4 with unit element 1 is a linearly independentsubset s of mutually anti-commutingelements of A, tbe squarea2 of any elementa e S being e 1 or -1.

Proposition 15.6 Let s be a subsetof mutually anti-commutingelements of the algebraA suchthat the squarea2 of any elementa € s is I or -r, Then S is an orthonormalsubset in A

All that has to be verifiedis the linear independenceof S. An orthonormal subsets each of whose elementsis invertible, as in Proposition 15.6,is said to be non-degenerate.rf p of the elementsof S have square*1 and if the remainingq havesquare -1, then s is said to be of type (p, q).

Proposition15,7 Let X be the linear imageof an orthonormalsubset s of a real associatiae algebraA. Then there is a uniquequadratic form on X suchthat, for aII a € S,sQ): -a2, and,if S is of type(p, q), X with this structureis isomorphicto Rp'q.If s also generatesA, then A is a clffird algebrafor the quadraticspace X. 15 CIffird algebras L27 The dimensionof a Clifford algebra There is an obvious upper bound to the linear dimension of a clifford algebrafor a finite-dimensionalreal quadratic space. It is convenientfirst of all to introducethe following notation. suppose that (ei : 0 < 1 < n) is an n-tuple of elementsof an associativealgebra A. Then, for each naturally ordered subset1 of n : {0, 1, ...,n _ I}, er will denotethe product fliaetwithe6:1, where0 denotesthe empty set.In particular,€n:flien€i.

Proposition 15.8 Let A be a real associutiuealgebra with unit elementI (identified with I e R) and supposethat (e; : I e n) is an n-tuple of elementsof A generatingA suchthat,for any i, j €n, eiel I ei e; €R.

Thenthe set {e1 :1 c n} spansA linearly. corollary 15.9Let A be q clffird algebrafor an n-dimensionalquadratic spaceX. ThendimA <2.

The following theorem gives the complete set of possiblevalues for dimA, whenX is non-degenerate.

Theorem 15.10 Let A be a CIffird algebrafor an n-dimensionalnon- degeneratereal quadraticspace X of signature(p, q). Then dimA:2, or 2-r, the lower ualuebeing a possibilityonly if p - q - L is diuisibleby 4, in which casen is odd and en : lI or -l for any orderedorthonormal basis(e; : i e n) for X.

Proof Let (ei : i € n) be an ordered orthonormal basisfor X. Then, for eachI c n) er is invertible jn A and,so is non-zero. To prove that the set {e7 : 1 c n} is linearly independent,it is enough to prove that if there are real numbers ,17,for each ,I c n, such that Dr-nhkt):0, then,for eachJ en, lt :0. Since,for anyJ cn,

D ttki: o + \ Ar@)(et)-1: 0, lcn lcn thus making )"t the coefficientof e6,it is enoughto prove that Dlrt",l: o=+ ls:0. 1cn Suppose,therefore, that fr-o )q@) : g. We assert that this implies 128 15 Clifford algebras

either that 10 : 0, or, if n is odd, that ).6* )."(e") : 0. This is because, for eachi e n and eachI c. n, ei either commutesor anti-commuteswith e1.So : : Dt,@,): o > l)rei@)e;t I (,)"1(e) o fcn Icn .Icn where (7.;: 1 or -1 according as ei commutesor anti-commuteswith e7.It follows that f, h@r) :0, with the summationover all l such that ei commuteswith esch e;. Now there are at most only two such subsets of n, namely 0, since€a : l, and, when n is odd, n itself. This provesthe assertion. From this it follows that the subset {e1 : I c. n,$I even} is linearly independentin A for all n and that the subset {e1 : I c n} is linearly independentfor all even n. For n odd, either {e7 : 1 c n} is linearly independentor enis real. To explorethis last possibilityfurther let n : p + q :2k * 1. Then (eo)2: ("zn+t)2: (-r)k(2k+r)+4.But, sinceen is real, (en)2is positive. Therefore("o)2 :1, implying that k(2k + 1)+ q is divisibleby 2, that is, 4k2+ p -l 3q - 1, or, equivalently,p - q - I, is divisible by 4. Conversely if p - q- 1 is divisibleby 4, n is odd. Finally, if eo : *L, n being odd, then, for each 1 c n with ffi odd, er : X€n\r.Sincethe subset{e1 : I c n,ffI even}is linearlyindependent in A, it followsin this casethat dimA:2-|. tr The lower value of the dimension of a Clifford algebra of a non- degeneratefinite-dimensional real quadratic spacedoes occur; for, as has akeadybeen noted, R is a Clifford algebra for R1'0and H is a Clifford algebrafor R0,3. Corollary 15.11indicates how Theorem15.10 is usedin practice.

Corollary 15.11Let A be a real associatiuealgebra *irl, o* orthonormal - subset{ei : I e n} of type(p, q), wherep+q : n. Then,if dimA 2n-1,A is a Clffird algebrafor Rp,qwhile, if dimA : 2n and if e" * I1., thenA is again a CIffird algebrafor Rp's,it beingnecessary to checkthat en + +1. only whenp - q - | is diuisibleby 4. For example,R(2) is now seento be a Clifford algebrafor R2,0simply 0\ 1 0\) ' becausedimR(2):22andbecauseth.*.t1l1 f i\o 1/'\o-r)J'" an orthonormalsubset of R(2) of type (2,0).

Proposition15.12 The real algebra2R is a Clffird algebraforRl,0. 15 Clffird algebras 129 UnhtersalClffird algebras The special role played by a Cliflord algebra of dimension 2n, for an n-dimensionalreal quadratic spaceX, is brought out by the following theorem.

Theorem 15.13 Let A be a Clffird algebrafor an n-dimensionalreal quadraticspace X, with dimA:2, Iet B be a Clffird algebrafora real quadraticspace Y, and supposethat t : X --+Y is an orthogonalmap. Thenthereis auniquealgebramap t1 :A--+B sendingl6y to lg1 anda uniquealgebra-reuersing map (taf : A--+B sending1711 to I611such that the diagrams

X_\ Y x -\v I' and J J, 1, tt J' A __:_+ B A 4n comrnute.

Proof We construct t4 the constructionof (taf being similar. Let (ei : I e n) be an ordered orthonormal basis for X. Then, if ta exists,ta(e7) : flier t(e), for eachnon-nuil f c n, while 2(11e;): 111;,by hypothesis.Conversely, since the set {e1 : -I c n} is a basisfor,4., there is a --+ uniquelinear map t1 : A B suchthat, for eachI c n, t1(e) : flia t(e). X -1 Y In particular,since, for eachi en,ta(e): t(e),the diagram f, l, A 4s commutes.It only remains to check that t1 respectsproducts, and for this it is enoughto check that, for any I, J cn,

t/(e 1)(e1\ : t1@ 1)t 1@ 1).

The verification is straightforward, if slightly tedious, and dependson the fact that, since I is orthogonal, t(e))2 : e?, for any i e n, and t(e)t(e;) : -t(e)t(e j), for any distinct i, j e n. The final details are left as an exerclse. n

Theorem 15.13is amplified and extendedin Theorem 15.32in the particular casethat Y : X and B: ,4.. Immediate corollaries include the following.

L- 130 15 Clffird algebras

Corollary 15.14Let A and B be 2"-dimensionalClifford algebrasfot an n-dimensionalreal quadraticspace X. ThenA= B.

Corollary 15.15Any Clffird algebraB for an n-dimensionalquadratic spaceX is isomorphicto somequotient of any giuen2"-dimensional Clffird algebraAfor X.

Proof What has to be verified is that the map (Lx)a : A'-+ B isa surjectiuealgebra map. This verification is left as an exercise. f-l

A 2'-dimensionalreal Clifford algebrafor an n-dimensionalquadratic spaceX is said to be a uniuersalreal Clifford algebra for X. Since any two universal Clifford algebrasfor X are isomorphic,and sincethe isomorphismbetween them is essentiallyunique, one often speaksloosely of the universalClifford algebrafor X. The existenceof such an algebra for any X has, of course,still to be proved. We shall denote the universal real Clifford algebra for the quadratic spaceRaq by Ro,q.

The constraction of uniaersal Clifford algebras Corollary 15.11 may now be applied to the construction of universal Clifford algebras for each non-degenerate quadratic space RP'<' The following elementary proposition is used frequently.

Proposition 15.16 Let a and b be elements of an associatiue algebra A with unit element L. Then, if a and b commute,(ab)z : a2b2,so that, in particular, a2:b2:-I - (ab)2:1, a2: _! and b2:l - (ab)2: -1, a2:b2:l + (ab)2:1, white, if a and b anti-comrrrute,(ab)2 : -a2b2, and q2:b2:-I - (ab)2:-1, a2 : -I and b2 : | - (ab)z:1, a2:b2:l - (ab)2:-1.

The first stage in the construction is to show how to construct the universal Clifford algebra Rp11,4+1for Rp+1's+1,given Rp,q,the universal Clifford algebra for Rp,q.This leads directly to the existence theorem. 15 Clffird algebras 131

Proposition 15.17Let X be an A-Iinear space,where A : K or 2K and K : R, C or H, and let S be an orthonormalsubset of EndX of type (p, q), generatingEndX qs a real algebra.Then the set of matrices '),o.s},{(91),f-1 {(: -t\10/'\L9 )} L\0 -a)'----) 0)J is an orthonormalsubset of End,X2 of type (p -l l,q I l), generating EndXz 8n R(2) as a real algebra.

Corollary 15.18For eachn, the endomorphismalgebra R(2') ls a uniuersql Clifford algebrafor the neutral non-degeneratespace Rn'n. That ls, R^, : R(2').

Proof By induction. The basis is that R is a universal Clifford algebra for R0'0,and the stepis Proposition15.17. n

We return to Proposition15.17 in Chapter18.

Theorem 15.19 (Existencetheorem.) Eueryfinite-dimensional quadratic spacehas a uniuersalClffird algebra.

Proof This follows at oncefrom the remarksfollowing Proposition 15.1, from Proposition15.8 and from Corollary 15.18. n

One might conjecturethat, for any finite p, e, Rq' is isomorphic to Ro,n.This is not quite the case,the true state of affairs being given by the corollary to the next proposition.

Proposition15.20 Let S be sn orthonormalsubset of type (pi-I,q), generatinga real associatiuealgebra A. Then,for any a e S with a2 : l, the set

{ba:bes\{a}}u{a} is an orthonormalsubset of type (q * l,p) generatingA.

Coroflary 15,21The uniuersalCIffird algebrasRpal,qand.Rq+r, ate isomorphic.

Proposition15.22 The uniuersalCIffird algebraR6,n ls isomorphicfo R, C, H, 2H, or H(2),according as q - 0, I,2,3 or 4. 132 15 Clifford algebras

Proof By Corollary 15.11it is enough,in eachcase, to exhibit an orthonormal subsetof the appropriate type with the product of its members, in any order, not equal to 1 or -1, for each algebrahas the correct real dimension,namely 2'. Appropriate orthonormal subsetsare

@for R, {i} for C, {i, k} for H, {(;-: ),(; -? ) (5 -t)}",'", *'{(;-i )'(d -?),( 5-l),(? 1)irorH(2)n Proposition15.23 Ro,r E C, R:,0 = Rr,z = C A R(2) = C(2), while R:.r ! R2,2= R(4).

This completesthe constructionof the algebrasRp,q, for p * q < 4. Finally, here is a more sophisticatedresult, leading to the 'periodicity theorem'.

Proposition 15.24Let S : {ei : i e 4} be qn orthonormalsubset of type (0,4) of an associatiuealgebra A with unit elementI and let R be an orthonormalsubset of type (p; q) of A such that eachelement of S anti- cornmuteswith eueryelement of R. Thenthere existsan orthonormalsubset R' of type (p, q) suchthat eachelement of S commuteswith eueryelement of R'. Conuersely,the existenceof Rt impliesthe existenceof R.

p' Proof Let a: e4: eoere2e3and let : {ab : b e R}. Sincea commutes with everyelement of R and anti-commuteswith everyelement of S and sincea2 : 1, it follows at once that R' is of the required form. The converseis similarly proved.

Corollary 15.25For all p, q,

Rp,q+4= Ro,nI Ro,+= H(2).

For example,by Proposition11.9, Ro,s = csH(2) = C(4), Ro,e 'r H8H(2) = R(8), Ro,z = 2HoH(2) = 'n(8), and Ro,a = H(2)sH(2) = R(16). 15 Clifford algebras 133

Corollary 15.26(The periodicity theorem.)For all finite p, q, Raq+s3Ro,n@R(16).

By putting together Propositions L5.22, 15.12,15.17, and.15.20, and theselast two corollaries,we can construclany Rp,q.Table15.27 shows themall, for 0 < p,q <7.

Trhle 15.27

q --+ pRc H 211 H(2) C(4) R(S) ,n(s) J2RR(2) c(2) H(2) zHQ) H(4) c(8) R(16) R(2) ,n(z) R(4) c(4) H(4) ,u@) H(8) c(16) c(2) R(4) 2R(4) R(s) c(8) H(8) ,n(s) H(16) H(2) c(4) R(8) ,n(S) R(16) C(16)H(16),H(16) 'g(z) H(4) C(S)R(16) znlre; R(32) c(32) H(32) 'rrg) H(4) H(8) c(16) R(32)2R(32) R(64) c(64) c(8) H(8) 2H(8)H(16) C(32) R(64)znlo+;R(12s) The pattern from the top left to the bottom right of the table is derived from Proposition I5.L7,while the symmetryabout the line with equation -p + q : -1 is derivedfrom Proposition15.20. Squares like thosein Table 15.27have already made a brief appearanceat the end of Chap- ter 11. There are clearly (non-unique)algebra injections Rp,q -Rp+r,q and R.n -+Rp,4+1,for any p, q suchthat the squarescommute. Table 15.27exhibits each of the universal Clifford algebrasRo,n as the real algebra of endomorphismsof a right AJinear spaceV of the form A', where A : R, C, H, 2R or 2H. This spaceis called the (reat\ spinor spaceor spaceof (real) spinorsof the orthogonal spaceRzq. It is identifiablewith a minimal left ideal of the algebra, and to that extent is non-unique.However it can be proved that any two minimal left ideals are equivalent,so that the slight ambiguity in the definition in practice is unimportant.

Proposition15.28 Let Rp,q: A(m), accordingto Table 15.27or its exten- sion by Corollary 15.26.Then the representatiuein A(m) of any elementof the standardorthonormal basis for Rp+sis orthogonalwith respectto the standardpositiue-definite correlation on A*.

Proof This follows from Proposition11.9 and Corollary 11.11,and its truth, readily checked,for small valuesof p and q. n 134 15 Cliford algebras 2H), When K is a double field (2R or the K-linear spacesV(I, O) and v(0,1) are called the (reat) half-spinorspaces or spacesof (real) half-spinors,the endomorphismalgebra of either being a non-universal Clifford algebraof the appropriateorthogonal space.

Complex Clffird algebras The real field may be replacedthroughout the above discussionby the field of complex numbers c, and indeed by any commutative field, though the caseof characteristic2 needscarefur handling - we ignore such matters here! The notation cn will denote the uniuersalcomolex Clffird algebra for Cn unique up to isomorphism.

Proposition 15.29For anyfnite p, q with n: p I e, Cn z Rp,q6lp C, = denotinga real algebraisomorphism.

Corollary15.30 For anyfinite k, Cx = C(2k)and Czn+t?rC(Z!).

The complexspinor and half-spinorspaces are definedanalogously to their real counterparts.

Saperfields A further generalisationof the conceptof a clifford algebrainvolves the conceptof a superfield.A superfield,Lo,with fixed field K, consistsof a commutative algebraL with unit element1 over a commutativefield K ii and an involution a of L, whose set of fixed points is the set of scalar Ir multiples of 1, identified as usual with K. (The algebraL need not be a 2Ro, liii field.) Examplesinclude R, e and eachwith fixed field R, and C and 'il 2co, eachwith fixed field c, where ii in thelattet two exampleso denotes the swap involution. Let X be a finite-dimensionalquadratic space over a commutativefield K. Let Ld be a superfieldwith fixed field K and let Abe an associative L-algebra with unit elemen! the algebra L being identified with the subalgebragenerated by L and the unit element.Then,4 is said to be an Lo'clifford algebrafor X if it containsx as a Krinear subspacein such a way that, for all x e x, x{2)- -xQ), provided also that .4 is generated as a ring by L and X or, equivalently,as an L-algebra by 1 and X. All that has been said before about real clifford algebrasgeneralises to Lo'clifford algebrasalso. The notationser' and2R;n will denotethe

l ii f

15 Clffird algebras 135 2Ro-Clifford universale- and algebrasfor Rp'q,and the notation 2Co,the 2c-Clifford universal algebrafor Cn, for any finite p, q, n.

Proposition15.31 Let Ld be a superfieldwith fixed field K, let X be a K- quadraticspace and let A and B be uniuersalK- and L-Clffird algebras, respectiuely,for X. Then,qs K-algebras,B = A @xL,

Note that, as complex algebras,e o,n and Cn are isomorphic, for any finite n, p, q with n : p * q. The detailed construction of the tables of L'-Clifford algebrasis left to the reader.

Inaolations und anti-im)olutions Certain involutions and anti-involutions of Clifford algebrasplay major rolesin the descriptionof their structure.These arise as corollariesof the followingtheorem which amplifiesand extendsTheorem 15.13 in various ways,in the particular casethat Y : X and B : A.

Theorem 15.32 Let A be a uniuersalLd-Clffird algebrafor a finite- dimensionqlK-orthogonal space X, Lu beinga superfieldwith inuolution u andfixed commutatiuefield K. Then,for any orthogonalautomorphism t : X -->X, there is a uniqueL-algebra automorphismtt i A--+A, sending any )" e L tu ), and a uniqueK-algebra anti-automorphism ti': A->A, sendingany A to ),o,such that the diagrams

X -\x x -\x and l, l, I' l, JJ tl A 4a A --:-+ A

commute.Moreouer, (Ix)n: 11 and,for any t, u e O(X), (ut)o: u4ttr: u;6

If t is an orthogonalinuolution of X, then ta is an algebrainuolution of A and tf'is an algebraanti-inuolution of A.

The involution of .4 inducedby the orthogonalinvolution -1x will be denotedby a -, d and called the main inuolutionor the grade inuolution of ,4. The anti-involutionsof ,4.induced by the orthogonalinvolutions 1a and -ly will be denotedby ar-t f and ar+ [ and called,respectively, reuersionand conjugation,d-being called the reuerseof a and q- the 136 15 Clffird algebras conjugateof a. (The reasonfor preferring d'tod and al to ? will become apparentlater.) Of the two anti-involutionsconjugation is the more important. Rever- sion takesits namefrom the fact that the reverseof a product of a finite number of elementsof X is just their product in the reverseorder. For example,consider a : I * eo* epz * ese1e2€ Ro,:.

Then 6: l-eo* epz- eoere2, ti-: | * eo* ezet* e2e1en: I t eo- eF2 - eoere2, while a-: 1 - eo* ezet- ezeleo: | - e0- ep2 * eoetez.

Proposition15.33 Let A be a uniuersalU-Clffird algebrafor a finite- dimensionalK-quadratic space X,Ln beinga superfieldwith commutatiue fixedfieldK. Then,for any a € A, a- : @f: fi

Proof Each of the three anti-involutionsis the unique anti-involution of .4 inducedby -lx. n

Eten Clffird algebras Let A be a universal Lo-Clifford algebra for a finite-dimensionalK- quadratic spaceX. By Proposition 1.6 the main involution induces a direct sum decomposition,40 @ Ar of A, wherc - -a). Ao : {a e A : 6 : a} andAr : {a e A : 6 Clearly A0 is an L-subalgebraof A. This subalgebrais called the euen CIffird algebra for X. It is unique up to isomorphism. Any element a e A may be uniquely expressedas the sum of its euenpart a0 e A0 and its odd part ar e Ar. Fot a : I I ea* etez* e6ep2,a0 : 1-* epz and ar : eo* eoetez. The even Clifford algebrasfor the non-degeneratereal or complex finite-dimensionalquadratic spacesare determinedby the next proposition.

Proposition 15.34 Let A be u uniuersalW-CIffird algebrafor a finite' dimensionalK-quadratic space X, Lo beinga superfieldwith fixed field K, and let S be an orthonormalbasis for X of type (p, q). Then,for any a e S, the set{ab: b e S\{a}} is an ofthonormalsubset of A0 generating Ao, and of type (p, q - I) or (q, p - I), according&s q2 : -l or l. In either case,moreouer, the induced.isomorphism of Ao with the uniuersa.l 15 CIffird algebras 137

U-Clifford algebraof a (p -f q - I)-dimensionalquadratic space respects conjugation,but not reuersion.

Proof The first part is clear, by Proposition 15.16. For the last part it is enough to consider generatorsand to remark that if a and b are anti-commutingelements of an algebrasent to -q and -b, respectively, by an anti-involution of the algebra,then, again by Proposition 15.16, ab is sent to -ab. On the other hand, if a and b are sent to a and b, respectively,by the anti-involution, then ab is sent to -ab and not to ab. n

Corollary 15.35 For any finite p, e, n, ! = Rnt R9.o+r- Ro,n, R9*,,0 --n -J) Ci'n*t t er', ci*t'n = ero (,R")9,n*,.r (2Ro)0,q, ('R')l*t,n = (2Ro)n*, C9*t = Cn and ('c')2*t = ('co)n. It follows from Corollary 15.35in particular, that the table of the even Clifford algebrasR!,n, with p + q > 0, is, apart, from relabelling, the same as Table 15.27,except that there is an additional line of entries down the left-hand side matching the existing line of entries acrossthe top row. The symmetry about the main diagonal in the table of even Clifford algebrasexpresses the fact that the even Clifford algebrasof a finite-dimensionalnon-degenerate quadratic spaceand of its negativeare isomorphic. So far we have consideredonly universal Clifford algebras.The use- fulnessof the non-universalClifford algebrasis limited by the following proposition.

Proposition 15.36 Let A be a non-uniuersalClffird algebrafor a non- degeneratefinite-dimensional quadratic space X. Then either 1.y or -ly inducesqn anti-inuolutionof A, but not both.

If lx induces an anti-involution of .4, we say that A is a Clifford algebra for X with reuersion,and, if -!a inducesan anti-involution of ,4, we say that A is a Clifford algebrafor X with conjugation.

Proposition 15.37 The non-uniuersalClffird algebrasfor the quadratic spacesp0'4k+3 haue conjugation,but not reuersion,while thosefor the quadraticspaces pak+1,0 haue reuersion, but not conjugation. t38 15 Clffird algebras Exercises 15.1 Let A be a clifford algebrafor a finite-dimensionarnull real, or complex, quadratic spaceX such that, for some ordered basis (ei : i e n) for X, fliunei f 0. prove that A is a universalClifford aTgebrafor X. (Try first the case that n: dimX : 3.) (The universal Clifford algebra,[X, for a finite-dimensional linear spaceX, rcgarded.as a null quadratic spaceby having assignedto it the zero quadratic form, is called the exterior or Grassmannalgebta fot X, Grassmann,sterm being the extensiue algebra for X (Grassmann(1844)). The squareof any element of X in A X is 0 and any two elementsof X anti-commute.) r5.2 Let X be a real or complex n-dimensionallinear space and let a be an elementof fi X expressiblein terms of some basis {ei : i € n} for X as a Tinearcombination of k-fold products of the e;'s for some positive integer /c. Show that if {/; : i e n} is any other basis for x then a is a linear combination of k-fold products of the fi's. Show by an examplethat the analogous proposition is falsefor an elementof a universalclifford algebta of a non-degeneratereal or complex quadratic space. 15.3 Let X be as in Exercise r5.2. yeify that the set of eremenrs of fiX expressiblein terms of a basis {ei : i e n} for X as a linear combination of k-forct products of the e;'s is a linear /n\ /n\ spaceof dimrtlr1on *t.t. the coefficientof xk in (r/ (;/ " the polynomial (1 + iX. ghis lin'ear space,which is defined by Exercise15.2 independentry of the choice of basis for X. is denoredby Au X.) 15.4 Let X be as in Exercise 15.2, let (a; : i e n) be a k_tuple of elementsof x, let (ei:ie n) be an orderedbasis for x andret --+ t : X X be the linear map sendingei to a;, for all i € n. prove that in /\X flo,: (deu)fle;. i€n i€n 15.5 Let X be as in Exercise15.2 andlet (ai: i e k) be a k-tuple of elementsof x. Provethat the elementsof (a; : i e k) arc \inearry independent in X if and only if, in I X, fli.r. ar * 0. 15.6 Let X be as in Exercise15.2 andlet (ai: I e k) and (b; : I e k) be ordered setsof ft linearry independenterements of x. prove that the /c-dimensionallinear subspacesof X spannedby these F

Exercises I39

coincide if and only if fl,.r b, is a non-zero scalar multiple of fli.xqi' (Considerfirst the casek :2.In this case a0a1: bsbl+ agalbs:0'

from which it follows that bs is linearly dependenton a6 and a1.It shouldnow be easyto completethe argument,not only in this case,but in the generalcase also.) 15.7 Constructan injectivemap 91,(X)--9{/\k X), whereX is as in Exercise15.2. (Use Exercise15.6. This is the link betweenGrassmannians and Grassmannalgebras.) 16 Spin groups

We turn to applications of Clifford algebras to groups of quadratic automorphisms and rotation groups in particular. The letter X will denote a finite-dimensionalreal quadratic space,whose elementswill be termed uectors,and A will normally denote a universalreal Clifford algebrafor X - we shall make someremarks at the end about the case where,4.is non-universal.For eachx e X, /2) : x-x:i.x: -x2. Also, since,4.is universal,Rnx: {0}. The subspaceRoX of ,4 will be denoted by Y, the spaceof perauectorsof X, and the letter y will generallybe reservedas notation for a paravector.The spaceY will be assignedthe quadratic form

Y-'R; ylY-Y.

It is then the orthogonal direct sum of the quadratic spacesR and X. If X =RP'q,then Y = RP'4+1.

Clifford groups our first proposition singlesout a certain subsetof a universal clifford algebraA of a finite-dimensionalreal quadratic spaceX that turns out to be a subgroup of .4.

Proposition 16.l Let g be an inuertibleelement of A such that, for each x € X, gxTr e X. Thenthe maP

Px6: x* gxTl

is an orthogonalautomorphism of X'

140 7

16 Spin groups r4l Proof For eachx e X,

@xg(i){2): (gft)g xTr :tis-t gxTl :}y - y(2), sinceix e R. So px* is an orthogonal map. Moreover, it is injective, sinceg x?-r :0 .> x : 0 (this doesnot follow from the orthogonality of px,s if X is degenerate).Finally, since X is finite-dimensional,pa,, must also be surjective. tr The element g will be said to induce or representthe orthogonal transformationpx,c and the set of all suchelements g will be denotedby f(X) or simply by f.

Proposition16.2 The subsetT is a subgroupof A.

Proof The closureof I under multiplication is obvious. That f is also closed with respectto inversion follows from the remark that, for any g € f, the inverseof pxs is py,r-r. Of course11zy € f. So f is a group

The group f is called the Clffird group (or Lipschitz group - seethe historical remarksfollowing Theorem 18.9)for X in the Clifford algebra .4. Sincethe universalalgebra,4 is uniquely definedup to isomorphism, f is also uniquely definedup to isomorphism. There are similar propositionsconcerning the action of A on the space of paravectorsY:R@X.

Proposition 16.3Let g be an inuertibleelement of A such that, for each y € Y, C yTl e Y. Thenthe map Pv,g:Y- gYTl is an orthogonalautomorphism ofY.

Proposition16.4 The subsetd2:{g e A: y eY - gyTl eY} is a subgroupof A.

From now on we supposethat X is non-degenerate,and prove that in this caseevery orthogonal automorphismof X is representedby an element of f. Recall that,by Theorem5.15, every orthogonal automorphism of X is the compositeof a finite number of hyperplanereflections.

Proposition16.5 Let a be an inuertibleelement of X. Thenq el, and the mqp px' is reflectionin the hyperplane(R{a\)r.

L- t42 16 Spingroups Proof By Proposition5.1, X : n{a} I (R{a})r, so any point of X is of the form )"a*b, where,l e R and b' a : 0' By Proposition15'3, b a : -ab. Therefore,stnce6: -a, -aQ' : -)' q' Px'o(Ia + b) : a * b)a-r * b' Hencethe result. n

The next proposition characterisesthe field of real numberswithin the algebra A.

Proposition16.6 Let a € A be suchthat ax : xA',for all x € X, A being a uiiuersalCtffird algebrafor thefinite-dimensionsl non-degenerate rcctl quadraticspace X. Thena eR.

Proof Leta:a0+a1,where aoeA0 andareA1 .Then,sinceax:xG', &ox: xqo andaL x: -xal , for all x e x, in particular for eachelement e, of someorthonormai b:-.-'. {ei:0

Theorem 16,7 With X as aboue,the maP px : l(X)--+O(X); se Px,g

is a surjectiuegroup map with kernelR. .

Proof To prove thal px is a group map, let g, g' e f. Then, for all x€x,

Px,gg'(x): gg'*6i)-t ^,-I-l : ggxg', g-

: Pxs Px,g'lx),

So px,ec,: px*px,g,,whichis what had to be proved. The surjectivity of py is an immediatecorollary of Theorem 5.15and Proposition16.5. Finally, supposethat py,s : px,s,, for g, g' € l. Then, for all x e X, gxg-r : g'x?-', implying that (g-lg')x : x(g-rg'), and thereforethat r

16 Spin groups 143

g-rg' e R, by Proposition16.6. Moreover, g-1g, is invertible and is thereforenon-zero. Hence the result. rt

We isolate part of this as a corollary.

Corollary16.8 Any elementg of the Clffird groupf(X) of a non-degeneratefinite-dimensionalreal quadraticspace x is representableas the product of a finite numberof elementsof X.

An element g of f(X) representsa rotation of X if and only if g is the product of an euen number of elementsof x. The set of such elementswill be denoted by l0 : r"(x). An elementg of f represents an anti-rotationof x if and only if g is the product of an odd number of elementsof X. The set of suchelements will be denotedby fl : f1(4. Clearly,f0 : f n,40 is a subgroupof f, while f1 : I o,41. Since,for any a e A0,6: a, the rctation induced by an elementg of fo is of the form X--+X; x-rgxg-1 .

Similarly,since, for any & e A1,d,: -a, the rotation inducedbv an elementg of fr is of the form

X --+X; x r--,-g x g-t .

An analogousdiscussion to thatjust givenfor the group f can be given for the subgroupo of the universal clifford algebraA for x consisting of thoseinvertible elements g of A suchthat, for all y €y, gy|l ey. However, the properties of this group are deducible directly from the precedingdiscussion, by virtue of the following proposition. The notations are as follows. As before,X denotesan n-dimensional non-degeneratereal quadratic space,of signature (p, q), say. This can be consideredas the subspaceof Rp,s+1consisting of those elementsof Rp,4+1whose last co-ordinate,labelled the nth, is zero. The subalgebra Ron of Rp,q11gen€rated by X is a universal Cliflord algebrafor X, as also is the even Cliflord algebraRl,n*r, by the linear injection

X-R!,n*, ; x> xen.

(SeeProposition 15.34.) The linear spacey : R @X is assignedthe quadratic form y e y- y.

Proposition 16.9 Let 0 : Ro,n- R9,n*, be the isomorphismof uniuersal 1.44 16 Spin groups

Clffird algebrasinduced, according to Theorem15.13, by ly. Then (i) the map u: Y --+R!'q+r;y- 0(y)eir is an orthogonalisomorphism, (ii) for any g e Q, 0(d € fo(p, q I I) and the diagram YLY I vvPo(c\ Rp,s+1 po,a+l commutes, (iii) the map d2--+lo(p,q* 1); g - 0(d is a groupisomorphism.

Proof (i) Since0 respectsconjugation, and sinceen- €r: l,

(0(y)eit) : !- !, for anyy e Y . (ii) First observethat, for any g e Ro,q,0(g) €n : €n0(f), for the isomorphism0 and the isomorphismge €n0(3)e;t agreeon X. Now let g € O. Then, for any u(y) € Rp's+r,where y € Y, e@)(0(y)e;\0(il-t : 0k) s(y)o(E)-t eil 0(SYT\";t : u P's(l). So 0(g) € f0(p, q + l), and the diagramcommutes. (iii) The map is clearly a group map, since 0 is an algebra isomorphism. One provesthat it is invertible by showing,by essentially the same argument as in (ii), that, for any h e I{(p, q + L), ?-t(fi e A. tr

Corollary 16.10 The orthogonaltransformations of Y representedby the elementsof Q are the rotationsof Y . Sinceconjugation, restricted to y, is an anti-rotation of I, the anti- rotations of Y also are representableby elementsof Q in a simple manner, It remains to make a few remarks about the non-universalcase. We suppose inat l, is a non-universal Cliflord algebra for X (you might have in mind H as a non-universalClifford algebra for R0'3).Since the 16 Spin groups 145 main involution is not now defined,we cannot proceedexactly as before. Howeverin the casethat we havejust been discussing,$ : I o, _g, fo, any g e f, according as g € f0 or fl. what is then true is trie fottowing.

Proposition16.rr Let g be an inuertibleelement of the non-uniuersarcrif- ford algebraAfor X suchthat,for alt x e X, SiS-t eX. Thenthe map X --,X; g x g-1 is a rotation of X, white the map-X --_>X; x e _g x g_t is an anti-rotationof X.

In this case| : fo - f1. The discussioninvolving y : R@X requiresthat conjugationbe defined,but if this is met by the non-universarclifford algebraA, rhen A may be usedalso to describe y. the rotations of The restriction to r of conjugationis, as before,an anti_rotationof y.

Pin groups and Spin groups The cliflord group f(X) of a quadraticspace x is rargerthan is necessary if our interestis in representing orthogonal transformationsof X. use of a quadraticnorm N on the clifford algebra,4 leads to the definition of subgroupsof f that are ressredundani for this purpose.This quadratic norm N : A--+A is definedby the formula N(a):aa,foranyaeA, A denoting,as beforethe universalcrifford algebraofthe non-degenerate finite-dimensionalreal quadratic spaceX.

Proposition16.12 (i) For any s e f, N(g) e R, (ii) N(1): 1, (tii) for an! E, gte f, N(gg,): N(g)N(g,), (iv) qny for s e f,N(g) f 0 and N(g-l): (lr(*))-t, (v) any g e l, for there existsa uniquepositiue real numberI such that I, lNe,ill: namety,t: ./(lN(c)t_l.

Proof rhat N(1) : 1 is obvious.Ail the other statementsfolrow directly from the observation that, by Theorem 16.7, any g € r is expressible (not necessarilyuniquely) in the form T-T : I I *r x0xr ...xk_2 xk_r, t€k

L r46 16 Spingroups where,for all i € k,4 e X,k beingfinite; for it followsthat s-:n'*u-t'-; t€k

and that N(e):s-s:fI"(r'), i€k

where,for each i e k, N(x;) : -xi € R. !

For X and f : f(X) as abovewe now define :1}' PinX:{g e r : lN(g)l:1} andSpinx: {g e f0 : lN(g)l

Proposition 16.13As subgroupsof I andf{ tespectiuelythe gtoupsPinx and SpinX are normal subgroups,the quotient groups l/PinX and : fo lspinx eachbeing isomorphic fo R+ {,1e R : ,t > 0}.

The next proposition assertsthe important fact that the Pin and Spin groupsd.oubly couer Ihe relevantorthogonal and specialorthogonal groups.

Proposition 16.14Let X be a non-degeneratequadratic space of positiue finite dimension.Then the maPs -+ PinX "+ O(X); Se Px,sand SpinX SO(x); g+ Px,s

are surjectiue,the kernelin eachcase being isomorphic to So'

j li.i, When X - \Pg the standard notations for f(X), f0(X), PinX and r't,l SpinX will be f(p, q),f}(p, q), Pin(p, q) and Spin(p,q)' SinceRlr = ,,.i R|,n,fo(qp) = fo(p, q) and Spin(q,p)= Spin(p,4). Finally, fu(O,n) is oftin abbreviatedto fO(n) and Spin(O,n) to Spin(n). That Proposition 16.12(i)is a genuine restriction on g is illustrated by the element1l eae R6,4,since N(l + ei :2(I * ei 4 R. That the sameproposition does not, in general,provide a sufficientcondition for g to belong to f is illustrated by the element I I ee e Rs,6,for, since N(1+e):(l-eoX1*e6):2,theelementisinvertible,but,byexplicit computationof the element(1* edes(I+ee)-r,it canbe shownthat it doesnot belongto f. However,the condition is sufficientwhen p*q < 5, as the following proposition shows. I ti ir li l;i Urli r 16 Spin groups 147

Proposition 16.15 Let A be a uniuersalclffird argebrafor a real non- degeneratequadratic space X with dimX < 5. Then

SpinX: {g e ,40: N(g) : +1}.

Proof rhe proof is given in full for the hardest case, namely when dimx: 5. The proofsin the other casesmay be obtainedfrom this one simply by deleting the irrelevant parts of the argument. From the definition there is inclusion one way (c). what has to be proved,therefore, is that, for all g e A0 such that N(g) : *1, xeX+gxg-1 eX.

Let.{e;: I e 5} be an orthonormalbasis for X. Then, sinceX c,41 andg e A0,x' : gxg-1 e A1,for any x € X. So there arercalnumbers a-i, bikr,c such that : / t o,",* e*er* ces. o=,*.,.rb,0,r, Now (x')-: (gxg-1)- : -yt, sinceg-1 : *g-, while (e;)- : -ei, (e1epe)-: €j€ke1,&t1d €s-: -es. So,for 0 < j < k

The argument ends at this point if n < 5. otherwise it remains to prove : that c 0. Now x'2 : x2 e R. Also f] e5 commuteswith eache; and so with x". So

x'Da2cx"(es) +c2(es)2 e R.

Sincex"2 and,c2(e)2 € R, and es #R, eitherc : 0 or x,, :0. Whichever is the correct alternativeit is the samefor everyx, for, if there were an elementof eachkind, their sum would provide a contradiction.since the map X --+A; x,-- g x g-l

is injective,it follows that c :0. Thereforeg x g-r € X, for eachx e X

To use Proposition 16.15we need to know the form that conjugation takes on the clifford algebra.This is the subjectof the next chapter.

t t7 Conjugation

This chapter is concernedwith classifyingthe conjugation and reversion anti-involutions of universalCliford algebrasof non-degeneratereal or complex quadratic spaces.Now, according to Table 1.5.27and corol- tarylS.:O any such Clifford algebraisrepresentable as an endomorphism 2K, : algebraA(m), for somenumbet m, wheteA : K or where K R' C or H. Moreover any correlation on the spinor spaceA', induces an anti-involution of the clifford algebra,namely the appropriate adjoint anti-involution, and converselyany anti-involution of ,4 is so inducedby a symmetric or skew correlation on the spinor space' So the problem reducesto determiningin each casewhich anti-involution it is out of a list which we essentially already have. The job of identification is made easierby the fact that an anti-involution of an algebra is uniquely determined, by Proposition 2.8,by its restriction to any subsetthat generates the algebra.

The algebrasRs,n For the clifford algebrasRqn the determinationof the conjugationanti- involutionis madeeasy by Proposition17.1.

: Propositionl1.l Conjugationon the Ctffird algebraR0,, A(m) is the adjoint anti-imsolutioninduced by the standardpositiue-definite conelation on the spinor spaceN".

Proof By Proposition l5'28,q €i: l, for any elemente; of the standard orthonormal basis for R0,r, here identified with its image in the clifford : -ei' algebraA(rt). Also by the definition of conjugacyon Rsn, ei- BvI

148 17 Conjugation r49 ef : -1. So, for all - i e n, e;- ef , ftom which the result follows at once, by Proposition2.8. n This indicates, incidentally, why we wrote a, and not a, for the conjugateof an elementa of a clifford algebraA, the reasonfor writing il and not d, for the reverseof a, being .i*ilur. The notation d is less harmful in practice, proposition for, in the context of 17.L atleast, ? in either of its sensescoincides with ? in its other sense. As an immediate proposition application of L|.r we have information on the groups Spin(n),for small n.

Proposition17.2

spin(l) = o(1) = s0, spin(2)= u(1) = sl, spin(3)= = = sp(t) s3, Spin(4) sp(l) x sp(l) _=s3 x s3, Spin(S)= Sp(2), Spin(6)is a subgroip of U@).

Proof Apply Propositions 16.15and I7.1. n In the caseof Spin(n), : .1.,2,3, for n 4, what this propositiondoes is to put into a wider relationshipwith eachother various resultswhich we have had before.It may be helpful to look at thesecases in turn. R2 : The universal clitrord argebraR6,2 is H, while the universal algebraR2,s llnora is R(2), the evenClitrord algebrasR$,, and n!-o .u.t being isomorphic to C. Supposethat : we useRs,2 H to describethe rotations of R2, R2 being identified with R{j, k} and R&, : C being identified with R{1, i}. Then the rotation of R2 representedby g e,Spin(2): U(1) is the map' xe gxg-' : gx!, that is the map

(xo+ xri)j : (xoj + xlk) ++ (a I bi)2(xs+ xri)j

: (a + b i)(xsj + xft)(a - b i), wherex : xoj *x1k andg : a* bi. On the other hand, by Corollary 16.10,we may use C directly,R2 being identified with c. Then the rotation of R2 repres""t.o iy i, tt. map f ylgyTr:gyg:g2y.

One can transferfrom one model to the other simply by settingx : ! i.

L l-- 150 17 Conjugation

R3: The universal Clifford algebra Rs,3 is 2H, while the universal Clifford algebra R3,sis C(2), the even Clifford algebrasR$,, and R!,0 each being isomorphic to H. Besidesthese, there are the non-universal algebrasR6;(1,0) and Rq:(0,1), alsoisomorphic to H. Any of thesemay be usedto representthe rotations of R3. The simplestto useis Rsj(l,0) = H, R3 beingidentified with the linear subspaceof pure quaternions.An alternativeis to useR$,, = H, in which caseR3 may be identified,by Proposition 16.3,with the linear subspace R{1, i, k}. In either caseSpin(3): Sp(1): 53. In the first of these two casesthe rotation of R3 representedby g e Spin(3)is the map

x+ gxg-t : gxg, while in the secondcase the rotation is the mao

yegyg':gyg.+l

One can transfer from the one model to the other by setting x : y j, compatibility being guaranteedby the equation

cyTt : cygj.

Ra : The universalCliflord algebrasR6,a and R4,6are eachisomorphic to H(2), the even Clifford algebrain either casebeing isomorphic to2E. There are various identificationsof R3 with a linear subspaceof 2H such that, for any x e R3, ;s(2): -x2: xx. Onceone is chosen,R4 may be identifiedwith R @R3, with y@ :y y, for any y € R4. One method is to identify R4 with the linear subspace

{('^Y),u.H} l\u v/ ) of 2H, R3 being identified with .{(;-: ),(d -:),(t -t)) rhen,for""r(31).,",

(q - 0\ /r o\ | l:l I \0 r) \0 q)' 17 Conjugation 151

: 'rr whlle Spin(4) {(s:) . : @l: lrl : 1). The rotationof Ra rerresentedby ( 6i)e Spin( ) is then,by Proposition16.9, the map (n;)* (6 :) (n ;) (*)': ("0"]u) This is essentiallythe map y+qy7,

which is what we had before,in Chapter 8. An alternativeis to identify Ra with the linear subspace ') (/v 0\ t(6;):tenj

The rotation induced bv e snin(+)is then,bv a similar ( e : ) argument,the map n{ (n; )- ( ,lu) and this reducesto the map ye qyV.

Proposition17.3 Spin(6)= SU(4).

Proof A proof of this may be basedon Exercise10.2. one provesfirst that if Y is the image of the injective real linear map y : C3-+ C(4) constructedin that exercisethen, for eachy €y,f y € R, and that if y is assignedthe quadraticform y--+R; yefy then y is an orthogonal map and r is the unit spherein y. The rest is then a straightforward checking of the things that have to be checked. Note that. for all t e Sp(2), f-: t-l

The algebrasR.r,n for small p, q f 0 For any g e Spin(n),N(g): 1. For g e Spin(p,e),on the other hand, with neitherp nor q equal to zero,N(g) can be equal either to L or to -.1. The subgroup{C € Spin(p,q) : N(g) : 1} will be denotedby Spin+(p,q). The imageof Spin+(p,q)in SO(p,q)by p is a subgroupof SO(p,q). This subgroup,called the (proper) Lorentz group of Rp,s,will be denoted by t52 17 Coniugation Rt: *.t11b1'shown on 22'48the Lorentz group of SO+(p,q). In Propositi of that preserveihe semi-orientations to be the set of rotations of Rp'q Rp'4. the inducedrotation Pe of F,rJ Example 17.4Let g € Spin(l'!\'.T'^" if and only / N(g) : 1' and reuerses preserDesthe semi-orie;t.t""; of'pt't 'them -I' if and onIYif N(g): physics' the casesof greatestinterest in The next proposition covers known enthe Pauti algebra' t;;i;;t" *?'^= c(2) being

Proposition17.5 2R:ad:t)= R. = GL(I; R)' spin+(r))={(6:)e : R) . *,,1 d't sP(2; spin+(r,2)={(; i) (; ;):1) ' :1) : sP(2;c)' = ecQ) d't (; andspin+(r,3) {( z ;) ;) may be give the proof for which Proof It is enough to .Spin+(l'3)'- L since R!'3 Rr2' Now' bv regarded as a subgro;0";ri;' :^9'' 15'34' rr"ofosition 16.15and Proposition : 1}' SPin+(!'3): {g € R1'2: 8-g anti to determining the conjugation so that the problem is reduced just to select a suitable copy R13' T; ;; ;" we have involution of in RL2 by is to represent ee'el afid e2 of-i R1,2in R1,2. Our choice -t\ mat- ; 0 \ i o ( 9 l ), r".p""tively,in c(2),these -1 ol ""0 \1u'l \o /'\1 and satisfyingthe equations rices being mutually u'nti"o*t)totative -l)':-1 (:9)':Lf 9 and(i;)':-'' \o -r)-"\r o) -* **' '' ( -i as is necessary.Now the anti-invorution( i ;) ) eachofthesethreematricestoitsnegatiie.This,byi,roposition2'8,is Since' the conjugationanti-involution' -r\fr:\:d"t(i1\, /ac\ ( d rorany €c(2), d):*""\b d./' li;) \-; ")\o the proPositionis Proved 17 Conjugation 153 It is natural, therefore,to identify the spinor spaceC2 for R1,2with the comolexsymplectic^plane cfo and, similarly,to identify the spinor 2R" spaceR2 for R1,ewith and ihe spinor spaceR2 for Rr,r wittr nfo. when this is done the inducedadjoint anti-involution on the ieal algebra of endomorphismsof the spinor spacecoincides with the conjugation antiinvolution on the Clifford algebra. Note, incidentally the algebrainjections -- Spin(2) Spin+ (1, 2) and Spin(3) -, Spin+ (t, 3)

induced by the standard (real orthogonal) injections

Ro'2-- R1'2and ao'3-- P1'3,

the imageof Spin(2): U(1) in Spin+(1,2)being SO(2)and the imageof Spin(3): Sp(l) in Spin+(t,3)being SU(2). The isomorphismsu(1) = so(2) and sp(l) = su(2) fit nicely,therefore, into the generalscheme of things. Proposition r7.6 is a step towards the determinationand classification of the conjugation anti-involutions for the universal clifford algebras Ron other than those already considered.

Proposition17.6 Let v be the spinor spacefor the orthogonalspace w,t withRp': EndV. Then,if p > 0 and if (p,q) * (I,0), the conjugation anti-inuolutiononRoq coincideswith the adjointanti-inuolution onEndv inducedby a neutral semi-Iinearcorrelation on V.

Proof By Theorem 13.8 there is a reflexive non-degenerateArplinear correlationon the right AJinear spacetrz producing the conjugationanti- involution on Rr,n as its adjoint. what we prove is that the correlation must be neutral. This follows at once from the even-dimensionalityof 2R 2H. V over A unlessA'p : R, C, E, or However, sincep > 0, there existsin everycase r e EndZ such that t-t - -1, namelyt: eo; for : -"3: eo-eo t) : -t. The existenceof such an elementguarantees neutrality when A'p : R, by Theorem 5.2g. The obvious analogue of Theorem 5.28guarantees neutrality in each of the other cases. n

Analogous results hold for the algebrasCr,n, where, as has already been saidin Chapter 15, ep' is obtainedby-tensoring Raq with the superfielde, that is c assignedthe conjugation involution, which fixes the field C. 154 17 Conjugation anti'inuolutioninduced Proposition 17.7Coniugation onG,n is the adioint spinor space.conjuga- by ihe standardpositiui-definite correlation on the adiointanti-inuolution tion oneo', whire p > 0 and (p, q) * (1,0)' is the spinor space' inducedby-a neutralsemi-linear correlation on the

Completion of the ctassificationof coniugation Tables3to8ofTheoremlT.Scompletetheclassificationoftheconjuga- tionanti-involutionsofthefiueCliffordalgebrasnaturallyassociatedto It is becausethere are the orthogonal spaceAr'4, for eachsignature P' Q' Ro'nfor the standard five and not just the one of thesethat the notation realCliffordalgebraforRp,qhasbeenchoseninthisbookforthereal Clifford algebrasrather than Cl(p, q), or somesuch' of all' the Clifford Besidesthe algebrasRo,n themselves we have' first with the identity as algebras Cn, obtained by-tensoring Ro,nwith C' by tensoring Ro'nwith involution, where p I q : n, ande o,n, obtained 17'7 above' Besides C, with conjugation as involution, as in Proposition 2R;n by tensoringR''n with thesewe consideralso the algebras obtained 2Ro with swappingthe slots it ,op.rn"td of diagonal2 x 2 reil'matrices, " zcowobtained by tensoring as involution, which also fixesthe field R, and C' Ro,nwith the superfield2Co , where swap fixes the field extends.both to In each of thesefive cases,minus the identity on Rp'a algebra'and also to an involution, the main or gradeinvolution on the In this way each an anti-involution of the algebra,namely conjugation' ofthefiveshouldberegardedasasuperalgebra,thatisasaZ2-.gtaded a|gebra,equippedalsowiththeconjugationanti-involutionasanintegral rank information part of its structure.Roughly speaking,C, encodesthe ,n;n It is of coursethe ilil *tit. encodesihe signatureinformation. with their assigned casethat as algebraseo,n and cn-areisomorphic, but conjugationsthey are not at all the same' typesdescribed By Table 15.27eachClifford algebra of any of thetve 2R, C' 'C,H, ot2FI' aboveis isomorphicto a full matrix algebraover R, themoduleonwhichthematrixacts,identifiablewithaminimalleft ideal of the algebra,being the spinot spacefor the algebra' NowbyChapter13anyanti-involutionofsuchanalgebra(inparticu- adjoint to some lar conjugation)may be regardedas the anti-involution space'There arc ten ,y-*etric or skew sesqui-linearform on the spinor 17'8 one must be such classesand to appreciatethe tables of Theorem familiarwithallofthem.ItisappropriatetorecallTheoreml3'7. 17 Conjugation 155

Theorem13.7 Let ( be an irreduciblecorrelation on a right a-linear space of finite dimension> 1.,and thereforeequiualent to o symmetricor skew correlation. Then ( is equiualentto one of the following ten types,these beingmutually exclusiue.

0 a symmetric R-correlation ; 1 a symmetric,or equiualentlya skew,2Ro-correlation: 2 a skew R-correlation; a J a skew C-correlation: 4 a skew fi-, o, equiualentlya symmetric,Fi-correlation; 5 a skew, or equiualentlya symmetric,2ff-correlqtion; 6 a symmetricH-, or equiualentlya skew, H-corcelation; 7 a symmetric C-correlation ; 8 a symmetric,or equiualentlya skew,e-coruelation; 9 a symmetric,or equiualentlya skew,2eo-correlqtion.

The logic behind the numbering of theseten types derivesfrom the order in which most of the casesappear in Table 3 below We begin by reminding the readerthat universalClifford algebrasR ,n for 0 < p, q <7 and Cnfor 0 < n < j areas in Tables1 and 2, where A(m) denotesthe algebra of m x ru matricesover A.

q --+ p+L R C H 211 H(2) C(4) R(8) 'n(s) zp JR R(2) c(2) Idle) 2IJe) H(4) c(8) R(16) 2R(2) C R(2) R(4) c(4) H(4) 'H@) H(8) c(16) H c(2) R(4) 2R(4) R(8) C(8) H(8) zn1Al nlre; 211 H(2) c(4) R(s) 2R(s) R(16) c(16) H(16)2H(16) 2np1 H(2) ng1 c(8) R(16)2R(16) R(32) c(32) H(32) 2H(4) c(4) H(4) H(8) C(16) R(32)2R(32) R(64) C(64) R(8) c(8) H(8)2H(8) H(16) C(32) R(64)2R(64) R(128) Table 1

q --+ ,c(z) +1 I czgc(2) c(4) ,c@) c(s) 2c(s) Table 2

Table 1 extendsindefinitely either way with period g, while Table 2 has period 2.ln eirhercase one obtains the even clifford algebrain any location by moving one squareto the left, if necessaryinto the additional column on the left. 156 17 Coniugation

Theorem 17.8Conjugation types for the algebrasR'r,n, and Cr, to be ouer- laid on the releuantpart of Tables1 and 2, are shownas Tables3 and 4. qmod8 pmodS zg 0842448020 0 + 0 | 23 45 67 8 2222 86266 8 4 3 210765 4 24 4802008424 4 5 67 0l23 4 8 6266822228 0 7 65 43 210 Table 3

nmod8 -+ z7 39727 | tg3z3 Table 4

By contrastthe tableforeo' is giuenin Table 5. qmod2 - pmod2 28 g2g J8 98 Table zR$,n To completethe setwe giue as Tables6 and 7 the tablesfor and 2Cr, the algebrasfor these,as algebras,being just the doublesof the algebrasRr,n and Cn' -P+qmod8 --t 21 | tg5z5 59121 Table 6

nmod2 -+ zg I qz9 Table 7

For the codes0, 4 and 8 in Tables3 and 5 thereis afurther classification by signature.Ihe choicealong the top row or down the extta columnon the left is the positiue-def.niteone. Elsewhere the choiceis the neutralone' r

17 Conjugation I57 Proof rhe first row of Table 3 is known to be correctby proposition 17.1. Tensoringby c, assignedthe identity (anti-)involution, then givesTable 4, by Theorem 13.11.The remainderof rable 3 then follows at once,also by Theorem 13.11.Tables 5, 6 and 7 follow in their turn by tensoring Table 3 by e and using Theorem 13.11,by tensoringTable 3 Ly 2R" and using Theorem 13.12and finally, by tensoringeither Table 5 or Table 6 by c and using Theorem 13.11.The neutrality statementfollows from Proposition17.6. n Tables3 to 7 may be appreciatedthe more if the variouscode numbers are replacedby the classicalgroups that preservethe sesqui-linearforms on the spinor spaces.We give them here for 0 < p, e 1g as Tables g to L2,contenting ourselves with the tablesup to n: p + q: 7. To save spacewe abbreviatethe notations slightly in obvious ways. q --+ p Or U1 Spr 'Spt Spz Uq ot 2os J GLIG) SpzR) Spz(C)Sp11GL2(H) o+(H) Oe(C) Sp2G)'SprG) Sp4(R)uz,z o4(H) 2o4(H) sp2(C) sp4(R) GL4(R) Oq,q Os(C) Spr,r Uz,z Oq1'oo,o GL2(H) Oq(H) Or(C) o4(H) 2o4(H) oe(C)

Table 8

o{C) Gh(C) sp2(c)2spz(c) sp4(c) GL4(C) O8(C) 20s(C) Table 9

q-+ 2Ur p Ut Uz 2(Jz Uq t(Jq Us 2(Jt I GkG) U2 GL2(C) U4 GL4(C) Us cL8(c) Uz 2(Jz U+ 2Uq Us ,U, GL2(C) Ut, GLI(C) U8 cr8(C) Uq 2Uq Ue 2Us GL (C) U8 GL|(C) Ua 2(Js GL8(C) Table 10 158 17 Coniugation

q --+ 2 2 p GL {R) GL {C) GL1(H) Gh(H) GL2(H) GL 4(C) GLsG) Grs G) 12Gr1G) GL2G) GL2(C) GL2(H\2GL2(H)GL4(H) GL8(C) Gr16G) etc.

Table 11

n ---+ 2 2 2 2 GL1 ( C ) GL (C) GL 2(C) GL 2(C)G L 4(C) GL 4(C)G L 8(C) GL 8(C)

Table 12

The dimensions of the groups in Table 8 are shown in Table 13. 013610162856 t3610162856 3610162856 610t62856 10 L6 28 56 t6 28 56 28 56 56 Table 13

Thesedepend only on the rank n: p + q and not on the index (p, q). The dimensionsof the groupsin Table 9 are twice those of the groups in Table 8.

Tablesof Spin groups For each n the quadratic norm g-g of any element g of the group Spin(n) : Spin(O,n) : Spin(n'0) is equal to *1, the group being a subgroup of the euenclassical group associatedin Table 8 to the index (0, n) or (n, 0). That group, being the part of the classicalgroup that lies in the even Clifford algebrafor the given index, lies in the table either in theposition p - 0, q:n-I or in theposition p:fl, Q: -1 (in an extra column on the left that matchesthe first row). For aaYP, q with neitherp \or q equal to 0 the quadraticnorm of an elementof Spin(p,q) may be equaleither to *1 or to -1. It is then the subgroupSpin+(p, q), consistingof thoseelements of Spin(p,q) with quadraticnorm *1-, that is 17 Conjugation 159 a subgroupof the evenclassical group for the index, namelythe crassical group in the positionp, q - l. The group Spin(n) . or Spin+(p,4), with n : p * 4, has dimension -1). It lr@ is the whole group foi n : p + qS 5, but is of dimension one less than this, namely of dimension 1.5,rather than 16, fo' n : p q :6. + In that caseit happensthat each argebrahas a real-valued determinant,and lowering the dimensionby 1 correspondsto taking the determinant equal to 1. In the caseof GL2(H) the determinantis defined by representing eachquaternionic 2 x2 matrix as a 4 x4 complexmatrix and then taking the determinant of that matrix, this necessarilybeing a positive real number.

Theorem 17,9 The groups Spin(n) for n < 6, as well as the groups Spin+(p,q) fo, p + q < 6, in the casethat both p and q are non-zero, are shownin Table 14. q --+ p +r o1 U1 Spr ,5p,. Spz SUq I Ot crl@) SpzG) Sp2(C) Sp1,1St2(H) 2spzG) u1 spz(R) spa(R) su2,2 Spr Sp2(C) Sp4(R) St4(R) ,Sp, Spr,r SUz.z Spz SLz(H) SUq

Table 14

The groupsSpin(n;C), for n 3 6, are shownin Table 15. n -'+ 2spz(C) +1 ol(C) GLI(C) Spz(C) sp+(C) sL4(C) Table 15 (Mgst recent writers do not give the whole of thesetables and some are in error. Note in particular that Spin+(3,3) = Sl,(4; R).)

Proof This theorem extendsthe results of propositions r7.2 and rr.5. The only difficulty is with the caseswhere p I q :6. A proof of the isomorphism ! Spin(6) SU(4) has already been sketchedas proposi_ tion 17.3, and an alternativeproof, basedon Diagram 24.5,wifl be given t, later as a lead-up to the discussionof triality. The first of theseproofs l is the more explicit and may easily be adaptedto provide proofs or tr. t.

; l; 17 conjugattoln 160 ^ ,.^,^.crry SL(4;c), aswe isomorphism:,f l,i"t{i,.ii#,lffi show'These i:j,;iKi"ill}:ii:,jffiare u;,.ffi-a*:d now the.relevant outsetthat in unitaryii?::i rr""T'^Ti"?#il;-,sroups' where T: ::";;ratsed of a modulet .r.-iru ,h"Ygl:,.-":'rh" of complnents " involves **upping conjugation of rnaps: looui. nrro. .left-cosetexact'sequences ^ .. -vingdiagram*-"e- of Lsiderthe follovlollowur5 Consider --'+__+ S1R3srp: x R3)n3) -'-"+ sL(3;a r /a. R)rt\ R): SL(2;R) Sp(z; I? :' g- sL1+;n) ! -' ---> sr'tl''l nl SP(4;R) IT :' 56+ x Ra)' 56utx not e R' xrl;J"'l;u1l S(R"x R') : {G'v) u"iOtn'in"O where,*'*ir, for anv n' oI tnl the secondrow *" ntont i' that exaat sequence J;;; leficoset :' SG3 xR3)' -:' SPin+(3'3) SPirf(3'Z) andhence :ri::,ril;,*,ll fffrii,1:::#*ki":m 3),on the point i"#iil,,]',i:'1""ff'#o^;;i'hi*-"q.'*',3)'througt'lt;"it ',"; -#.;';;';t(3' I ', ((r) [i)) = 1' that is with equationx'y x R31in n3 x fP of the quadricSGJ : 4' (xz+ yzf * (xt + vz)2 - - y + lxr* yr )2 + - - (x' - yzf (xz z\2 -(xr y)2 [:il1.}lilli#,:::'i'ii;i4^l;ii the.left ,111,il:iilH;:hru:;1";fiof themap i arcatt ""i.:Lli, * *l i' defined flbres""r"o the map Jl il;in bYtou'n to the diagranr' (fi)) ff;Til)'((fr) 17 Conjugation 16I the first componentof the image being the last column of the matrix a and the secondcomponent the last column of the the -urririo"j:i, with aozAozI anAn * a22A22: deta : l. The fibres of this map are the left cosetsin SL(3; R) of the subgroup sz(2;R), identined with the inverse image or ((i) (i)) ,* map SL(4; R) - S(Ra x R4) is analogouslydefined. The map n : sL(4; R) * g _ c sL(4; R) is defined,as is the mapn in ExerciseL0.2. bv ^:::i\( -x -r;\ ,_(::,2: _3,:-::;_::: -r; \ ;il::: :: :::) \ _:": _::: )' with fibres the left cosets in Sf(a; R) of the group Sp(4; R) identified with the inverseimage by z of the unit matrix. Note that the inverseof any elementof Sp(4; R) is then explicitly given. The -- map Sp(a; R) S(Ra x R4) is definedbv "- ( n i: ii r,) -( ( j!,) ?:il) - with ao2&B anaoz * ay2a9 - a32a43 : _a32, (-aD, a.02, ct22). (a6, ag, a2, ay) -1.

That the map between the two copies of s(Ra x R4) is the identity is theneasily verified' By elementarydiagram-chasing, u. in eroporiiionz.q, it followsthat thereis a linear map R3 x n3 Jn1+;,"r,ifJr* 16 bijectionbetween s(R3 x R3) and th" i-ug" e of z, that makesthe " diagramcommute. Explicitly this is the map ((l)ff:))-U j^rl)

E--- 162 1.7 Coniugation The image of this linear map may be taken to be the spaceof paravectors in the even Clifford algebra R!.r. Identification of SL(4; R) with Spin+ (3,3) followsdirectly. The whole of the aboveargument goes through unchangedif the field R is replacedby the field C. Accordingly Sf( ; O) may be identified with Spin(6;C). As a matter of fact it is enough simply to prove this last result. Then all the real casesfollow by restriction. n

Tablesfor anti'inaolutions other than coniugation Anti-involutions of clifford algebrasother than conjugationmay also be classified.

Theorem17.10 The reuersion anti-inuolution ofRo,n, induced by the identity endomorphismof the quadraticspace RP'e, is classifiedby a table similar to Table 3 of Theorem17.8 which classifiesconjugation, namely Table 16.

qmod8 - pmodS 07 6543 2I J zgo 84244 80 01ZJ 45 67 82222 86266 432l 07 65 244 80zgo 84 4567 0123 86266 82222

Table 16

In this case positiue-definitenessof the spinor product and compactness of the associatedclassical group reign down the left-hand column,with neutralityerserywhere else where there is a choice.

Note that Table 16, like Table 3 of period 8, is just Table 3 translated one squareto the'South-East'. More generally any orthogonal involution of Rp'q induces an anti- involution of Rp,q,and one would like to classifyall such. The relevant table for the algebrasRs,, just looks like Table 3 turned upsidedown, as displayedin Table 17. 17 Conjugation

0942448020 7 6543 210 6266822228 5 67 01234 4902008424 3 210765 4 2222 86266 8 123 45 67 0 Table L7 Here the first row classifies conjugation and the first corumn classifies reversion. The transition from conjugation to reversion as fewer and fewer in turn of the standard basis vectors of Raq have their signs changedby the involution is made by traversingthis table steadilyfrom the first row to the first column in a south-westerly direction- i, ult intermediatestages neutrality is the rule. th: al8ebrasRo,n_o comprising . l:t the pth row of Table 3 there is a similartable. For example in the casethat p :2 *"have, for the algebras R2,'-2,Table 18, which is just Tabre 17, borderedby twoadditional rows on the top and two additionalrows on the left that respectthe periodicity 8.

2222 86266 8 012345 67 0 o20o84244go20 1 07 65 43 21 0 286266822228 3 45 67 0r 23 4 4244 8 o20o g 424 5 43 210765 4 6 82222 86266 8 7 01234 5 67 0 Table 18 Here also the transition from conjugationto reversionas fewer and fewer in turn of the standard basisvectors of Rp,qhave their signschangeo uy the involution is made by traversingthe table steadilyfrom the first row to the first column in a south-westerrydirection. The spinor product is positive-definiteifthe involutionchanges the signofall vectorsofsquare -1 while all those of square*1 remain fixed. It is neutral otherwise. much -"**** ,u"^rl"!*?""'!^*rn"rvhorethingisso I C' coincides classificafionof the atgebras ;:". fi.ll Thereuersion *iri rlrcconiugation classificafion' I HurwitzPairs I conjugauon':o1 Clitro:dalgebra with Let ,4 be a possiblynon-universal spaceX' embedoeo | lt""t-i'"'g'nit"q tt1l,:"adratic a finite-dimention"r letthe sPrnor 'ffi:,J..T1*:1t(",,1]""fi I .4in thestandard'*"tl in (for""0 example' as those | *tlt*"t"i* idealof '4 ortheSffi;;;.'u*n'last corumn ttttn'rner.' I $J1T::T:#;;;;;;;exception l :t)-f xnt: (x-x)(u u/) (xu-1-(xu') a .:L- | sucha pairingof correlated'P:':-I;;:#:H:I :;;{;X:; I r;r;l;*:ffii#l'111:[:iff'"i';1ii:911t111'**** ilff'*: **l'lq"rfi(1e86)'i::li:;"ill1r#tkt#zi:JfrRandriamih"t'::". ti:;:i"- al- n t"iiaiatti in.- t""-ittiversal or Animportan' "il'*nt;;-;;d ?* -ry:#;;::""1ff:l ffispaces elements',with the l*',lJl:Ui::f,i';"'d'['i"-?,'i,i;;;;u'oouorGL(s;n)u"ing'of nG) arr of *ho.. by definition, r;;:;u*pu., " inR(2) isa linear or copieslubsPaceof H in h***tiii-ll;iJl'T#lr*t'il"'t1111"ii"; standard GL(2;R) t u*i't'i*'i' R(a)isatine2r.,"u',"n"*.lriii^r.1:f t}:T ,Xi#*Ah1ilt:i$il"l'11*;I dimensionsrea'[er ii ;:'JF'!:lxH"';;; elementsa artdb of ;lilJ ii#i linearly ino"ntnOttt than 1 thto tht"' ;tol}ia"t)(i" andtherefore such ul1€ R' GL(s;R)to"t'ttiot' ro' "*iit"iitt;R) therundamental il' -n"* ;: : thatc * ' o'ii'' !t)Zio*"ut'''uvtott' ttt;t * '11) 0' " urg"iJ' it'*' it 1 Til it*b"-i 1t:fof odddegree' theorem"r t t) ueinea polvnomialmap themap * -' *ii '-;;i1; subspaces tt*ru;l*:li;:;t;*H: a methodof constructinglinear

of GL(s;R). r

17 Conjugation 165 Proposition17.12 Let EndKm be a possiblynon-uniuersal clffird arge- bra the positiue-definite for orthogonarspace Rn, for any posiiiueinteger n. Then R R, O is a linear subspaceof AutKm : ef(*; K) and there_ of GL(m; R), fore GL(2m;R) or GL(4m;R), accorilingas K: R, C or H. Moreouerthe conjugate of any eremento/ R @Rn is the conjugatetrans- poseof the representatiue in GL(m;K) or, equiuarently,the tmniposeof its representatiuein GL(m;R), GL(2m;R) or GL(4m;R).

Proof Ler y: A*x e R@Rr, where,l e R and x e Rn. Then y- y : - : Q' x)Q"* x) A2a yQ'sis real, and is zero ifand only if y : g. Thereforey is invertible if and only if y + 0. The last statementof the proposition follows at oncefrom Proposi- tion 17.1. n The following is an immediate corollary of proposition 16.r coupred with the explicit information concerningthe criflord algebrasRqp contained in Table 15.27and its extensionby Corollary 15.26.

Theorem 17.13 Let (1a(k))be the sequenceof positiueintegers defined by j, :0for X$p+q):4p* wherej e:0,1. for q: L,2forq:2 or 3 qnd 3 q : for 4, 5, 6 or 7. Then if )x(r')diuides s there existsa k-dimensioncil linear subspaceX of GL(s; R) suchthat (i) each -x, for x € x, xr : x" x : -x2 being a non-negatiuereal sI, multipleof zeroonly if x:0, (ii) R @X is a (k + I)-dimensionallinear subspaceof GL(s; R). The sequencex is called the Radon-Hurwitzsequence (Radon (rg23) and Hurwitz (1923)).It can be provedthat thereis no rinearsubspace of GL(s; R)) of dimensiongreater than that assertedby Theoremri.r:(ii). proposition As a particularcase of 17.r2there is an eight-dimensional linear subspace of GL(8; R), since R(g) is a (non-universal)clifford algebra for R7. This remark provides a route in to the study of the algebra of Cayley numberswhich we undertakein Chapter 19.

Terminology Becausethe theory of spinors has developedpiecemeal, driven by ap_ plications in theoretical physics,an extensiverange of different types of spinor have appearedin the literature_ Dirac spinors,Majorana ,pino.r, weyl spinors - and the like in the descriptionof which both the complexifications 'natura| of the rear,complex or quaternionicspinor spacesplay 166 17 Coniugation important roles. Seefor exampleBenn and Tucker (1987)or Trautman (1t93). My personal view (as a pure mathematician)is that given the overviewof all possiblecases given by the tablesof this chapter,most of this often very confusingterminology canwith advantagebe dropped! 18 2x2 Cliffordmatrices

Let Abe a universarreal clifford argebrafor a finite-dimensionalnon- degeneratereal quadratic spaceX, with Clifford group f. Then by Proposition 15.17the real argebraA(2) of 2 x 2 matriceswith entries in .4 is a universal real crifford algebrafor the real quadratic space x o R1'1,where elements of x o R1,1are representedin A(2) by matrices of theform(x -xL x€xandp,v€R,suchmatricesbeing-----r' \ l /'),*n.r" referred to below as uectorsin A(2). Let f(2) then denote the clifford group of A(2). For many applicationsone would rike to characterisethe elementsof f(2) in terms of X and f. In the casethat X is positive- definite such a characterisationwas given by vahlen (Lgo2),andhis work was re-presented in a seriesof papersby Ahlfors in the early 19g0,s.for example(1985), (1986).The indefinitecase is somewhattrickier to handre. The characterisationwe give hereis that of Jan cnops (Lgg4),developed from earlierwork of Maks (19g9)and Filrmoreand springer(1990). For a parallel account, involving paravectors,see Elstrodt, Grunewald and Mennicke(1987). See also Waterman (1993).

The charucterisstionof Cnops We begin by characterisingconjugation and reversionon A(2\.

Proposition l8.l Let A be a uniuersarctffird argebrafor a finite-dimen- sionql non'degeneratereal quailraticspace x and let A(2) be the uniuersal Clffird algebrafor X O R1,1as in proposition 15.17.Then _il and : (;;)-:Cr- (; ;)_ (I_: )

167 Zx2Cliffordrnatrices 168 18 of '4(2)' il in A(2) and so for the whole Proof Theseholdfor vectors i' a'n*o to u' the matrix (; Thepseudo-deterrninant A of ;) of A(2) rts pseudo- of the Clifford group I(2) a[- cf.For any element det"rmirrant is equal to the Product -?) (z3)G;)-:(;)(-r

( t groupY(2) a \ a, qn elewlentof the Clffird Proposition18.2 Let \, d t Then ittl' pseudo-determinantL' of the Ctffird olg"b'otilzl' L: a[- cf-- {a'{b'

Proof For such a matrix F-"\:( c\ (a '\f L-'\(od-i d/' \b d)\+ F): \-n \b il being equal to A' either side of the equation group I of a finite- element Clifford By Corollary 16'8 any C ?f the as the spacex is representable norr-orgrrr"rut"r"at quadratic dimensionar Following Cnops of invertible elenents of X' product of a finite elementsof "";;; @ of all finite products of (1994)we introduce tf'""t"""10 with an rnonoid'being a set furnished X, whether inve*ible''lr-;;t' " @ : I u {0}' fo"it" tu'" that X is poJitive-definite associativeproduct' of @' rir, severalelementary properties In the next propositioo *"

a e @ and eachx' y' z € X ' Proposition18'3 fbr each if and only if a is inuettible' (i) a{: {a eF-,being non-zero $\ axfeX' g € I' or e:ugfar sonneoeX and (iii) eitherau[:0for altu€X (iv) 1+ xY € @, (v)xY+Yze@, -fothenxaraye@, that aua (vi) if there existsu e X such (vii) x *Yxz e@' l"

18 2x2Clffirdmatrices r69 Proof

(i) This is clear. (ii) It is sufficientto prove this when a € X. But then

axf: axe: (ax * xa)a_ x1a2)e X. (iii) This is obviouswhen a is invertible,when the secondalternative is true. Otherwisea can be written as a product uru2._ukof elements of X, where we can arrangefor all the non-invertiblefactors to come first. We prove the proposition explicitly for a product of two non-invertiblevectors, the induction being easy.So let a: uru2,where \ and u2 &re not invertible. If theseare not mutually orthogonal then u1* u2is invertible and up2 : ut(ar + u2),giving the secondalternative. Otherwise uez*uzut: 0, and then,for any : u € X, u2ltt)2 (u2uI uu2)u2,while up2u1 : (upz * u2u1)q: e, from which it follows that auf:O, the first alternative. (iv) Supposefirst that at least one of x and y, say x, is invertible. Then 1 *xy: x(x-l +y). Secondly,suppose that neitherx nor y is invertible,but that x.y I 0. Then -l (xy yx)(1+ xy) : xy + y x+ xy xy : (x * y * xyxXx*y).

Finally, supposethat neitherx nor y is invertible,but that x.! : 0, though x * y.Now X is non-degenerate,so by proposition5.4 thereexists a non-invertibleelement z of X suchthat z.y:0 blt z. x + 0, and we may choosez so that z. x : 1. Then x2 : y2 : z2 : 0, xz * z x : -2 and (x * z)(x * z - y)(x - z - y)(x - z) : (-2- (x * z)y)(2- y(x - z)) : -4(I * x y). (v) Sincexy*y z: (xy-tyx)+y(z-x) it followsat oncefrom (iv) thatxylyze@. (vi) By (iii) a: ug,where u is a vectorand g is invertible.Then

x a * a ! : (xu * ug y g-\c : (xu I uy,)g, wherey' : g y g-t e X, by (v). (vii) supposefirst that x is invertible. It then follows that x * v x z : x(1 * x-l y x z), whichis in f by (iv). Next, supposethat x is not invertibleand that x . z :0. Then x I y x z : (1 - y z)x, whichis in f by (iv).

I lii 170 18 2x2Clffirdmatrices Finally, supposethat x is not invertible and that x'z f 0' Choose k e R so that kx * z is inVertible.Then

(x * y x z)(kx + z) : xz * y(k(x z + z x) + z2)x,

which is in f by (v). since kx * z is invertible it follows that x*lxzalsoisinf. rr

We shallrequire Lemma 18.4during the proof of Theorem18'5'

Lemma 18.4Let A be a uniuersalreal Clffird algebrafor a finite-dirnen- sionqlreal quadraticspace X, not necesserilynon-degenerste, and let u, D, w,yeX.Then uuywl)-uYwuueX.

Proof Choosean orthonormal basisfor X such Ihat u, w andy are in the subspacespanned by es,e1 and e2. Then the even elemento w y o must commutes be of the form a* D,;.s bi1eie1,where a and bi1€ R. Now ek - wilh eiel tf i, j,k are iistinct, and otherwiseepeiei leieiep e X' The assertionfollows. !

Theorem 18.5 Let A be a unioersalclffird algebrafot a finite-dimensionalnon-degenerate real quadtaticspace X, with CIffird groupI and Ctffird rnonoid@, tet A(2) be the uniuetsalCtffird algebrafor X @R1'1 constructeilas in Proposition15.17, and let G be the set of all matrices ( ? ", 1 of AQI suchthat \b d/ (a) a, b, c, d e @, (b) att-,c il, fc, fd e X, (c) A : a [- cb-e R''

Then G is the CIffird group f(2) of A(2).

entry of a rrratrix ^ G can be Proof weprove first that each (f, i) written in the form o g, where u e X and g e l, and that there exist p, Q,t, s € x such that in each case one of the following alternatives holds:

a:bp or b:aP', A:eC Or C:qAi c: dr of c: dt; b: sd or d: sb. 18 2x2CIffirdmatrices t71

Supposefirst that eithet a or b, say a, is invertible. Then b : u a with u : aT/(ail €.X, and b : &p, wherep : e-r1)a. Otherwiseneither a nor b is invertible, but neither is zero, with

(atr- cb-)b : a({b) and (df- b {)a: b(-{a), neitherof thesebeing zero.Now a(d-b) : a(6-d): (ali-)d. So ati:* 0. It follows that a(Td)tr:af(dtr-b{) +0, from which it follows from (iii) of Proposition18.3 that a: ng for w e X and g e f, a similar result clearly holding for b. The caseof c and d is similar. To provethe statementfor a with c or for b with d one usesthe alternativeexpression for the pseudo-determinant A givenin Proposition18.2. Next we provethat, for any x e X, ax[+ cxFe X. Supposefirstly that all four entries in the matrix are non-invertible and, necessarily, non-zero. Then in particular there exist u e X and g € | such that &: Dg and explicitcomputation shows that

(; i):(:n -:i,)(s:), wherew : L-rg(Fb)g-l and u: c{ both are in X. Then

a x F+ c x Li": -(u y w u)u I u(uy w u), wherey : gx{, and this is in X, by Lemma 18.4. Alternatively, at least one of the entries of the matrix, say a, is invertible.Then b : &p: apa-t e, c: ra andd -- (1+apa-r r)a,where p' : ap a-r G X, r e X and A,: ).a. Then

ax F* c xF: yQ,+ r p')* r y p' : )"+ (y r * r y)p', wherey : &xa-r, and this is in X. Simitarlyit may be verifiedthat, for anyx e X, ax{* cxf e R and bxF* dxTeR. Now considerthe product ' '). (::)f -"/\b-"\({ \b d/\p o-)' wherex e X and p, v € R. Now, by property (b) of G either a and d are in @0and b and c are in @1,or vice versa.So up to sign this product is r72 18 2x2Clffirdmatrices equal to (;;) (; -; ) ( -r-?) which is equal to

( axtr+ pctr- vaf* cxf -ax{- pc{-lva{- cxrf'\ -bx{- ' \ bxd-+ pdtr- vbF-f dxr pd{* vbf- dxri- ) and this is the matrix of a vector in A(2). From this it follows at once thal G c f(2). Finally, we have to prove that f(2) c G. For this it is enoughto prove that the product (::)f ' v) \bd)\p-*) is in G, where the first matrix is in G and the secondis an invertible vector in A(2), the condition for this being that -xz - vpr I 0. None of the verificationscauses any trouble, though at one point one requires againto use the fact that ax[* cx6:is in X, for any x e X. tr

An exampleof an elementof f(2) none of whoseentries is invertible has beengiven by Maks (1989).Consider generators es and er of R1,1 with ef,: -1 and e?: l, and take the standardmodel of R42in R1.1(2). -l Then the rotation of R2,2that sends,o to ( ? ), unO that vector \r v/ to -es,and similarlysends r, ,o andthat vector to (? l), inducedby the matrix (; ?)(; ) : ( of f(2), none of whoseentries is invertible. Cnops(L994) has given examples to show"!;':,'that none of the requirements (a), (b), (c) in the characterisationof f(2) can be derivedfrom the other two. His examplesare

('t*,i,,),(l'T')and ([ ",:,",), which violate (a), (b) and (c), respectively,but not the other two. None is in f(2), sinceeach is the sum of an even part and an odd part, both different from zero. 18 2x2Clffirilmqtrices 173 Theorem 18.6 Let A : Ro,r, with X : f,p,Q,Clffird monoid @, and Clifford group l. Then the Clffird group l(p, S + l)/is representableas -D^ the subsetof A(2) consistingof matricesof theform | ! |, ,h"re \o a/ (a) a and b are in @, with either a or b in l, (b) af e X and (c) aa +6li-: aa-*bb- € R..

For matricesin fo(p,q + l), 6 : a € @0andi : -b e @r. Moreouer,in the casethat a e fo, there existsp e X suchthat b: ap -" and,for any sufficientlysmall |eR, ( :, ) € f(2)0. \,iD a /

Proof Everythingfollows directly from Theorem 1g.5. tr

Groups of motions of obvious practical importanceis the group of rigid motionsof R3, that is the group of rotations of R3 extendedby the group of translationsof R3. More generallywe may considerthe group of rigid motions of any finite-dimensionalquadratic spacex, this being the group of rotations of X extendedby the group of translationsof x. such rigid motions are readily representablewith the aid of the Clifford group f : f(X).

Theorem18.7 Let A:Ro',with X: f,A4and Clffirit groupl, and let ( : i\ * A(2) representan elementof the Ctffird group lo(p,q r), \r 4/ " + with a e 10. Then the map X --+X; x ,--t axq-r * ba-l is a rigid motion of x and any rigid motion of x may be so represented,the representation beingunique up to non-zeroreal multiplesof a and b.

Strictly speakingwhat is involved hereis the subgroupof A(2) consist- ^ / a b \ ing of all matricesof tlheform aer.o andb:ap,wherc (, ; J,with p e x. In the particular casethat x: R3 spin(4) is most frequently 2H. identified with the group s3 x s3 c An alternative to 2H c;nsists ' \ of the matricesof H(2) of the form ( : with the injective algebra \ o o ) 2H_,H(2) map sending , , it tr,en (f :) ( +[I:i] til;:l) beingthe casethat if lql : lrl and r + -S then (q -r)(q+r)-r is a pure matrices 18 2/2 cliffofi I14

of theform t Hfr#H#dd#{*ff*-11:"tt"esfi'ili,*:di."ryTt: \ Tfi *"O'''"o*l*tine Thesubalgeb'"

important we discussthis ;,i'H?:,T:Tt;[ii,#t;:'yrm*l;lri'rj::il"?;;the mauix (; * i) represented Chapter24' "o'o' ut't'" of specialcase fufiher "oO

pfffianchorts ,-_^onhisflI1-.s - is

iiff#*r"u:'xlff$"$'srn - I (1 * sXl s)-t iJ":ff;Hl --+SO(n);s ''-' EndlR') quesuon t* -" *" "" "#*i* injective,*i* *"i'i' isi'r$I,1:'li;xlii'li *::::: )',o,o,,r: (,r ur,;-,1ji1

:#HtT,::!,{;;:rrij:)!iu'if!'*'""'k*';"be.'henLv'wv' - t Wgnet f;'; € R' and Litisiieiei -tt r .^-.-c whefe o' ir i i e ie i s:::*::'ff T:''{ll:';li pr o or Letr : : :t,' jJ'q'= i-"'*"**' Hi Ji;";'' il:'li; "'t. ot,-ll': o*'+e*" i

f sgnn fltn1r01,,1**r1 neP i€m

if n:2m is even,P beingthe set of all permutationsof 2m for which

(i) for any h, k e m, h < k - n(2h) < n(2k), (ii) for any k e m, n(2k) < n(2k + t).

For example,if n:4, pf s: s01s23-s02sr: *ss3s12.By convention, pfs : 1 if n : 0, in which cases : 01: 0. For any t - n, let s1 denote the matrix (s;7 : i, j e I\. Then sr e End-(R'), where k : #L The completepfffian of s, pf s, is, by definition"the element !nf ',", /cll

of the Clifford algebraRs,,. Sincepf s7: 0 for ffI odd, pf s a R&,. In fact Pf s e fo(n), as we now state formally.

Theoreml8-9 Let s e End-(Rn). Thenpf(s) e yl(n) and is the uniqueelement of fo(r), with real part L, thqt induces the rotation (1+s)(1-s)-1.

Proof A start to the proof has alreadybeen provided by proposition 1g.g. It then follows by applicationof part of rheorem 1g.6that the coefficient of e7 is a polynomial in the terms of the matrix sr, obtained from s by deletingall the rows and columns of s exceptfor those with i, j e 1. one proceedsby induction, and by verifying at eachstage that the coefficient of highestdegree in either erer or elesq is zero. Moreover, by the final part of rheorem 18.6,this polynomial contains exactly one term from eachrow and eachcolumn of s1,so that the terms of the polynomial are, up to real multiples,the terms of pf s7. Finally, considerany one such term,

).sg1sns45egere2e3e4es, fOr example.

This term will be equal to the correspondingterm in pf s, where s, € End-(Rn) is definedby

s6r: sor: -s'ro, s\l: szt- -s!zzand s!5: s45: -s/54, 176 18 2x2CIffirdmatrices all the other termsbeing zero. However,

Pf s' : (1 * soreoer)(lI sye2q)(l * s+sea,es) : I I soteoet* szsezelI s45e4e5+... + s06xs4se0ep2e3e4e5,

sinceeach of the factors is in f0(6), the real part is 1 and the coefficients of the terms eiei ata correct. So, in this case,I : 1 in accordancewith the statementof the theorem.The other terms are handled analosouslv.n The map End-(R') --+Spin(n); s -r Pf s/lPf s)l

will be called the Pfffian, or Lipschitz,chart at I on Spin(n). Lipschltz (1880),(1884) shares with Clifford the discoveryof the Cliflord algebras. 'Correspondence See,for example, from an ultramundanecorrespondent' (1959). For further details of the early history of Clifford algebrassee Van der Waerden(1985). and a forthcoming book by Pertti Lounesto. For the alternative presentationof the Lipschitz chart that involves the exterior exponential of the biuector Dxi siieiei, see, for example, Lounesto(1987) and Ahlfors and Lounesto(1989). The following property of the Pfaffianis sometimesused to characterise it. Seefor exampleArtin (1957).

Theorem18.10 For qny s e End-(R"), (pf s)2:6s1r.

Proof Let s e End-(Rn), Then, for any / e R(n), frsl e End-(Rn). Now, for any such s and f, Pf(t"st): dettPf s. To show this it is enough, by Proposition 1.8, to verify that pfs is invariant under an elementarycolumn operationcoupled with a matching elementaryrow operation. The matrix s inducesa skew correlation on Rn with product

R'x Rn--+R;(x, x')*> yxsyt.

LeI 2m be the rank of this correlation. Then, by a slight generalisation of Theorem 6.7 to include the degeneratecase, there existsu e GL(n;R) such that

(u" su)21r,21ra1: L : -(u" su)s,21ra1 r

Exercises l7j for all k € m, and (u" su);;: 0 otherwise.It follows from this that

Pf(u"su): fl(t * e21,eq,a1). k€m There are two cases.lf 2m < n, pfQf su): 0, implying that pf s : 0, sincedetu 10, while det(u"su): 0, implying that dets : 0. lf 2m: n. pf(u"su): 1 and det(u"su):0, implying that

(det a)2(pf s)2 : | : (det u)2det s. In either case,(pfs)2 : dets. n

Exercises 18.1 Let p, be the rotation of Ra induced by an elementg of f0(R4) with real part equal to 1.. Prove that p, is expressibleas the composite of two hyperplane reflections (cf. Theorem 5.15) if and only if g is of the form

1 * soreoer* ss2ese2*ss3e6e3 * sne&z * snetezI s4e2e3 where (e6,ey €2,e) is the standardbasis for Ra. Deducethat 1* soreoer* sozeoez+ s$elq * spepzl sselq* szzezet is the product in the Clifford algebraR6.a of two elementsof Ra if and only if

Pf s : sgls3- s02s13* s63s12 : S. 18.2 Provethat an invertible elementof the Clifford algebraCa 1 *soreoer* sozeoez*ss3ese3 *snetez* spelqlsztezez is the product of two elementsof C4 if and only if pf s : 0. 18.3 Provethat an element

soreoer+ s02e0e2-f soreoe: * snegz * spe14 I szlezet of [21X41,where K : R or C, is the product of two elementsof K4 if and only if pf s : 0. Deduce that the image constructed in Exercise 15.7 of the Grassmanniang2(Ra) in the projective space9(f12(Ra)) is the projectivequadric with equation

S01Szl- S02Sr:* sO:srZ- 0.

18.4 Define an analogueof the Pfaffian chart for Spin(p,q),pq * 0. 19 The Cayleyalgebra

In this chapter we take a brief look at a non-associativealgebra over R that neverthelessshares many of the most useful properties of R, C and H. Though it is rather esoteric,it often makes its presencefelt in 'responsible' classificationtheorems and can ultimately be held for a rich variety of exceptionalcases. Most of these lie beyond our scope,but the existenceof the algebraand its main propertiesare readily deducible from our work on Clifford algebrasin previouschapters.

Resl diaision algebras A diuisionalgebra over R or real diuision algebra is, by definition, a --+ finite-dimensionalreal linear spacex with a bilinear product xz x; (a, b) ++ ab suchthat, for all a, b € X, the product ab : 0 if and only if a : 0 or b : 0, or, equivalently,if and only if the linear maps

X --+X; xe xfi srtf,x,t--+ ax

are iniective when a and b ate nofl-zeto,and thereforebijective. we are akeadyfamiliar with three associativereal division algebras, namely R itself, C, the field of complex numbers, representableas a two-dimensionalsubalgebra of R(2), and H, the non-commutativefield of quaternions,fepresentable as a four-dimensionalsubalgebra of R(4). Each has a unit elementand for eachthere is an anti-involution, namely conjugation,which may be made to correspondto transpositionin the matrix algebrarepresentation, such that the map of the algebrato R,

N; ar-+N(a):4q,

is a real-valuedpositive-definite quadratic form that respectsthe algebra

r78 r 19 The Cayleyalgebra r79 product, that is, is such that, for all a, b in the algebra, N(ab): N(c)N(b).

A division algebrax furnishedwith a positive-definitequadratic form N : X--+R suchthat, for all a, b e X, N(ab): N(a)N(b) is saidto be a normeddiuision algebra.

Alternatiue dioision algebras An algebraX suchthat, for all a, b e X, a(ab) : a2b and (ab)b : ab2 is said to be an alternatiuealgebra. For example,any associativealgebra is an alternativealgebra.

Proposition l9.l Let X be an alternatiuealgebra. Then, for all a, b e X, (ab)a: a(ba).

Proof For all a, b e X.

(a * b)2a : (a + b)((a+ b)a) - (a2+ ab * b a + b2)a: (q + b)(a2+ b a) * a2a* (ab)a* (bq)a I b2q: aa2+ a(ba) * ba2 +b(b a) + (ab)a: a(ba). n

Proposition 19.2 Let x be an alternatiuediuision algebra. Then x has a unit elementqnd eachnon-zero a e X has sn inuerse.

Proof rf x has a singleelement there is nothing to be proved.so suppose that it has more than one element.Then there is an elementa e x, with a f 0. Let' e be the uniqueelement such that e q : a. This exists,since the map x 3 xa is bijective.Then e2a: e(ea): ec. So e2: e. Therefore, forallxe X,e(ex): e2x:ex and(xe)e:xe2:xe.Soex:xand xe: x. That is, e is a unit element,necessarily unique. Again let af 0 and let b be suchthat ub: e. Thena(ba): (ab)a: ea: ee. So ba : e. That is, b is inverseto a. n

The Cayley algebra Thereare many non-associativedivision algebrasover R. such an algebra may fail even to be power-associatiue,that is, it may contain an element 19 The CaYleYalgebra 180 exampleis' given o(o')' exotic for example'@\a * *less division a such that, o1" il non-associative in Exercise; 19.1. .,,,,*'[ji;ffiH"ffi;;'ln! '#;:X".y;;;11ii'**""llJl:f ;ir"*,", or al CavleYalgebra ;,^ ;l; :#;:.d;ot *,*l-il'jfrbv any algebraof octuoes"{ffi:tiT;[I* subalgebra generated ltfi;;;;tment' the co-mmutativitythere ri ana so is associative' ; ffi;'';* as two of its elements P l:";;*hn'ing the sarneproperties and therei' o to"iojitioi'"iii-i""or*tion for R,:.$*i* thlt thematrix ton:ugution dependson thefact of the algebra algebrafor **i*-j:,S*[rn*#t'iilllm;r{**;:*:lfire existence :1*t".?'::: i#]ffi;;sal) clifford ;'of the;;'"i"u'"1"1i'l$ Clifford alge -p"t' iffi';Lilii**m:iI#J {t"Li!?This eight- together than '"t"' rt invertible' -J*Jll'"T11ii#;;iiiii*n"i"t'l; elements,other be denoted bvY' proposition.r?.r1,Jrr:iij"rriirir:*:':;,:#:,?i,^i::,T!)i'!::::, (c, anv Rs x R8 --+R6; : 0 ot b : 0' Moreouet' o-:o i' oia *ty if a that,for alla, b t il'T non-zeroelementJt'i{'"'i'*oa't't'iutritel'*'ntfotsuchaproduct inuerseof the isomotphism by choosingt' to uu-'i) Y--R8; Ye Ye' eleme": rhedivision algebra with unit i:':,l5lrtl Tli""#"'l 3'i

o:::5? bv o (for-octonions)' ;":*"i"*":#'H:':::{x'?l:l*'Tq"::i";il''1".1r'"r15'i-'or the J;;6;**l speaksimpl v unimp-ortant,it is advantageous ,rnougtrthe choice l; esslntialry we eo' "; ii r'or definiteness .select elernenr *r* I l* . denotebv to selecran "i basisf* Rt' we then element d;'T;J"rd * ;'l*;,,",,),'the zetoth r:=::iffi"; rJT,T',il':ffffl.'positive-deflnite:,:,:,'' v whictr iit standard "o' ?ttoli?Ifg.it1y assignedto n" Here we have rmP form ,t uctur", with quadratic -6'a:da' , N 'R8--+R;ar-N(a) induced by conjugation' an orthogonal structure' The spaceY also has 19 The Cayleyalgebra 181 namely transposition,on the clifford algebraR(g), with quadratic form Y --+R{e}; y + y"y. The cayley algebra o inherits both, the one directly and the other via the isomorphism u. As the next proposition shows,the choiceof e as an elementof length 1 guaranteesthat thesetwo structurescoincide.

Proposition19.4 For all a e R8, (u1a;;"u1a): N(aXs1).

Proof For all a e R8,a: u(a)e.So N(a): a"a: e"(o(a))"u(a)e.Since y y"y e R(81)for all y e and sincee"e: 1, it followsthat u(a)'o(a):N(aX81). tr Conjugation on R(8) inducesa linear involution O--+O; aF-+a: (u(a))'e which we shall call conjugationon o. This involution inducesa direct sumdecomposition O: R{e} O O, in whichO, : {b e O :6: _b}. The following proposition lists someimportant propertiesboth of the quadratic form and of conjugation on o. The product on R(g) and the product on o will both be denotedby juxtaposition, as will be the action of R(8) on o. It is important to rememberthroughout the discussion that, though the product on R(g) is associative,the product on O need not be.

Proposition19.5 For all a, b e O, N(ab): N(a)N(b), implyingthat O is a normeddiuision algebra, (;a'b)e: \(ab +6 Q, implyingthat O' : 1R{e})a, (N(a))e:da: ad, : and i6 6a, implying that, conjugationis an algebra anti_inuolution. Moreouer, . for all a, b, c e O, a. (bc) : 6 @a) : E . (ab).

Proof For all a, b e O,

N (ab) : N(u(a)b): b"o(a)" o(ct)b : b"(N (a)(8t))a : N(c)N(b). : Alsod b16 a a(be) *6@ e) : o(a)" o (b)e * o(b) o(a)e : 2(a.b)e,implying that, if a e R{e} and if b e O,, then2(a. b)e: ab _ bq : 0,sincee, and thereforc any real multiple of e, commutes with anv element of 19 The CaYleYalgebra l8Z N(o) : da and' since sinceN(a).: a'a' that O. It implies,secondly' : thatad: N(a)' ,Ort y o(a)"o(a), (b(hc)r\ :6:T '. {ck a) and thisth' we *'at,io i- n,i' ' Nextwe prou" "tt ,S 2'a''c' is equal $ (6d) Firstly do by proving that each c' :do@)'c : @a)' a'(b c):dv(b)c 1o@)a)"c: On the other hand' when :6 ' (ca) when a e R{e}' Secondly,6a)' ' 'e :0 and a e O',de: a :: b' (ca) - (6a\' c @d.E+(6a)'c'o by the argumentused above' 1ei)+ a' (bc), a.@6+bc) a'2(z'b)e : 0. a,b andc cyclically for al a e_o,a.(bc):a.k!).permuting the equation so, Finally*" *"i': e in we alsoobtain a t'olZ'z'Oal' e.tub\ :d'(bc).Then e(ab)* aTd: a(bd+TEa' n - thataE --5a' Thatis, ab * ab ab +6:a'so productof rhe saidto be.lle scalattriple The realnumber a'(bc\is the scalar that order' This generalises Cayleynumbers o'a )ii'"'in afterProposition 8'17' o'rJi*""" H, defined Thealgebraott;;;i;*iio*t"'ative'sincedimo/>1'Norisitfollowing' Neverthelesswe havethe associative,u' *t 'f'utt *e' algebraO is alternatiue' Proposition19'6 The Cayley (za)b.So u@)u(a)b:o(a)"o(a)b: proof For anya,b eo,d(ab): a(ab) a*a€ R{e}' =ri:x;;:'&::,''since that are equal it follows likewise By proving that their conjugates (ab)b: ab'' tr r

19 The Cayleyalgebra 183

Throughout the remainder of this chapter we shall identify R with R{e}. In particular, we shall write 1 in place of e for the unit elementin O, being careful to distinguishthe numeral 1 from the letter l.

Hamilton triangles It has been remarked that two elementsa, b e O, are orthogonal as elementsof R8 if and only if they anti-commute. An orthonormal orderedsubset (i, j, k) of O', with i : jk, j : ki and k : ij, therefore spans,with 1, a subalgebraof O isomorphicwith the quaternionalgebra H. Such a subsetwill be said to be a Hamilton triangle in O and will be denotedby the diagram A j4t

in which eachvertex is the product of the other two in the order indicated bv the arrows.

Proposition19.7 Let a and b be mutually orthogonalelements of Ot and let c : ab. Thenc e OI and is orthogonalboth to a and to b.

Proof First

a.b:0-ab*ba:0 - ?: fr:Ea: (b)(_a): _c _ceO,. : Also e..c l1a{ab)+(ffi)a) : L(N(0)U+6 tt1a11,by Proposition19.6, : 0, sinceb *6 :0. Similarly,b.c:0.

Corollary 19.8Let (i, j) be qn orthonormalordered pair of elementsof O, and let k : ij. Then(i,j, k) is a Hamiltontriangle in Ot.

From this follows the assertionmade earlier that the subalgebragen- eratedby any two elementof O is isomorphic to R, C or H and so is, in particular, associative. 19 The CaYleYalgebra 184 CaYleYtriungles or- m.a1be extendedto a useful any Hamilton triangle in Finally, !r' a Cayley ffiangle ol o'''*t"i"'in by defining 'rn thonormal basis for that c is also Lipl"'1o'b'c) n o' such to be an orthonormaio'd"'"a orthogonalto ab' in Ot' Then, c) be a Cayleytriangle Proposition 19.9Let \a, b' c) (ab)c:0, exhibitinr (i) a(b * 'ri*""ts:,':.^:::t':fa' b' c formi],T al[ir:, cavtev *ianste fJ ;1b?,:t"o:':*;,;;;;'i*-'ii ate listed' '::';::;,"'t;;,ry,"t''i,itn'whateuerotder theY - -' : v, unp.LyLtltLtLwL 1a,\w) b,e, c.). $fi ab. b c ! " ?i!?::::*' fiiangle. that(ab'bc' ac ) ii a Hamilton il"i(iv) irrltoc): ac,implving

Proof triangle' (i) Since(a,b, c) is a Cayley : 0' : c cb : (ab)ct c(ab) ab * ba : a c I c a b *

-a(cb)- c(qb) a(bc) * (ab)c : : (a2+ c2)b- (a* c)(ab* cb)

-0. :0 and (i) it followsby conjugatiolfhat@E\d+e(6a1 (ii) From (ab)c: *c(ab) : 0' it that(b cia i therefore "@u)-:0 1i""" thata' (bc) :0' thara(b tAti" : o' implying follows "lI

(bcxb 4) 2ab.b c : @a)(bc) + - : (ba)z + (bc)2 : (b(a c))z : -b2& -b2c2+b2(a- c)z : -bz(ac+ca) : 2b2a'c :0.

(rOAppry (i) to thecuvl:Y.t'11nt|-\"^,':^o-1) -1' -'a(b(b : ac' sinceb2 : Then (ab)(b c) : c)) 19 The Cayleyalgebra 185

We can reformulatethis as follows ('1' being the letter'1').

Proposition19.10 Let (i, j,l) be a Cayleytriangle in O' and let k: ij. Then{i,j, k, l, il, jl, kl} is an orthonormalbasis for O', and if theseseuen elementsare arrangeil in a regular heptagonas shown then each of the

seuentriangles obtained by rotating the triangle (i,j,k) throughan integral multiple of 2n/7 is a Hamilton triangle,that is, eachuertex is the product of the other two uerticesin the appropriateorder.

This heptagon is essentiallythe multiplication table for the Cayley algebraO. From this it is easyto deducethat there cannot be any division algebra over R of dimension greater than 8 such that the subalgebragenerated by any three elementsis isomorphic to R, C, H or O. Such an algebra A, if it exists,has a conjugation anti-involution, inducing a direct sum decompositionR @A' of A in which ,4' consistsof all the elementsof ,4 which equal the negative of their conjugate. Further details are in Exercise19.3. The following proposition then settlesthe matter.

Proposition l9.ll Let (i, j,l) be any Cayley triangle in A' , let k : ii and let m be an elementorthogonal to eachof the seuenelements i, j, k, l, il, j I and kl of the Cayleyheptagon. Then m: O.

Proof We remark first that parts (i) and (ii) of Proposition 19.9hold for anya, b, c € ,4.'such that a'b : &'c: b'c: ab'c: 0. Usingthis several 're-bracketing times,we find, on making a circuit of the pentagon',that

(ij)(l m) : -((i j)l)m : (ifi l))m : -i(CI1)m) : iCIfim)) : -(i j)(1m).

So (ii)(lm) : 0. But ii f O:so lm : 0, and therefore,since I f 0, m : 0. n 186 19 The CaYleYalgebta Further results There are various strongerresults, for example

(i) Frobenius'theorem(1378) that any associativedivision algebra over R is isomorphic to R, C or H, (ii) Hurwitz' theorem (1898) that any normed division algebra over R, with unit element,is isomorphic to R, C, H or O, (iii) the theoremof Skornyakov(1950) and Bruck, Kleinfeld (1951) that any alternative division algebta over R is isomorphic to R, C, H or O; and (iv) the theoremof Kervaire(1958), Milnor, Bott (1958),and Adams (1958),that any divisionalgebra over R has dimensionI,2' 4 ot 8.

The first two of theseare little more difficult to prove than what we have proved here and can be left as exercises.The starting point in the proof of (i) is the remark that any elementof an associativen-dimensional division algebramust be a root of a polynomial over R of degreeat most n and, therefore,by the fundamental theorem of algebra,must be the solution of a quadratic equation. From this it is not difficult to define the conjugation map and to prove its linearity. Result (iii) is harder to prove.The discussionculminates in the following.

Theorem 19.12Any real non-qssociatiueq.lternatiue diuision algebra is a Cayleyalgebra.

Proof Let Abe a real non-associativealternative division algebra,and, for any x, y, z e A, let - lx, Yf : xY - Yx and fx, Y,zl : (xY)z x(Yz)' It can be shown that if x and y are such Ihat u : [x y] f 0, then there exists z such that u : lx, !, zl + 0. It can then be shown that uu * u u : 0 and therefore,by the previous remark, that there exists I such that w - luutl + 0. One can now verify Ihat u2,u2 and w2 ate negativereal numbersand that i:ulJ4, j:u/J-rP andl:w/J-wz

form a Cayley .triangle. Then .4 contains a Cayley algebra as a subalgebra.It follows, essentiallyby Proposition 19.11,Ihat A coincideswith the Cayleyalgebra. 19 The Cayleyalgebra LB7 The details are devious and technical, and the reader is referred to Kleinfeld (1963)for a full account. n

Finally, (iv) is very hard indeed. Its proof uses the full apparatus of algebraictopology. see Adams (1958),(1960), Kervaire (195g)and Milnor and Bott (1958).

The Cayleyprojectiae line and plane Most of the standardresults of linear algebrado not generaliseover the cayley algebra o, for the very definition of a linear spaceinvolves the associativityof the field of scalars.Nevertheless we can rcgard,the map --O') O'x O (@i:i en),y)* Qiy :i en)) as a quasi-linearstructure for the additive group On. 'projective It is also possibleto define a line' and a 'projective plane' overO. The Cayleyprojectiue line opl is constructedby fitting together two copiesof o in the manner of Example 14.2 for the projectiveline Kp1, : for K R, C or H. Any point is representedeither by [1, y] or by [x, 1], with fl, yl : [x, 1] if and only if y - x-r, the squarebrackets here having their projective-geometryconnotation. There is evena 'Hopf map, h : O2\(0,0)--+Opl definedby h(yo,yr): [yoyrr,l),whenever y1 10, and by h(yo,y): fl, lr y;1], wheneverlo * 0. Sinceany two elements of o (for example,)ts and y1) generatean associativesubalgebra, it is true that !o)t;r: Oryo1)-1, and so the two definitionsagree, whenever /s and y1 are both non-zero. The Cayley projectiueplane OP2 is similarly constructed by fitting together three copiesof 02. Any point is representedin at least one of the forms Ll, yo, zof,fx1, I, z1]or fx2,y2, Il. The obvious identifications are compatible, though this requires careful checking becauseof the generallack of associativity.What we require is that the equations : xt !i1, zt : zo!01 and x2: x4ll, lz: ztl be compatiblewith the equations

X2: ziI, yZ: yOzil. But all is well, since

xrztl - yil(zoyir)-r : tir and z;r : (zoyi\-r - yozil, 188 t because ffffffi:: r^":*:;::::",:::" gcnerated by any two elementsis ^ ,tlsefulanarogues over o . 2 do not""t.,. ""-:- spac il" ,."rtj-9-ofprojectiveo"risr"reiJ' ;;ii;ff i,riTH::Xffi:nT*

rg.1 Exercises Let x bea fou '::l-t'al reallinei o"not"J-{,'i,l-f unou, i.i prescribin!,#'""J il:;rff 1H:'"XT;fH,' i'-_it=k2=-1, -' J'\jk*kj-1. rAJ-Kr+ik:ij+ji:o and

jk:ai, ki:pi, ij:yh rearn V;::,r^{ i,#; }:--:.o umbers, 19.2 ffr;'::;'rffi q: 19.3 r,etXbeadivisionfrii,'ru r:ry ol"' rhereexisr *;;;:,,1'"*ll'1 R suchr

rch iJ,,I'i r'{.,'ndln'i;i*;nTi;, *{i;1that *2 + 6,fp :0, x'is a linear ,r#lff#; subspaceofx, that x- R @x,and R@f,'+R @X,;A*x,r_-+ 11_x,, where,t e R. , 1s.4 r"t ananri-invotution i;"=J;J,:,f1*-a, alternative ofx. z e division algebra, A, bt and, for dfl! x, !, h, yJ : x! _ y x and [x, y, z] _ (xy)z prove _ x(y z). that in ttre sign, any two reters ;;ft*"nging in [x, y,e] changes zl- xu' 'l - [*' z1!: 31x' 21' Hence,0"* ,ll' v' tll ra* oi", , ,,:'t,:td, f-^,commuhrive,r l- alsois associatjve. tx * h x f y,"t = u:lr, !*2,y121.1

---- Exercises 189

I9.5 Let A be a real alternativedivision algebra,and, for any w, x, y, z e A,let

fw, x, y, zl : lw x, !, zl - xlw, y, zf - fx, y, z]w. Prove that the interchange of any two letters in [w, x, !, zf changesthe sign. (For all w, x, !, z e A, wfx, l, zl - lw x, !, zl * fw, x y, zf - lw, x, y zf I fw, x, yfz : 0.) 19.6 Let A be a real alternativedivision algebra,le| x, y e A and let -uu. u : fx, !1, u : fx, y, zf.Prove that fu,x, yf : u r,t: I9.7 Prove that the real linear involution O --+O; a r-+ Z, sending j,1, jl to -j, -1, -jl, respectively,and leaving 1, i, k, il and kl fixed, is an algebra anti-involution of O. 19.8 Verifythat themap f :H2-+O; xr--+xo*lxr is a rightH-linear isomorphismand compute f-r(F(*)B(y)), for any x, y e }d2. (Cf. Exercise19.7.) Let Q: {(r, y) e (H\2 : iolo * iryt: 1}. Provethat for any(a, b) e o. x}J, (P-1@),B-r1a-r(l + lb))) e Q andthat the map

O* x H --+Q; (a,b) => (P-r@\ P-r @-r(I + I b))) is bijective.(Cf. Exercise9.4.) y --+O; x" xo*j is a I9.9 Verifythat the map i C4 ltJ.!*r*jkx: right C-linearisomorphism and computey-'(y(x)y(y)), for any x,yeCa. Let Q: {(r, y) e (C4)2: D;.1 xilr : 1}. Provethat, for any (a, (b, c, d)) e O. x C3,

(y-t@), y-r @.-10 + ib + lc + ik d)))e Q and that the map O- x C3--+Q;

(a, (b, c, d\) v-+(y-r @), y-' (o-t (l + j b + I c + j k d))) is bijective. 19.10 Show that the fibres of the restriction of the Hopf map

02 --+OP1 ; (yo,y) - [yo,yr]

to the sphereS15 : {(yo,yr) e 02 : foyo+ fth : 1} are 7-spheres,any two of which link. (Cf. Exercisesl4.I and 14.2.) 190 19 The CaYleYalgebta t9.1.1 Let a, b,c € O. Provethat : (ab)(ca)' a(b(ac)) : ((ab)a)c,((ab)c)b : a(b(cb)),a(b c)a Theseare known as theMoufang identities(1935) for an alternative product. They are most easily proved,for O' as exercises on the rebracketingPentagon. 20 Topologicalspaces

In this chapter we shall simply summarisethose topological concepts that will be required in the sequel.For most of the proofs the readeris referredto this book'sparent, porteous (1931).

Topologicalspsces cohesion may be given to any set X by singling out a subset{ of the set SubX of subsetsofX suchthat (i)0e{andXeT; (ii) for all A, B e f, AnB e f; (iii) for all I c. f , the union of all the elementsof g is in {. The set 7 issud, to be a topology for X, and the elementsof { are calledthe opensets of the topology.Atopological space consists of a set X and a topology { for X. A set X may have many topologies.For example,for any set X, both Subx and {0, x} aretopologies for x. However,most of the setsthat we shall be concernedwith will be subsetsof a finite-dimensionalreal linear space,and for any such set there is a natural choicefor its topology. we start by remarkingthat anynorm on a finite-dimensionalreal linear spacedetermines a topology for the space.In chapter 4 we definedthe norm lxl of an elementx of a positive-definitereal quadratic spaceX to be Jlxv-11,and in Proposition 5.31 we listed some of the properties of the map X--+R; x r-r lxl. Theseincluded the following:

(i) forall xe X,lxl> 0,with lxl:0if andonlyif x:0:

191 24 Triality

At the beginning of chapter 19 we remarked that the cayley division algebra 'responsibre' o can ultimatery be herd for a rici variety of exceptional phenomena. Among these is the triality which we study in this - chapter an automorphism of order 3 of Spin(g) that does not project to an automorphism of so(s). As a byproduct we make acquaintance with the fourteen-dimensionarLie group Gz, the group of automorphismsof the Cayley algebraO. Triality has somethingof interestto say about the projectivequadrics ,rf {cs) and Jr{Fi4A). This quadric triality seemsfirst to havebeen noted by study (1903), 'triality' (1913),though the word is due to Elie cartan (1925), who placed the phenomenonin its proper Lie group context.

Transitiaeuctions on spheres pot the group To Sp;n(8)in context we begin by looking at a]l thegroups Spin(n),with n < 10. By virtue of earlier work we know that

Spin(I)=o(1) c R8.r=R, Spin(2)=U(1) c R8z=C c R(2), Spin(3)=sp(l) c R83=H c R(4), Spin(4)=Sp(l)xSp(l) c R8r=2H c ,R(+), Spin(S)=Sp(2) c R85=H(2) c R(8), Spin(6)=SU(4)cU(4) c R86=C(4) c R(8), Spin(l)co(8) c R8.z+R(8), Spin(8)cO(8) x O(S) c R8.e=2n(s), Spin(9)c0(16) c R8.r=R(16), Spin(lo)cU(16) c R8.,0=C(16),

256 F

24 Triality 257

and so on. The induced Clffird or spinor actions of Spin(L) on S0, Spin(2)on Sr, Spin(3)and Spin(4) (in two ways) on 53, Sptn(5),Spin(6), Spin(l) and Spin(8)(in two ways)on 57 and Spin(9)on Sr5 are,moreover, all transitive, although the Clifford action of Spin(10)on S31is not, as we shall see- a good reasonfor stopping at this point! Apart from these Clifford actions of the groups Spin(n)on spheres there are the standard orthogonal actions. In studying the standard orthogonal action of Spin(n* 1) on Sn, for a positive integer n, it is appropriate to work in the Clifford algebra Ro,, ! R8,n+r,identifying Rn+l with r = R e R', R and R' being embeddedin Rqn in the standardway. Then, for any y e Y,9 : y-,

S' : {y e Y : y- y: 1} and Spin(n): {g e Spin(n+1) : !': 91. It is worth a passingmention that I is closed under the operation of squaringand thereforecan be assignedthe bilinear product

Yz --+Y ; (yo,y) e lo lr * yr yo.

This gives Y the structure of a Jordanalgebra (Paige(1961)). Moreover the squaringmap Y ---,Y ; y rt y2 is surjective,since any elementof Y with non-zero real part (and there are such, since n > L) generatesa subalgebraof R6,, isomorphic to C. The standard orthogonal action of Spin(n* 1) on I is

Spin(n* 1) x Y --+Y;(h, y)r, hyi-r,

the map Y -->Y; y > hyfi-r being a rotation of Y, for eachh e Spin(n* l). ri ; Proposition24.1 Any l elementof Spin(n* 1) is expressiblein theform z g, t: wherez€S',geSpin(n). ir

Proof LetheSpin(n*1). Since1e Snso alsohi-t eSn.Let z eY be suchthat z2 : hi-l and let g : z-r h :?h.Such z exists,since squaring on Y is surjective,while, since ni-r e S', so also z € Sn. Moreover, ^-l gE-': z-'hh-t-t r i-t z-r:1.-t Sog:$ implyingthatg € Spin(n). n

It is easyto verify that, for any n > 0, the sequence

Spin(n) -----+Spin(n* 1) ---+ Sn h*, hi-l fll I l.; 258 24 Triality !r li, is left-cosetexact, and projectsto the left-cosetexact sequence

SO(n)-----+ S O(n * 1)-----+ S'

studied in Chapter 14. Thus Spin(n* 1) acts transitively on S,, each isotropy subgroupof the action being isomorphic to Spin(n). All thesetransitive actions of the groups Spin(n)on spheresbear closer study,not only independentlybut in relation to eachother. Of particular interest are the isotropy groups of the various Clifford actions. The story is summarisedin the following sequenceof commutative diagrams:

Diagrtm 24.1

Spin(l) Spin(2) : in R6.1! Q, hn 5-o 5-r - Sr

involving the Hopf map hp, the restriction to Sl of the Hopf map R2-+ RP1, composedwith a stereographicprojection;

Diagram 24.2

Spin(2) Spin(3) ------' 5z = = : in R6,23I{, 5t 5: I 5z

involving the Hopf map hg, the restriction to 53 of the Hopf map 92 --+CPt , composedwith a stereographicprojection;

Dizgram 24.3

5: 5: I Spin(3) Spin(4) 5: in R6,3! 2fl, i J 5: 5:

involving various transitive actions of Spin(4) on 53, Spin(4) being iso- morPhicto Sp(l) x Sp(1)c'53 x 53; F

24 Triality 259 Diagram 24.4 5: : sp(1) JI Spin(4) ----+ Spin(S) -----) 5+ in Rry =H(2), JI: 5: -----' 5z J\ 5+ involving the Hopf map hs, the restriction to 57 of the Hopf map *--+\lpr, composedwith a stereographicprojection, and the isomor- phismSpin(S) = Sp(2);

Diagram 24.5

Sp(l) -----) SU(3) ---+ 5s JJtt: ----+ Spin(S) Spin(6) -----) 5s in Rqs = C(4),

5z:gztl involving rz the isomorphisms Sp(l) SU(2), Spin(S) = Sp(2) and Spin(6)= SU(4);

Diagram 24.6 SU(3) -----) Gz -----) 5r J1 Spin(6) Spin(1) ----+ 5o in Rq6 = R(8), ll 5z:gz introducing G2, the automorphismgroup of the Cayley algebraO, and involving the transitive action of Gz on 56;

Diagram 24.7

G2 Spin(1) -----+ 5z t_ J J_ Spin(l) Spin(8) ----+ 5z in Rs,7= 2R(8), I J J 5z 5z

- IIl 260 24 Triality involving various transitive actions of ,spin(g)on s7 and the associated triality automorphism j, of Spin(g) of order und finally,

Diagram 24.8

Spin(1) Spin(1) J J Spin(8) ----' Spin(9) 58 in Rqs = R(16), I t_ J J_ 5z S15 I 5s

involving the Hopf map hs, the restriction to ,St5 of Oz + the Hopf map Opl, composedwith a .r"rrogr;lri; projection.

Thefirst few diagrams Diagrams 24'1 to 24.5 can be deart with fairry rapidry, for much detail has appeared of the above,in chapter to, o. eadier. "uen Diagram 24.1

in Rqz:_C. The map ftp ^,*.^Iotk : ,S1--Sr is the restrictionto sr of the map c-+c; ot,'fquiuur"ntly,the map (x, y) * (*2 - 1.? ",, R2_-R2; y2, 2yx), which admitsirre iactorisation 5t ----+ RPI '----r sl (x, y) *' [x, y] H (2xr -l,2yx) withx2ly2:l

r4)^.: lZx',Zyx) (x2- y2, 2yx). (at least rrhen x # 0) The map Rp1--+5t may be interpreted as stereographic from (-1, 0) in R2.

Diagram 24.2

The Clifford algebrain which we work is Rq2 3 H, with the real linear spaceR3 = R o C embedded in H c Ce;;y rhe real linear map -;), Re C--+C(2); (),,r)r- ( : 24 Triality 261 t. 52 in R3 being representedby those (21,z) such that A2* zZ :1. The map hg : 53 -* 53 is the restriction to 53 of the map H-+H; -;) n: (*, ;),- s4: (r, i) (r,

I ww-zz -2w u"r)' :l 'w-w: \ 2zw or, equivalently,the map

gz -, C2| (w, z) r+ (ww - zz, 22fr), which admits the factorisation

5: -----+ CPt -----+ 5z (w,z) H fw,zl r+ (2wW_ 1,22fr\ withwil * zZ :1 : l2wW,2zWl (wW- zZ,2zfr), at least when w f 0

the last of thesemaps being stereographicprojection from (-1, 0) e C2. Diagram 24.3

The Clifford algebra in which we work is R6 = 2Id^,with the real = 2H linear spaceR4 H embeddedin c H(2) by the real linear map

H--+H(2);s,- (t 9 ), \ v q)' s3 in H beingrepresented by thoseq suchthat q4: !. with this choice R&, : H is embeddedin 2H by the real linear map H.-+H(2);s*,(58)

The diagram is

5: --- 53 t: Spin(3) -----) Spin(4) ----+ 5: I J J 5:: 5: 262 24 Triality where the horizontal maps are

53-- 53; r p-+7, wherer F : 1,

spin(3)--+ spin(4)q-, 9 , *r,.'"q4: r, ( 3 ) and spin(4)._s,; : (s :)_.(3 :) (l ;) (n; l), whereqQ : rT : l. The central vertical maps are, simply,

--+Spin(4); 53 r,- ( ; : ), and spin(4)-'s';', ( i),-t The diagram relatesone of the clifford actions of Spin( ) on s3 to the standard orthogonal action and in so doing delatestwo distinct product structureson Spin(4),the group isomorphismSpin(4) = Sp(1)x Sp(l) and the diffeomorphismSpin(4) = Spin(3)x 53 with (4, r) corresponding to (q' qF)' A similar dagram relatesthe other cliflord action of spin(4) on 53 to the standard orthogonal action. 'vertical' One way in which the embeddingsof 53 : Spin(3)in Spin(4) difler from the'horizontal'oneis that they do not projectto embeddings of so(3) in so(4). we refer to theseembeddings in the sequelas the clffird embeddingsof spin(3)in spin( ).It is, in a sense,fortuitous that the clffird homogeneousspace spin(4)/spin(3) is homeomorphicto the standardone, the real Stiefelmanifold (cf. Exercise22.5) o(R3,R4) : o(4)/o(3): So(4)/So(3): Spin(4)/Spin(3).

The force of this remark will becomemore evident in the sequel. Diagram 24.4

The Clifford algebrain which we work is R6,a=H(2), with the real = linear spaceR5 = R @R4 R OH embeddedin R6,aby the real linear map -u^) R@ H=+H( 2); (1,s) *, ( I , 24 Triality 263 54 in R5 being representedby those(,1, 4) such that 12I qe:1. With 2H this choiceR$,0 is the standardcopy of inH(2),namelythe subalgebra of diagonal matrices.The diaeram q' : splt)

Spin() ----) Spin(s)-----) 5+ ll ', ------' j J\ 5+

relatesone of the cliflord actionsof spin(4) on s3 and the clifford action of spin(S) on s7 to the standardorthogonal action of spin(S)on s4. The horizontal maps are

spin(4)=sp(1)xsp(l)--+spin(S)=sp((:'rr;(o/ a 0 \ /4 0\ d)- \o r/' o c z -b spin(S)=sp(2)--so; ( : : ),-- ( \ / \ \Dd/ \bd/\-. A)

_ (2aa-t -2a6\ \ 2bd 2aa-I J'

,-(o r\/a6\_(r o\ 'o' : (.; t,a6\(o c\ (u a)\r r/ ; ):\:, ; )\; ; ),

53 c H--57 c. H2 ; a++ (a, 0), with aa : I, and

57 c H2 ------) 7gPz -----r 5+ (a,b) r--+ [a,b] H (2ad_I,2ba). withaa+b6:1 - : l2ad,2bal (aa- b6,2ba\ at leastwhen afO The vertical maps are, simply,

s3= splr;---+spin(4)= sp(1)x sp(l);a- ( 1 I ), \v e/ and this composedwith the inclusionSpin(4)---+Spin(S), spin(4):sp(l) x sp(1)__r,, _ ( t 2) ,, 264 24 Triality and ll i1 spin(5)= sp(2)-s'; e (a,b). I ( ; 1) li The vertical embeddingof Sp(1) in Sp(2) is a standard one, but the induced embedding of Spin(3) in Spin(S) factors through one of the Cliflord embeddingsof Spin(3) in Spin(4) and is not standard. We refer to it as a Clffird embeddingof Spin(3)in Spin(S).The Clffird homogeneousspace Spin(5)/Spin(3) is homeomorphicto Sp(2)/Sp(L\= 57. On the other hand it can be shown, by methods of algebraic topology (see,for example,Steenrod and Epstein (1962), Theorem 4.5 or James(1976), that the standardhomogeneous space Spin(5)/Spin(3), homeomorphicto the real Stiefelmanifold O(R3,R5) = SO(5)/SO(3),is not homeomorphicto S7,nor to the product 53 x Sa. There is, of course,an analogousdiagram involving the other Clifford actionof Spin(4)on 53.

Diagram 24.5

We have akeadymet this diagramin Proposition17.3 where we proved that Spin(6)= S U(4). We give below a slight variant of that proof. The Clifford algebrain which we work is Re,5= C(4), with the real linear spaceROR5 = R6 = C3 embeddedin it by the real linear injection

C3-- C(4); (zo, z1, rrr,- -jir) (:, :^ i

With this choice,Rfi., is the standardcopy of H(2) in C(a). The sphere55 in R6 is representedby those(zo, zt, z2)such Ihat zoZo* zt4 * z2Z;: 1. The determinantof the matrix representing(20, zt, z2) is easilycomputed to be (zszo+ 44 * zz4)2, which is equal to 1 when (20,zr, z2) e 55. Since,by Proposition24.1, any elementof Spin(6)c U(4) is of the form zg,where z c 55 and g e Spin(S)= Sp(2),and since,by what we have just provedand by Corollary 10.14,both z and g have determinantequal to 1 (as elementsof C(4)), it follows that Spin(6)c SU(4). Since both these groups are connectedand of the same dimension,namely 15, it follows that they coincide. Filling out the rest of the detail of the diagram is then straightforward. For any t e C(4),r-: ? is given as in Exercise1.0.2. A direct computation 24 Triality 265

(in which Exercise10.1 is relevant)shows that, for any u € SU(3), 0 uo2 g\ (T@ utz -T,") (;l) (3 ?)-: -TA uzz \ _7,:uo2 0 which is the identity matrix if and only if u : f I ? ), with ueSU(2)=sp(l). So,finally, we obtainthe commutativediagram sp(1)= SU(2) ---+ sU(3) ----' 5s ll Sp(2)= Spin(S) ----' Spin(6)= SU(4) -----+S) I +I T s7 J': o wherethe maps not explicitly describedabove are all standardones, each row and eachcolumn being left-cosetexact. The embeddingof SU(2) in SU(4) here is the standardone. It is therefore a corollary of the diagram that the complex Stiefel manifold rJ(c2,c4) = su(4)lsu(2) is homeomorphicto s3 x s7. since sU(4) = Spin(6) and SU(2) = Spin(3), this complex Stiefel manifold may also be regardedas a Clifford homogeneousspace Spin(6)/Spin(3), the embeddingof Spin(3) in Spin(6)being a Clifford one, as it factors through a Clifford embeddingof Spin(3)in Spin(4). By contrast, it can be shown,by methodsof algebraictopology, that the standardhomoge- neousspace Spin(6)/Spin(3), homeomorphic to the real Stiefel manifold O(R3,R6) = 50(6)/SO(3),is not homeomorphicto 55 x 57.

Getting further To get any further it is appropriateto jump a stageand to take a look first at Spin(8).Any linear automorphismof R' inducesan automorphismof Spin(n),which projects to an automorphismof SO(n), sincethe original automorphism of Rn commuteswith -n1. Of all the groups Spin(n)the group Spin(8) is unique in that it possessesautomorphisms of order 3 that do not project to automorphismsof SO(8). To construct such an automotphism we begin with Spin(8) as a subgroup of O(8) x O(8), or tather, since Spin(8) is connected,as a sub- fl 266 24 Triality

group of SO(8) x SO(8). The Clifford algebrain which Sptn(8) lies is 2R18;, R8,r= Rq7 3 where we may supposethat Rs is embeddedby the injection '(:) o Rs-.2R(8):--\-/' o,---- ( ). \ 0 o(a)')' u(upsilon): O : R8--+R(8) being the injectionwith imagey, with which we becamefamiliar in our discussionof the Cayley algebra O early in Chapter 19. Here, as on that occasion,the product on R(8) and the product on o will both be denotedby juxtaposition, as will be the action of R(8) on O, the unit elementin O being denotedby e. One technical detail is worth isolating as a lemma.

Lemma24.2 Let x eY,let g eR(8) andsuppose that gye: xye,for all y €Y. Theng : x qndg y e : (geXye).

our purposein singling this out is to emphasisethat it is incorrect to contract (g eXy e) to g y e or to expandg y e to (g eXy e), unlesswe know that g e Y.

The companioninaolution conjugation on the cayley algebn o is associatednot only with the conjugation anti-involution of the Clifford algebra R3., = R(g), namely transposition,but also with an involution of R(8), which we term the companioninvolution of R(8), and which restricts to an involution of so(8). This involution is definedby meansof the elementof o(g) which inducesconjugation on o (by left multiplication), namely the symmetric (t anu-rotatton^,i-t--; 0\ |,.0 _rt ).

Proposition24.3 The map - /1' 0\ /r o\ R(8)-+R(8'');s+t:\o _7t)sIo _rr/

is a linear inuolutionof the algebraR(8)which commutes with transposition and restrictsto a groupinuolution of SO(8)and is suchthat

Ey e : gTe,for alt g e R(8) andall y ey, or, equiualently,

E b : 56, for alt g e R(S) and att b e O.

I 24 Triality 267 In particular,by settingy : I, or b: e,

Ee : gE,for all g € R(8).

Moreouer,for all g € SO(8),

d:gege:eele:e,

g, in sucha case,being of theform ( : ? \, *nrr" h e So(7). \0 h)

The elementf will be called the companionof the elements.

The triality automorphism / o^ n \ considernow "-, an elenIent of Snin(8),gs and 91being elements ( 6 ) of SO(8) and f1 being the companionof 91. Its action on R8 and in particular on 57 is given by (: o-),.-({.g \f v o \/d'o \ \oy'l \o d,/\oy'/\o so,),

wherey eY or 57, that is by y + go!g'1-1,since gI: g{: gp1;the correspondingaction on O beinggiven by yee goydr-t.. In this way the pair (go,gr) of elementsof SO(8)defines a third elementSz e SO(g) by

goy it-t e : dzy e, for all y e y.

An orderedtriple (96,gt, gz)of elementsof SO(g)such that

(r"' 1 ).spin(S), \u 8r/

or, equivalently,such that gsydfl e y for all y ey, and such that : y, goy {fr e {zy e, for all y e will be calleda T-triad of SO(g),g, the triality automorphismof sprn(8),being the automorphismof order 3 ( s^o ( q' o: spin(8)--.spin(U;""\0 ? ),-- I \. *) \0 dz)' whichexists by virtueof the followingtheorem.

L 268 24 Triality 24.4,tet (go,.gr, Jheorem 92)be a 0_triadaf s,o(g); ako Then(g1, 92, g0)and ft'u,#"'uii;.';:::nas o" 1g;"-s;"s;'), t ;| ii"iil) and 0:spin(B)_-spin(U;(? "'\o t ),*/r,0-\o \ gil ir) is an autumorphism o.fSpin(g) of order 3.

Proof The key to the proof is the scarartriple product on o with which inPropositio;il: whar we [:,3X""fi"|TjiS: "",Ji,l".,rr i, a'bc:6.ca:v.ab, where . denotesthe standardscalar product on so let (96, R8 gt, gz)be a g+riad. Then goydlr €y, with goyd.-te:g_zye, forallye y. Then, by I emma 24.2, goy {[1 ze: (goydt-t e)(re): ({zye)(ze), for all y e y. Since ge is orthogonal it follows, propositi by on 24.3,that i.e. (y e)({fl z e) : goTE.go y {[t z e : . g2fe(z ffi.xe- e), for all x, y, z e y, and so, by proposition 19.i as we promisedabove, tE' (dl-t z e@e) : sztrE.(z e)(fex e) : yE' gi' z {sxe, for all x, !, z € y, gz being orthogonal.Therefore

@1-rze)(xe) : gzr z {sxe, for all x, z e y. So, by Lemma 24.2 yet again, gzr z ioe Y, with gzl z : ior g-r-, ze, for allz ey. That is' (gr1' gil,cfl) is a '-triad. Repeatingthe whole argument tnrc d-triad as startingpoint with r.vr'L vuaone d"d;;;

E 24 Triality

With this we have the companiontheorem:

Theorem 24.5 Let (go,gt, g) be a \-triad of SO(g). Then so also is (il, io, g-2).Note the changeof order!

Proof Let (go,gt, g) be a ?-triadof SO(8).Then r goy ir-1 e Y, with goyd e : {zy ee, forall y e y. Then g'1y- git : Goy dar)- e y, with

iry-gir e: (soyd[1)-e:ioyi.,-i:6w: s2!-e, for all y- € Y. That is, (dt, do,$2) is a l-triad. o \ It is therefore appropriate to call ( I' the comPanionof \ J' ;, ) /o^O\"" | :" I in Spin(8). \u 8t/

Corollary24.6 Supposethat (g, E, d) ts a 7-triadof SO(g).Then !: I so that (g, g, g) is a ?-triad o/SO(g).

Proof Since(s, E, E) is a O-triad,so is ([, !, g) : G, t, il. So [ : g. n

Theorem 24.7 The triality automorphism0 of spin(g) doesnot project to an automorphismof so(8). Howeuer,it doesproject te an automorphism of So(8)/So.

Proof lJnderthe projectionSpin(8)--+SO(8) the elements(f;o t) . /-sn o \ and projectto the sameelement of SO(S).How- ( 0" _d, ) {2 :l'l'1(i?):(l?),*nu"'(-; -?):(-; l),,in.. (1, 1, 1), and therefore(-1, -L,l), is a g-triad of SO(g).Fromihe fact /1n\ r_1-; 0\ that ana project to distinct elementsof so(8), ( ; ; ) ( ; ,) namely 1.and -1, it follows that g doesnot project to an automorphism of So(8). Under the projectionSptn(8)--+SO(8)/S0, however, the four elements (*so-0" o \ to the sameelement +f2 of SO(8)/S0,while under ( fi, )project

L 24 Triality .), ( tf I 0 they map to the four etements -rg2/,X which then project to the \ u sameelement *{6 of SO(S)/S0.The automorphismg thereforeprojects to the automorphism

SO(8)/S0--+ SO(8)/S0; *do - *fs (and * gsr+ tg2),

where(96, gt, gz)is a O-triadof SO(8). n

It is incorrect to say that 0 and 0-t are the only automorphismsof order 3 of Spln(8)that do not projectto automorphismsof SO(8),for if { is the automorphismof Spin(8)induced by a changeof coordinateson R8 then 0 0 O-t wiil also be an automorphismof order 3 that doesnot project to an automorphismof SO(8),and not all such @commute with 0. Essentially,however, 0 is unique. The proof that Spln(8) is the only one of the groupsSpin(n) to admit a triality automorphismdepends on a much deeperanalysis of the structureof the groupsSpin(n) and their Lie algebrasthan it is possibleto give here.See, for example,Loos (1969).

The group G2 Let G be any group and let p : G --+G be an automorphismof G. Then the subset{e e e : Vk): g} of elementsof G left untouchedby rp is clearly a subgroupof G. Consider,in particular, the group Spln(8)and its triality automorphism 0. The subgroupof Spin(S)left untouchedby 0 consistsof thoseelements I o^ {l \ |:" \u 8t) ,(3'3,,):(e'3,):(r'3,),

/o O\ that is, those ( e Sptn(8)such that (g, g, g) is a g-triad of SO(S). \u: c; / l Clearly this group is isomorphic to the subgroupof SO(8) consistingof thoseg € SO(8)such that (g, g, g) is a O-triadof SO(8).This group we define to be the grotp G2. (The name derivesfrom the classificationof Lie algebras.There is no group G1 !)

Theorem24.8 Let g € Gz. Theng : ! and ge : e.

* 24 Triality 271 Proof Let g € G2.Then

c yt-l € Y, with c y|-l e : Eye, for all y € y. particular, y In by setting! : l, g|-t e and CE-le: f e, from whichit followsthat i-l gf-t e: e, so that, by the last part of proposition24.3, c-r\g-r:E-rg!-1, implyingthat (gE-1)3:1. Let x:gf-l. Then x e G2and xy*-l e: *y e, for all y eY.But i : x-1 andx3 : 1..So !xe: xye,for ally e Y; thatis (ye)(xe): (xe)(ye),for all y e y. So xe: *e; thatis x: t1.But (-1, -1, -1) is not a 0_tiadof SO(g).So x: 1.That is g: E.Then!e: e.So ge: e. ! The next theorem characterisesG2.

Theorem 24.9 G2 is the group of automorphismsof the cayrey diuision algebraO.

Proof suppose first that g € G2, acting on o by left multiplication. Then, for all b : ! €, c : z e € O,

e(bc): gyzs:gyg-lgze : gyg-Legze,sinceg:f, : gyegze, againsinceg:f, (eb)(s c),

with, in particular, g e : e. Thus g is an automorphismof O. Conversely,by the argument of Proposition 8.1.8,applied to O rather than to H, aly automorphism or anti-automorphismg of O is of the / 1 n \ form ( i : ) : RORT--+R@R7;a e reatt(pua),wherer is an \u r/ orthogonalautomorphism of O. Then

g(bc) : (eb)(Sc),for all b : ! a, c : z e e O, thatis gy ze : gyegze, for ally, z €Y, thatis Cy{-t gze:Eyegre, for ally, z €y, since! : g, thatis CyE-Le:Eye, for all y € Y, thatis geGz. n Since! : g, for all G e G2,it follows, from the last part of proposition 24.3,that G2actually is a subgroupof SO(7).We shall prove shortly that Gz is in fact a Lie group of dimension L4 (SO(7) and SO(g) being ! 272 24 Triality I .1j Lie groups of dimension 21.and 28, respectively).This we can do after :l we have establishedDiagrams 24.6 and 24,7. Before turning to these we prove one further result about the way that the group G2 lies in Spin(8). In doing so it is helpful to think of S/n(8) as the group of 0-triads of SO(8) themselves,under the group multiplication

(go,gr, c)G:o' ei' gL): (goglo, gr g\, gzgL), with Gz the subgroupof triads of the form (9, g, g).Now, for any 9-triad (go,gt, gz),

gog"r-1: I + g2e: e + {2e : e.

Bearingthis in mind, we define,for each i € 3,

Hi : {(go,gr, gz)e Spln(8): g;e : e}.

Theorem 24.10For eachi e 3, Hi is a subgroupof Spin(8) isomorphicto Spin(1) the three subgroupsbeing permuted cyclically by 0, namely

O(Ho): H1, 0(H): lt2 and 0(H): go.

Moreouer.

Hr A Hz : Hz(\ Ho : Ho i H1 : Qt'

Proof lt is clear that Hz is the isotropy subgroup at 1 of the standard orthogonalaction of Sp;n(S)on 57, this subgroupbeing isomorphicto Spin(1).It is clear also that the three subgroupsare isomorphic and that they are permuted cyclically by 0. To provethe last part, supposethat (gs,gr, gz)€ H1OH1 Then 92 : {o and gs : f1, implying that (go,gr, gz) : (go,do, do) and therefore,by Corollary 24.6,that go: gr: 92.That is I11O H2 e G2.Conversely it is clear that G2is a subgroupof eachof the II;. So I11O I12 : Gz. Likewise Hz n Ho: IIo 1-.1Hr : Gz. !

We are at last in a position to appreciateDiagrams 24.6 and,24.7. Paradoxically it is convenientto considerDiagram 24.7 first. a it '"'i ail 24 Triality 273 Diagram 24.7

The diagram is

Gz ---> SPin(1): Po -----+ 5r

+*tt: Spin(1): P, -----+ Spin(8) ------) S7 '

J+tl 5z:52 where, as has been explicit throughout the precedingdiscussion, except momentarily in Theorem24.10, Spin(8) lies in the Clifford algebraR0,7 : 2R(8),^ l/o^ O\ any element{ l" ; I of Spin(S)being such that (96,91, 92)is a \u 91/ ?-tliad of SO(8). The diagram relates two of the three actions of Spin(8) on 57, the standardorthogonal action and one or other of the two Clifford actions of Spin(8)on 57. Supposethat we choosethe action

Spln(8)x 57-' 57 = (( r ,1,),,) sov with isotropy subgroupat 1 the subgroupI{o definedabove. The central vertical sequenceof maps is then the correspondingleft-coset exact sequence,while the central horizontal sequenceinvolves the standard orthogonal action with isotropy subgroupat 1 the subgroupII2. In view of Theorem 24.10the whole structure of the diagram should now be clear. An analogousdiagram relatesthe standard orthogonal action to the other Clifford action of Sp;n(8)on 57. It is the casethat, under the standardprojection p : Spin(8)--+ SO(8), (1,o\ (-1 o\ with o(.; : e(. t, p(H) = so(7), while i ) ; _i ) p(Hd=p(Hr)=Spin(1). In line with our practice abovewe refer to the horizontal embedding of Spin(1) in Spin(8), with image H2, LS the standard embedding,the vertical embeddings,with imageHs or Hy, being the Clffird embeddings of Spin(1)in Spin(8). It is a corollaryof Diagram24.7 that Spin(8)lGz= 57 x S7. 274 24 Triality Diagram 24.6

The details of Diagram 24.6 can now be inferred. From Diagrams 24.5 and 24.7 and the standard left-cosetexact sequence

Spin(6)--+ Spin(I) --+56 we have the diagram

SU(3) ----+ G2 -----) 5o tl Spin(6)----> spi"(t) -----+56, tt 5z:52 where the elementsof Spin(6) are those of Spin(1) which lie in C(4), regardedas a subspaceof R(8) in the standard way. It follows at once that the group SU(3) coincideswith the subgroupof G2 consistingof all those automorphismsof O which belong to C(4) rather than to R(8). Moreover the sequence

SU(3)--+ Gz- 56 is left-cosetexact, the sphere56 being thus representableas the homogeneousspace G2/SU(3).

Diagram 24.8

Diagram 24.8, concerning Spin(9), is now easy to establish. The Clifford algebra in which we work is Ro,s3 R(16), with the real linear spaceRe : R @R8 embeddedby the real linear map

R o Ro -----+R(16) -'y, tt,D* (,!u1 ) , whereu : O =R8--+Y c R(8) is the standardembedding of O in R(8), 2 - the sphere58'being representedin R(16) by the *utrice, ( ,." ), wherel, € R, y e Y and 12* y y" : I. 24 Triality 275 With this choicen$,, is the standardcopy of 2n1a;in R(16). Any elementof Spin(9)is of the form (^-{")(i'1) \v ,t/\o il/

.,, ( eoo \ -- . *itn . Spin(8),and ),2 * y y" :1. The detail of the diagram, ( 6' g, ) namely

Spin(1) Spin(1) TJ Spin(8) -----+Spin(9) ---+ 58, : tt ------, sL I 5s is then very similar to that of Diagram 24.4, with O replacingH. The map

1 -{' o \ / 1 -v" \2 spine-+s';( (i' '^)' \, .1i) \o d,)-(, with isotropy subgroupat 1 the subgroupSpin(8), determines the central horizontal exact sequence.The lower vertical maps are

sp,n(8)-- s', - ,or, ( 3o ,9,)

I -{" spin(e)--s";'\v( ) ( fl' ? e.gse,ysse) ^ )\o ilt)'- and the restriction of the latter to 58, while the lower horizontal maps are 57 c O --+S15 c 02; go€ r+ (gse, 0) and

515- 92 ____+ Opz ___+ 5l (Agse,y gse) r-> [Lgoe,y goe] H Q 12- 1,2)"ye) , where),2 + (y eXy o)" : 1,

llgoe,1lBoel : l2(lSoeXTgoe),2(ygseXIg0e)l, 276 24 Triality at least when ,t { 0, and go : (2 )'2- l' 2)"v e) t2(,1go eXZ go e), 2(v goeXI e)l : ((1'-yy")e,21Ye).

Herewehaveassumed,forthesakeofdefiniteness,thattheleft.hand clifford action of spin(8) on column of the diagram collespondsto the of the two clifford spin(1)',s s7 with isotropy group at 1 equal to f16,one involving the other in Spin(8).There is of'course-ananalogous diagram Clifford action of SPin(8)on S'' ItisacorollaryofthediagramthattheCliffordhomogeneousSpace to Stt' ol the other Spin(9)/Spin(7): Spin(9)/Hois homeomorphic to the real Stiefel manifold t und, spl(qj/rrr, *t i.il'i' holn"o-orphic (by Steenrodand Epstein (1962)' b6t, nn), is not homeomorphicto S1s Theorem4.5).

The action of SPin(I}) on 531 up until now have been All the Clifford actions on spheresdiscussed of spin(1o) on s31 is not, for transitive. By contrast, the ciifford action a Clifford copy of Spin(l) as the isotropy subgroupat 1 at least contains dimensionof the orbit of I is a subgroup,from wtrich it follows that the at most equal to -21 :24' dimsPin(1O)- dimsPin(l; : 45 quotient topology,can be shown In fact the spaceof orbits, assignedthe the real line, one end-point to be homeomorphicto a closedinterval of 21' homeomorphicboth of which representsan orbit A21of dimension the embeddinsol Spin(6) io Spin(9)/3pin(6) and to Spin(10)lSU(5)' one' while the other i" Sprigj'in the' former case being a Clifford 24' homeomorphic to end-point representsan orbit B2a of dimension in Spin(1O)being Clifford' SpinitO)lSpii1l1, tUeembedding of Spin(7) atL'Each of the interior S'pinill:I/e beingindeed the isotropy subgroup pointsoftheintervalrepresentsanorbitofdimension30,homeomorphic to Czo: SPin(!0)/SPin'(6)7 A21x Se

(the embeddingof Spin(6)in Spin(10)being Clifford)' orbit A21 ard various rela' More information about Spin(\})' the found in Exercises24'2 to tionships betweenAzr, Cn and Bzq will be 24,4.ThestandardtotondifrerentiablegroupactionsisBredon(|9.72), 24 Triality 277 though the above example is not to be found there. I am grateful to ChristopherSpurgeon and to Dr Hugh Morton for establishingmany of the details of the action as part of an M.Sc. project in 1978.

G2 &s a Lie group In our treatment of the groups Spin n so far we have regarded them certainly as topological groups and not just as groups, but, apart from one brief argument when discussingDiagram 24.5,we have disregarded the fact that they are Lie groupsand that the variousmaps between them and the homogeneousspaces formed from them are not only continuous but smooth (in fact C@).What about G2?

Theorem24.11 The group G2 is a compact,connected Lie group of dimension 14.

Proof ConsiderDiagrams 24.5 and 24.6.It is enoughto provethat the vertical map, the surjectionSpin(1) -- 57; g r-+ g e of Diagram 24.6,is a submersionand to provethis it is enoughto provethat the tangentmap ar 1, namelyT(Spin(I))1-- T(S7)r;y + y e,is surjective.However, this map composedwith the injection T(Spin(6))1--+T(Spin(7))1is the tangent map aI 1 of the standardsubmersion SU(4) -' 57 of Diagram 24,5. Hence G2is a smooth submanifoldof the Lie group Spin(1),of dimension dimspln(I) - dimST :2I -7 : 14.The group is clearlyclosed in the compact grotp Spin(1), so is compact. Finally, since SU(3) and 56 are connected,so is G2. The group G2 is one of a clutch of five compact exceptionalsimple Lie groups all associatedin one way or another with the Cayley algebra O, the other four being known as F4,Ea, Et and Es, of dimensions 52,78,133and 248,respectively. For the definitionsof serri-sjmpleand simple Lie algebrasand Lie groups the reader must refer to one of the standard texts on Lie groups, such as Helgason (1962),Gilmore (L974) or Jacobson(1962). Most treatmentsconstruct the exceptionalgroups by first constructingtheir Lie algebras.An elementaryaccount of them, as groups,is hard to find. n

Other aspectsof triality For any positive integer n the Lie group surjectionp : Spin(n)-' SO(n), with kernel S0, induces an isomorphism Tp1 betweenthe Lie algebras \

278 24 Triality

T(Spin(n))1 and, T(SO(n))r, the latter normally being identified with End-(R'), by the remark following Proposition22.28.What about 0?

Proposition 24.12 The triality automorphism0 : Sptn(8)--+ Sptn(g) is smooth(indeed C* ) and inducesa Lie algebraautomorphism Tfu : T(Spire(8))1-----+ T(Spin(8))1

of order 3. Although 0 doesnot project globally ro so(8) its restrictionto a suitablysmall neighbourhoodU of I in Spin(8)does project to a smooth map 0y : V --+Y,where V : p(U),the diagramof Lie algebram&ps

r(Spiz(8))1 g T(Spin(8))1 r ptl= , r,I= r!9' r(so(8))1 r(so(8))l being commutatiue.

Triality may be formulated entirely in terms of the action of SO(8) on the Cayley division algebraO as follows.

Theorem 24.13 The triple (gs,gr, gz) is a 7-tiad of SO(g)if and only if io@b) : (gp)kzb), for all a, b e O.

Proof (gs, gt, gz)is a 0-triad of SO(8)

+forallye Y, goyit-Le Yand goydlre:iz!e, e for all y, z € Y, goy d{l2 e : (iz y e)(ze), by Lemma 24.2, e for all x, y € Y, gsyxe : ({zy e)(S1xe),setting - xe {;rze, za: {1 xa, e' for all x, y € Y, g'g(xe y e) : (gr xqGzTC), conjugatingboth sides, e for all a, b e O, {s(ab) : g a)(gb), setting Xe: A. Ve: b, n Theorem24.13 is dueto Elie Cartan(1925) (page 370). The Lie algebra versionis known as Freudenthal'sprinciple of triality (1951):

Theorem24.14 For an! To€ f(SO(8))1 (= End_(R8)),there existunique It, tz €. f(SO(8))1such that

fo(ab): (y a)b* a(yzb),for alt a e O.

I 24 Triality 279

Proof For existencelet V be as in Proposition 24.12and take tangents at 1 of eachside of the equation

do@b) : ((0y2 gs)a)((0v sdb) for eachgo e V and eacha, b e O. Then -+ fo@b) : (((T0,2) tyda)b a(((T0 v\ndb), for eachyo e T(SO(8))rand eacha,b eO; for Cayleymultiplication is bilinear, while the companioninvolution and evaluationat a or at b are restrictionsof linear maps. So take,t: 1T-0v2)r?oand yr: (T7y)fly. For uniquenesswe haveto provethat if yr yz,yl, yi e T(SO(8))r are suchthat (ytb)c-tb(yzc): (y\b)c*b(y;rc),for all b, c € O, then71 :71 andy2:yi.It is clearlyenough to provethat if (y1b)c-f b(y2c): Q, for all b, c e O, then71 : lz:0. Let a: y1e. Then e. &: e .?re : -yle. e (sincey1 is skew): -a.e. So a is a pure Cayleynumber. However,we have 0:aclYzc, forallceO. So 0 : (7rb)c - b(ac), for all b, c e O. So 0:?rb -ba,forallbeO. So 0 : (ba)c - b(ac),for all b, c e O, which is not the case,unless a : 0. So y1b:0, for all b e O andlz :0, for all c eO. Solt : lz:0. n It is more usual to start the entire discussionof triality by first proving Theorem 24.1,3directly and defining (?0,yr, y2) to be a triality triad of r(so(8))1 if

iob c : (mb) + b(yzc), for all b, c e O. See,for exampleLoos (1969),Vo1.II.

Quadric triality In Exercise22.4 we noted that either componentof vf+(Rf,u)is home- homeomorphicto SO(4) by Exer- omorphic to ",t/l@flb),each being cise22.3, and we askedthe questionwhether or not either componentof d

280 24 Triality

IcGI|r is homeomorphicto 'f {Cl,). Now, back in Theorem I4.L7 we have representedeach of thesequadric Grassmanniansas follows: .'{+(PJ'a) = ,?"+(Rf,u) ., (o(a) x O(4))lo(4), ,rf {R4,4) = x{Rf;i ., e@) x o(4))/(o0 x o(3) x o(3)), ..trq(C8) = q(Cl,) = o(8) x u(4). "f ,'{(CB) = rlr(Cl,J = o(8)xU(1)xo(6). Henceone componentof ,4ra(Rflo)is homeomorphicto the homogeneous space(SO(4) x SO(a))/SO(4),clearly homeomorphicto SO(4), while ,t/r(Rf,u) is homeomorphicto (So(4)x so(a))/(sOx So(3) x so(3)) = (S3x s3)/s0 = Spin(4)/So= so(4). Likewise one componentof cQlil is homeomorphicto SO(8)/U(4), "f while I{Cf;o) is homeomorphicto SO(8)/(U(1)x 50(6)). It looksat first sightas though the isomorphismof U(4) to U(1)xSO(6) is a necessarycondition for the homeomorphism qGIa) of tf to "4rlcf;b) - yet it can be shown by methods of algebraic topology that these groupsarc not homeomorphic! However,SO(4) is not homeomorphicto S0x SO(3)x SO(3)- thoughSO(4)/S0 is homeomorphicto SO(3)x SO(3) by an obviousisomorphism, Spin( ) beingisomorphic to Spin(3)xSpin(3). A better questiontherefore is: Are U(4)/S0and (U(1) x 50(6))/50 homeomorphic? Triality providesan affirmativeanswer. Once again we work in R(8) 3 Ro,o,with R6 embeddedin this Clifford algebrain such away thal the even Clifford algebraR8,r h the standard copy of C(a) in R(8). It is easily verified that the product of the basis elementsfor R6 is then either of the diagonal elements*i of C(4) and, we so order them that the product is in fact i. The elements1, i and the six basiselements of R6 then span the copy of R8 in R(8) that we have found it convenientin Chapter L9 and,in this chapter to denote by y. With thesenotational conventionswe can now srare

Theorem24.15 Let g e U(4) c SO(S),let z be the inuerseof either of the squme roots of the determinantof g, regardedas an elementof C(4), so that z\g, definedup to sign, is an elementof SU(4), and let p : Spin(6).. SU(4)--+50(6)be the standardprojection. Then / /, o \\. ls,zE,| ^, !, | | isa o-triadolso(8). \ \u Plzzc))/ 24 Triality 281

In parttcular, the projection of 0 ro SO(8)/S0 maps the subgroup Ug)/50 o/SO(8)/S0 to the subgroup(UQ) x 50(6))/50 by the isomor' phism (, o \ *s*'t\ootrirl/

Corollary 24.16The triality automorphismo/SO(8)/S0 permutescyclically the two componentsof lf aQf;) with the proiectiuequadtic .'f {Cf;) itself.

In fact triality also is involved in the caseof the real quadric ./r(Rf,u). We note first the following.

Proposition24.17 Let SO(4) be embeddedin SU(4) = Spin(6)in the obui' ousway. Thenp(SO(4)) is a copyof SO(3)x SO(3)in 5O(6).

With an obviousreordering of basiselements where necessary for sense we then have the following.

Theorem 24.18The triality automorphism0 of Spin(8)restricts to an automorphismof p-r(SO(4) x SO(4)), the inducedautomorphism of the group SO(8)/S0likewise restricting to an a.utomorphismof (SO@)x SO(4))/S0. Moreouer,for any g € SO(4)c SU(4), ( ( +\ rs, .. 9., . )) ,,Lra,-*iadofu a- soc,)xso(a). ls, I o- r\\:-'p(lr)iil o' \ \ )/ )/ In particular,the projectionof 0 to (SO(4)x SO(4))/S0maps the subgroup SO(4)/S0of 60() x SO(a))/SOto the subgroupSO(3) x SO(3) by the isomorpkism+e- p(:il.

Corollary 24.19 The tiality automorphismof (SO() x SO(4))/S0per- mutescyclically the two componentsof -,f 4(Rlf;)with the projectiuequadric ",f (Rlf;) itself. We do not wish to deny the reader the fun of filling in the details of the proofs of theselast few theoremsfor him- or herself' We have noted severaltimes that the real projectivequadric v|/r(Rflu) is homeomorphicto SO(4). Study'sinterest in this quadric first arosein (1891)in connectionwith the problem of representingthe group of rigid motions of R3. As we saw towardsthe end of Chapter 18 such rigid 282 24 Triality motions can be represented,uniquery up to non-zeroreal multiples, by pairs quaternions of (u, B) with a. f :0 but with a + O, so that the group is representableby the quadric with one of its isotropic half spacesremoved (the group "rilRl) productis (u,B)(y,d): (ay, a6_l Fil, and the unit elementis (1, 0)). The relationship of this representation to the representationof the group so(4) by the whole quadric, which we have explored in a wider setting in the section on groups of rigid motions in chapter 18, is hinted at in Study (1903) and siated quite explicitlyin Study (1913).The samepassage in Study (1913)contains a cTearstatement of what is now known as study's principle of triality, but which he called the Reziprozitiitsgezetzor reciprocitylaw, namerythe existenceof an anaTytichomeomorphism between the quadric ,lrriRfro) and either component of the quadric Grassmannian-'[4(Flf;) and also between the quadric and either component "f {Cs) of i+,(CB). For an exhaustivetreatment of quadric triality in a generalsetting seeTits (1959).

Exercises 24.1 verify that the setof copiesof the algebrac in the cayley algebra 0 canbe representedas the homogeneousspace G2/SU(3), while the set ofcopies ofthe algebraH in o can be representedas the homogeneousspace G2/ S O(4). 1/l a Provethat the map

RoRe=C@o-- C(16) '{:)' ((, x) : ((, c) ' (,t,

(,4 where € R,(: (+i4,c € o.'R8, x:(q, c),2: u(c)e y and I (iota) denotesthe square root of -1 in the coefficients of the elementsof C(16), is a real linear embeddingof R O Re : in Ro,s C(16)such that n$,, is the standardcopy of R(16) in c(16), Henceprove that any elementof Spin(I}) c C(16)is express_ ible in the form ';: -{)(tr;) (,t, ) (] Exercises 283 where

(r;)e Spin(8),with gs,8r € SO(8),

(|-{) € S8, with,l. e R, y € y and ).2* y y" : 1, and ';: So, ( (,t, ) € with e C, z ey and(C, + z z, : lo the image of such an element by the standard projection ';:)' Spin(rl)--sebeins ( ,( With the elementsof Spln(lO) representedas in Exercise 24.2 acting on R10 = C o O c C(16) by left multiplication, prove that the isotropy group at 1 is isomorphic to Spin(1) (in fact to the Clifford subgroup Ho of.spin(g) - cf. Theorem24.L0) and, t that the isotropy at j,. sroup ( " consistsof those t , /) elementsof Sptn(l0)for which / cos9 - sin0 co:lsinO cosg \o o ,1),,.rgieso(6), with (:2(cos0*ising) andz: y(sing_icosg),the latter group having dimensionequal to

dim50(6) * dimse : 15* 9 : 24, and acting transitivelyon Sewith isotropy group at 1 isomorphic to Spin(6)= SU(4). (By a theorem(Montgomery and Samelson (1943))that lists all compact Lie groups acting transitively on spheresit follows that this latter group must be isomorphic to su(5).) Establish the following commutative diagrams for the orbits Aa.,Bz+ and C3sof the Cliflord action of Spln(l0) on ,S31: Spin(6) Spin(6) J J --+ Spin(1) Spin(9) Sl5 , j J -. 5o Azt -) Srr 284 24 Triality Spin(l) : SPin(1)

Spin(9) -+ Spin(t0) -r se JI- S15 -) Bu --, se

Spin6=SU(4) -+ su(5) -) se J J Spin(9) -' Spin(|0) + 59'

v .t Azt --+ Azt

implying that Cn = Spin(10)/Spin(6)= An x Se, Spin(6) : SPin(6) JI Spin(9) -+ Spin(LD) -+ 5r, JI Azt --+ Cso --+ 5e

Spin(6)'r SU(4) : Spin(6) t- SU(s) --+ Spin(t0) o: iJ 5l -+ Czo Azt and Spin(6) JJ Spin(1) + Spnz(10) + u: JJ 5o Czo -) Bz+

n References

Adams,-J.F.(1958) 'on the nonexistenceof elementsof Hopf invariant one,, Bull. Am. math.Soc. 64,279-282. Adams, 'on J'F. (1960) the nonexistenceof elementsof Hopf invariant one,, Ann.Math 72,20-104. Ahlfors,,L._(1985)'Miibius transformationsand clifford numbers,,in I. chavel, H.M._Parkas Dffirentia.l \9ds) Geometryand ComplexAn Dr*, ta.ai*trO to H.E. Rauch) Springer-Verlag, Betlin, 65_73. Ahlfors, (1986)'Miibius L. transformationsin R, expressedthtough 2- x' 2- matricesof Clifford numbers',Complex Variablis 5,2I5_i4.'-- p .Some Ahlfors, L. and Lounesto, {19g9) remarks on Clifford algebras,, Cornplex Variables 12, 2Ol-209. Arnol'd, UJ. (1974)Mathematical 'Nauka" Methodsof ClassicalMechanics,Izdat. Moscow.English translation by r. vogtmann and i. w.inrt.in, GraduateTexts in Mathematics,60, Spiinger V6rhg Xew york _---- Heidelberg(1978). Arnol'd, 'The V.I..(1995) geometryof sphericalcurves and the algebraof quaternions', . . UspekhiMa, Nauk IO,3_69 : RussianUotlr. S"iriyi SO. Artin, E. (L957)Geometric Algebra,Interscience, New york. Atiyah, M.F., Bott, R. and Shapiro,A. (1964).Clifford modules,,Topology 3 (Supp.1),3-38. Benn,I.M' and rucker. R.w. (1987)An Introducionto spinorsanril Geometry wih _ Applicationsto Physics,Hilger, Bristol. Brackx, F. Delanghe,R and Serras,H. (1993)Clffird Algebrasand their Applicationsin Mathematical physics,Kiuwei Acadeiric puurisrrers, Dordrecht, Netherlands. Bredon' G'Y G972) Introductionto compact TransformationGroups,Academic Press,New York - London. 'Recent Bruck, R.H. (1955) advancesin the foundationsof Euclideanplane geometry',Am. Math. Mon. 62.2-17. Cartanoan, E. (1925)'Le principrincipe de dualit6 et la th6orie des groupessimples et semi-simples',Bull. Sci.Math. 49,361-374. Cafian, 'Sur P. (!94-7) I'espaceanallagmatique r6el d n dimensions,,Ann. polon. Math.20,266-278. Cartan, ii. (tSCg)'Deux th6o_rdmesde g6om6trieanallagmatiqu e rielle d.n dimensions',Ann. Mat pura Appl.-(4) 2g, l-L2.

285 1 286 References

Cayley,A. (1845)'OnJacobi's elliptic functions, in reply to Rev.Brice Brownin and on quaternions',Phil. Mag 3,2t0-2I3. Chevalley,C. (1946)Theory of Lie groups,Princeton University Press,Princeton, N.J. Chisholm,J.S.R. and Common,A.K. (1986)Clffird Algebrasand their applicationsin MathematicalP hy sics,Reidel, Dordrecht, Netherlands. Clifford, W.K. (1876)'Preliminary sketch of biquaternions'Proc. LondonMath. Soc.4.381-395. Clifford, W.K. (1876)'On the classificationof Geometricalgebras', published as PaperXLIII in MathematicalPapers, (ed. R. Tucker),Macmillan, London (1882). Cnops,J. (L994)Hurwitz pairs anil applicationsof Mdbius tansformations, Thesis,Universiteit Gent. 'Vahlen's Elstrodt, J., Grunewald,F. and Mennicke,J. (1987) group of Cliflord matricesand spin groups',Math.Z. 196,369-390. 'Mtibius Fillmore, J. and Springer,A. (1990) groups over generalfields using Clifford algebrasassociated with spheres',Int. J. Theo. Phys.29, 225-246. Freudenthal,H. (1951)Oktauen, Ausnahmegruppen und Oktauengeometrie,Math. Inst. der Rijk Univ. te Utrecht, Utrecht. Fuchs(1963) Partially orderedalgebraic systems, Pergamon, Oxford. Gilmore, R. (1974)Lie groups,Lie algebrasand someof their applications, Wiley-Interscience,New York. Grassmann,H. (1844)Die Wissenschaftder extenswenGrdsse oder die Ausilehnungslehre,eine neuemathematishcen Disciplin, Leipzig. Graves,J. (1848)Trans. Irish Academy21,p.338. Haantjes,J. (1937)'Conformal representationsof an n-dimensionaleuclidean spacewith a non-definitefundamental form on itself', Proc. Ned.Akad. Wet. (Math ) 40, 700-705. Hamilton, W.R. (1844)'On quaternions,or on a new systemof imaginariesin algebra',PhiI. Mag.25,489-495,reprinted in The mathematicalpapers of Sir William RowanHamilton, Vol. III, Algebra,Cambridge University Press, London(1967), 106-110. Hampson,A. (1969)CIffird algebrasM.Sc. Thesis,University of Liverpool. Helgason,S. (1962)Dffirential Geometryand SymtnetricSpaces, Academic Press,New York. Hermann, R. (1979)Appendix'Kleinian mathematicsfrom an advanced standpoint,A: Conformal and non-Euclideangeometry in R3 from the Kleinian viewpoint',bound with F.Klein Deuelopmentsof Mathematicsin the 19th Century,translated by M. Ackerman,Math. Sci. Press,Brookline, Mass.,367-376. Hopf H.(1931)'Uber die Abbildungen der dreidimensionalenSphdren auf die Kugelfliiche',.r\4ath Ann. 104,637 -665. Hopf H. (1935)'Uber die Abbildungen von Sphdrenauf Sphdrenniedrigerer Dimension',Funilam. Math. 25, 427-440. Hurwitz, A. (189S)'Uber die Komposition der quadratischenFormen von beliebigvielen Variabeln', Nachrichten oon der KdniglichenGesellschaft der Vl/issenschaftenzu Gdttingen,Math. Phys. K1.,308-316. Hurwitz, A. (1923)'Uber die Komposition der quadratischenFormen', Math. Ann.88,294-298. Jacobson,N. (1962)Lie algebras,Interscience,New York.

i References 287

James,I.M. (1976)The topologyof Stiefelmanifolds, L.M.S. LectureNote Series, 24, Cambidge University press,Cambridge. Kaplansky,I. (1963)'Lie algebras',Lectures on ModernMathematics (ed. T.L. Saaty),Vol. I, Wiley, New York. 'Non-parallelizability Kervaire,M. (1958) of the n-spherefor n > 7,, proc. natn. Acad. Sci. U.S.A.44,280-283. Kleinfeld, E. (1963)A characterisationof the Cayley numbers',Studies in Moilern Algebra (ed.A.A. Albert), Vol. 2, MathematicalAssociation of America, Prentice-Hall,Englewood Cliffs, N.J. 'On Kuiper, N.H. (1949) conformally-flatspaces in the large',Ann. Math. 50, 9t6-924. l-awtynowicz,J. and Rembieliiski, J. (1986)'pseudo-euclideanHurwitz pairs and generalisedFueter equations',in J.S.R.Chisholm and A.K. Common (eds):Cffird algebrasanil their applicationsin mathematicalphysics,Reide, Dordrecht,Netherlands, 39-48. Liouville, J. (1850)Appendix to Monge, G. Applicationde I'analysed la geomttrie,56d. par Liouville. 'Principes Lipschitz,R. (1880) d'un calcul alg6briquesqui contient cornme elqdcesparticulidres le calcul des quantit6simaginiires et des quaternions', C.R.Acad. Sci.Paris 91,679-621,660-664. (Jntersuchungen Lipschitz, R. (1886) ilber die Summenoon euadraten,Max Cohen und Sohn, Bonn. Loos, O. (1969)Symmetric Spaces, Vols. I and II, Benjamin,New york. 'Cayley Lounesto,P. (1987) transform, outer exponentialand spinor norm', Suppl.Rend. Circ. Mat. Palermo,Ser. II, 191-198. Maks, J. (1989)Modulo (l,l) periodicityof clffird algebrasand the generalized ( anti-)Mdbius transformations,Thesis, Technische, Universiteit Delft. 'Clitrord Maks, J. (1992) algebrasand Mdbius transformations',in A. Micali et al. (eds),Clffird algebrasand their applicationsin mathematicalphysics, Kluwer, Dordrecht,Netherlands. 'M6moire Meusnier,J.-B.-M.-C. (1785) sur la courbure des surfaces',Mbmoire Dio. Sao.,I0,477-510. Micali, A., Boudet,R. and Helmstetter,J. (1991)Clffird algebrasand their applicationsin mathematicalphy sics, Kluwer, Dordrecht, Netherlands. 'On Milnor, J. and Bott, R. (1958) the parallelizabilityof the spheres'Bull. Amer.Math. Soc.64,87-89. 'Transformation Montgomery,D. and Samelson,H. (i943) groups of spheres', Ann.Math. 44,454-470. 'Zur Moufang, R. (1935) Struktur von Alternativkiirpern', Math. Ann. ll0, 416-430. 'Jordan Paige,L.J. (1961) algebras',Studies in ModernAlgebra, ed. A.A. Albert, Vol. 2, MathematicalAssociation of America, Prentice-Hall,Engelwood Cliffs, N.J. Pontrjagin,L.S. (1946)Topological groups,2nd edition, Gordon and Breacn, New York (1966). Porteous,I.R. (1969)Topological geometry, 1st edition, Van Nostrand Reinhold, London. Porteous,I.R. (1981) Topologicalgeometry,2nd edition, with additional material on triality, CambridgeUniversity Press,Cambridge. L

288 References

Porteous,LR. (1993)'Clifford algebrutables', in F. Brackx et al. (eds)Clffird algebrasand their applicationsin mathematicalphysics Kluwer, Dordrecht, Netherlands,13-22. Radon, J. 'Lineare Scharenorthogonaler Maftuen', Abh. math. Semin.Uniu. Hamburgl, l-14. RandriamihamisonL.-S. (1990)'Pairesde Hurwitz pseudo-euclidiennesen signaturequelconque', J. Phys.A: Math. Gen.23,2729-2749. Ryan, J. (1995)'Some applications of conformal covariancein Clifford Analysis' .{3 in Clffird Algebrasin Analysisand RelatedTopics, ed. J. Ryan, CRC Press, I Studiesin AdvancedMathematics, Boca Raton, FL, 128-155. Segre,B. (1947)'Gli automorphismidel corpo complessoed un problemadi Corrado Segre',,4tti Accad.naz. Lincei Rend.(8) 3, 414-420. .l i'.4 Spurgeon,C. (1979)Clffird algebras,M.Sc. Thesis,University of Liverpool. Steenrod,N.E. and Epstein,D.B.A. (1962)Cohomology Operations, Princeton i:l University Press,Princeton, N.J. ; Study,E. (1891)'Vonden Bewegungenund Umlegungen,Math. Ann.39, 44r-566. Study, E. (1903) Geometrieder Dynamen,Teubner, Leipzig. Study (1913)'Grundlagenune Zieleder analytischenKinematik', Sitzungsberkhteiler BerlinerMath. Gesellschaft12, 36-60. Tits, J. (1959)'Sur la trialit6 et certainesgroupes qui s'en d6duisent',Inst. iles HautesEudes Sci. Pubt.Math.2, t3-6O. Trautman A. (1993)'Geometric asp€cts of spinors',in F. Brackx et al. (eds) Clffird algebrasanil their applicationsin mathematicalphysics, Kluwer, Dordrecht, Netherlands333-344. Vahlen,KTh. (1902)'Uber Bewegungenund complexeZahlen', Math. Ann. 55, 585-593. van Waerden,B.L. (1985)A hisnry of algebra,Springer-Verlag, Berlin' der ;_i Wall, (1968)'Gradedalgebras, antiinvolutions, simple groups and i_ C.T.C. I symmetricspaces', BuIl. Amer.Math. Soc.74,L98-2Q2. l! 'Mijbius Waterman,P.L. (1993) transformationsin severaldimensions', Ailuances *. in Math.,l01,87-113. r' 2 Weyl, H. (1939)The ClassicalGroups, Princeton University Press,Princeton, N.J. ,!' lr* *. * x €t 1 r

Index

absolutevalue of quaternion,60 Arnol'd, V.1.,66, 112,285 action,Cliflord, 257 Artin, E., x, 176,285 action,spinor, 257 Atiyah, M.F., x, 124,285 Adams,J.F., x, 186,l8'l, 285 atlas,2l5 adjoint,27,53, 96, 101 atlas,standard, 111 adjoint representation,238 automorphism,algebra, 13 admissiblechart.2l7 automorphismof H,62 affineform, 112 automorphism,linear, 3, 5 affine map, 6 automorphism,orthogonal, 27 affine space,6 automorphism, tr iality, 267 affne subspace,6, 212 axis of rotation. 35 Ahlfors, L., 167,176, 254, 285 algebr4 alternatl e, l7 9 Banach space,202 algebra,Cayley, ix, 180 basis,2 algebra,Clifford, 123 basistheorem for hermitian spaces,54 algebra,division, 178 basistheorem for 2K-linearspaces, 92 algebr4 extensive,138 basistheorem for real quadraticspaces, 34 algebra"exterior, 138 basistheorem for symmetriccorrelated algebra,geometric, 123 quaternionicspaces, 77 algebr4 Grassmann,138 Benn,I.M., 166,285 algebra,Jordan,257 bijection,2 algebra"Lie,239,242 bijective,2 algebra, linear, 9 bilinear map, 4 algebra,Pauli, 152 biquaternions,174 algebra,tensor, 124 bivector, 176 algebraanti-isomorphism, 9 Bott, R., I24, 186,187, 285, 287 algebraisomorphism, 9 Boudet,R., x, 287 algebra map, 9 boundary point,192 algebra-reversingmap, 9 boundedlinear map, 194 alternative algebr4 I79 Boyer,R.H., x analytic,209 bracke! Lie,239 angle,absolute, 39 Brackx,F., x, 285 angles,preservation of , 245 Bredon,G.E.,277,285 annihilator, dual, 4 Bruck,R.H., 186,188, 285 annihilator,orthogonal, 32 bundle,tangent, 219, 220 anti-automorphism; algebra, 13 anti-automorphism of H, 62 Cartan,fr.., 247, 256,278, 285 anti-involution,algebra, 13 Cauchy-Schwarzinequality, 39 anti-involution of H, 64 Cavley,A., 180,286 anti-rotation,27, L43 Cayley algebra,ix, 180

289 il lt ri I I 290 lndex

Cayleychart, 118,174 commutative diagram, 19 Cayleyheptagon, 185 compactspace, 198 Cayley numbers,165 compactification, conformal, 247 Cayleyprojective line, 187 compactification, projective, 246 Cayleyprojective plane, 187 companion,267 Cayley triangle, 184 companioninvolution, 266 centre, 40 complement, linear, 3 centreof algebra,10 complemen! orthogonal,32 chainrule,203,207 complex field, as rcal algebr4 9 chafi,176,275 complexgroup : symplecticgrotp,]9 chart, admissible,217 complex orthogonal group, 50 chart, Cayley,118,174 complex quadratic space,49 chart,Pfaffian, 176 complexStiefel manifold, 265 charts on quaddc Grassmannians,114 complex symplectic plane, 50 chart, standard,l1l, 215 complexsymplectic space, 50 Chevalley,C, 124,243, 286 complexification,86, 105 Chisholm,J.S.R., x, 286 composite,L choice,axiom of, 2 conformal c ompactification, 247 circle, unit, 40 conformalgroup, 248 circlegroup,56,57 conformal map,245 classicalgroup, 100,103 conformal spheres,248 classificationtheorem for comolex conformal transformatiorl special,252 quadraticspaces, 49 conjugateof a quaternion,59 classification theorem for comolex conjugation,135,148 symplecticspaces, 50 conjugationanti-involution, 59 classificationtheorem for hermitian spaces, connected,l99 continuousmap, 193 classidcation theorem for 2K-correlations, continuously differentiable map, 205 98 contour, l classificationtheorem for 2K-semiJinear contractionlemma. 211 maps,94 correlated2K-space, 95 classification theorem of real quadratic correlated linear space,24 spaces,first pafi,34 correlatedquaternionic space, 74 classification theorem of real quadratic correlation,52,74 spaces,second pafi,36 correlation, linear, 24 classificationtheorem for real symplectic correlation,semi-linear, 95 spaces,47 correlation,symmetric, 24 'Correspondence classificationtheorem for symmetric from an ultramundane correlatedquaternionic spaces, 78 correspondent',176 Clifford,W.K., I23,286 cosetspace representation, 20 Clifford action,257 cover,double, 146 Clifford algebra,complex, 134 cover,induced, 197 Clifford algebr1 even, 136 cover,open, 197 Clifford algebr4 real, I23 crossproduct, 60 Cliflord embedding,262 Clifford group, 141 decomposition,orthogonal, 32 Clifford homogeneousspace, 262 decomposition,Witt, 37, 109 Clifford matrix, 167 Delanghe,R., viii, 285 closedmap, 196 determinant,6, 98 closedset, 192 determinant,absolute, 70 Cnops,J., 164,167, 168, 172,255, 286 diagram chasing,18 code numbersfor ten types of classical diffeomorphism,21 l, 21.8 group, 103 differentiable map, 205 code numbersfor ten typesof matrix differential,higher-order, 209 algebrawith anti-involution, 155 difrerentialof a map,205 cormage,I dimension,2 Common,A.K., x,286 dimension of affine subspace,212

T Index 29L

dimensionof classicalgroup, 103 form, sesquilinear,95 dimensionof Grassmannian"111 free, linearly, 92 dimension,quaternionic, 68 Freudenthal,H., 278,286 dimensionof smooth manifold, 217 Freudenthal's principle of l'riality, 278 dimensionof submanifold,212 Frobenius,G., 186 Dirac spinors,165 Fuchs,L., 13,286 direct product, 3 direct sum"3 generallinear group, 7, 98 disconnected.199 generallinear group, thought of as a dissimilar.14 unitary group, 91 distance,39,192 generators,set of, 12 division algebr4 178 geometric algebr4 723 division algebra.normed, 179 Gilmore, R., 243,277, 286 domain"1 gradeinvolution, 135 double cover,146 gradient norm, L94 double field. 5. 91 Grassmann, 2[, H., 138,286 double field as real algebra,g Grassmannalgebra. 138 dual. 52 Grassmannian,110 dual,linear, 68' Grassmanniansas cosetspaces, 113 dual of semi-linearmap,73,95 Grassmannian, quadic, 112 dual annihilator,4 Graves,J., 180,286 dual linear map, 4 group, circle,56, 57 dual linear space,4 group, classical,100, 103 group, Clifford, 141 Elstrodt, J., 167,255, 296 group, complexorthogonal, 50 embedding,196,2L3 group, conformal,248 embedding, Clitrord, 262 group G2, ix,270 i endomorphism,linear, 5 group, generallinear, 7, 98 Epstein,D.B.A, 264,2'7 6, 288 group, Lie, viii, 100, 103,231 equationof map,2 group, Iinear,7 equivalentatlases, 216 group, Lipschitz, 141 i equivalentcorrelations, 52 group, Lorentz,238 equivalentnorms, 194 group, Mdbius, 248,255 equivalentsemi-linear correlations, 74, 95 group,orthogonal,27 equivalentsmooth manifolds,217 group, Pin, 146 euclidean,tocally, 27 5 group, quaternionic,78 euclideanspace,39 group, real syrnplectic,47 evenClifford algebra,136 group, special linea4 7, 99 exact,Ieft-coset, 20, 108 group, specialunitary 55 exactpair, 18 group, Spin, 146 exact sequence,18 group, topological,225 exact sequence,short, 18 group, unitary, 55, 104 existencetheorem for Clifford algebras,131 groupaction,2l extensivealgebra, 138 group action, Lie,233 exterior algebra,138 group map : group homomorphism,20 group map, Lie,231 fibre,1,212 group map, topological,225 field,,2 Grunewald,F., 167,255, 286 field, complex, 2 field, double,5, 91 Haantjes,J.,246,286 field, real,2 half-spinorspace, 134 Fillmore, J., 167,255, 286 Hamilton, Sir WR., 58, 286 finite-dimensional,2 Hamilton triangle, 183 five, 154 Hampson,A., ix, 286 form inducedby correlation,74 Hausdorff spac,e,792 form, hermitian. 52 Heine-Boreltheorem, 197 form, qtadratic,22 Helgason,5., 243,277, 286 I.t 292 Index

Helmstetter,J., x, 287 Jacobianmatrix, 205 heptagon,Cayley, 185 Jacobson,N'277,286 Hemrann,R'247,286 James,I.M., 264,287 hermitian form, 52 Jodan algebr4 257 hermitian product, 52 juxtaposition,266 hermitian space,52 higher-order differential, 209 Kaplansky,L,243,287 homeomorphism,193 kernel.3 homogeneousspace,226 kernel of quadratic space,25 homogeneousspace, Clifford, 262 kernelrank, 3,51,73 Hopf, H., 122,286 Kervaire,M., 186, 187,287 Hopfmap, 111,258,259 Kleinfeld, E, 186,187, 287 Hurwitz, A., 164, 165,186,286 Kuiper,N.H.,247,287 Hurwitz pair, L64 hyperbolig 14 Lagrang1an,ll2 hyperbolic plane,26 Lawrynowicz,J., x, 164,287 hyperbolicplane, standard, 23 left linear spacn,67 hyperplane at infinty, 246 left-coset exact, 20, I08, 227 Leibniz notation, 205 ideal,left, 11 length,39 ideal, right, 11 levelset, 1,212 idempotent,14 Lie algebr4 215,239, 242 idempotent,primitive, 14 Lie bracket.239 image, I Lie group,ix, 100,103, 215,231 immersion,213,222 Lie group, exceptional,277 immersive,213 Lie group action,233 increment, 209 Lie group map,231 incrementformula" 211 line complex,79, 113 index, 37 linear algebra,9 I index,Witt, 37, 109 linear map, 2 t induced cove4 197 linear map, left, 68 1 inducedtopology, 193,196 linear map, right, 68 ! infinity, hyperplane at, 246 linear space,1, 2, 5 t infinity, point at, 111 linear space,left, 67 T F inflation, 254 linear space,right,67 'It injection,2 linear subspace,3 injective,2 linear subspaceof generallinear group, 164 injectivecriterion for smoothness,212 linked spheres,42 inverse, 2 Liouville,J.,245,287 inverse of a vectot, U Liouville's theorem,245 inversefunction theorem.211 Lipschitz,R, 176,287 inversion.254 Lipschitz chart,176 invertible vector,24 Lipschitz group, 141 involution, algebra,13 Liverpool, the University of, x involution, companion,266 locally euclidean,215 involution, grade,135 Loos,O., 270,279,287 involution of H, 64 Lorentz group,238 involution, linear, 5 Lounesto,P., ix, 176 involution, main, 135 irreducible,15 Miibius gronp,248,255 irreduciblesemilinear map, 95 Miibius map,246 isomorphic quadratic spaces,26 Miibius transformation,250 isomorphism,semi-linear, 73 main involution, 135 isotropic quadratic space,23 main involution of H, 64 isotropy subgroup,21 i Majorana spinors,166 i Maks, J., 167,172, 255, 287 Jacobi identity,242 manifold, smooth,215, 217 ;

tl :!l i ai ta fls Index 293 manifold, Stiefel,244, 262 orientation-reversing,7 manifold, topological,215 orthogonalcomplement, 32 map, 1 orthogonaldecomposition, 32 map, affine, 6 orthogonal grotp,27 map, bilinear, 4 orthogonal group, special,27 map, conformal,245 orthogonalisomorphism, 26 map, Hopf, 258,259 orthogonal map,25 map, identity, 2 orthogonalquaternionic group, 78 map,linear,2 orthogonalvectors, 23 map, Miibius, 246 orthonormal subset,126 map, orthogonal,25 map, semilinear, 72 matrix, 3 Paige,L.J., 257,287 matrix, Jacobian,205 paravector,l40 matrix, quate-rnionic,69 partition,1,l95 matrix algebr4 full, 9 partner, l0 mean value theorem,211 Pauli algebra,152 Mennicke,J., 167,255, 286 pentagon,rebracketing 2 Meusnier,J.-B.-M.-C., 246, 287 periodicity theorem for Clifford algebras, Micali, A., viii,287 IJJ Milnor, J., 186,187,287 Pfaffian, 175 minimal left ideal, 12,45, 133 Pfaffian,complete, 174 module,5 Pfaffian chart. 176 2K, module, 5 physics,x module map, 5 physics,theoretical, 165 modulus of quaternion,60 Pin group, 146 monoid, 168 point at infinity, 111 MontgomeryD.,283,287 Pontrjagin,L.S., 238, 243, 287 Morton, H.R.,277 Porteous,LR., ix, x, 14, l9l,202,287,288 motion, rigrd, 173,281 positive-definite,23, 55 Moufang, R., 190,287 positive-definitecorrelated space, 78 Moufang identities,190 power-associative,179 preservationof angles,245 negative-definite,23 primitive idempotent,14 neighbourhood,192 product, 5 neutral quadratic space,23 product induced by conelation,T4 neutral space,38 product, algebra,9 non-degeneratecorrelation, 25, 52,7 4, 98 product,cross,60 non-degenerateorthonormal subset,126 product, non-degeneratequadratic form, 25 direct,3 product, norm, 191 hermitian, 52 norm, gradient,194 product, scalar,22 norm, product, 194 product,tensor,8l norm, quadratic,145 product norm, 194 norm of quaternion,60 product topology, 196 normed division algebr4 179 projection,3,196 North pole, 40 projection, stereographic,41, 249 null quadratic space,23 projection, tangent, 220 nullity, 3 projective compactification, 246 projectiveline, 110 octaves,180 projective1ine, Cayley, 187 octonions,180 projectiveplane, 110 open cover,197 projectiveplane, Cayley, 187 open map, 196 projectivepoint, 110 open set, l9l, I92 projectivequadrig 112 orbit, 21 projectivespace, 110 orientation-preserving,7 pseudo-determinant,I 68 T

294 Index

pure part of quaternion,58 scalarproduct, reflexive,43 pure quaternion,58 scalarproduct, skew,43 scalar product, symmetic, 22 quadratic form,22 scalartriple product, 61, 182 quadratic norm,22, 145 section.2 quadraticspace, complex, 49 Segre,8.,13,288 quadraticspace, null, 23 self-adjoint,53,75 quadratic space,real, 23 semi-linearmap, 5L,72 quadric,projective, 112 semi-linearmaps over 2K, 94 quadric Grassmannian,112 semi-neutral"112 quadric Grassmannian,affine, ll2 semi-neutralquadric Grassmannian,112 quadric Grassmanniansas cosetsDaces. semi-orientation-preserving,36 716 semi-simple,277 quasi-sphere,106, 229, 248 Serras,H., x, 285 quaternion,pure,58 sesqui-linear,52, 74 quaternionproduct,58 Shapiro,A., 124,285 quaternionicgroup, 78 signature,37 quaternionicgroup, orthogonal,78 signatureof correlatedspace, 78 quaternionicgroup, symplectic,78 signatureof hermitian space,55 quaternions,algebra of, 58 signaturetheorem, 36 quotient, 195 similar, 14 quotient topology,196 simpIe,277 skew,16,52 radius,40 skew correlation,74, 95 Radon,J., 165,288 skew scalarproduct,43 Radon-Hurwitz sequence,165 skew-adjoint,53, 75 Randriamihamison,L.-S., 1,64,288 Skornyakov,L.A., 186 rank,3,51,73 smooth manifold, 215,217 rank, kernel,3 smooth map, 209,218 rank of quadraticspace, 25 smooth structure,216 real part of quaternion,58 smooth submanifold, 2I2 real quadratic space,23 smooth subset,212 reciprocitylaw,282 smooth,infinitely, 209 recognitionof subalgebras,88 source,1 reducibleanti-automorphism, 15 sourceofa map,794 reducibleautomorphism, l5 South pole, 40 reduciblepermutation, 15 spaceof half-spinors,134 reduciblesemi-linear map, 95 space,linear, 1, 2, 5 reflection,34 spaceof spinors,133 reflection, hyperplane, 34 specialcomplex orthogonal group, 50 reflexivecorrelation, 75, 96 speciallinear grotp, 7, 99 reflexivescalar produc| 43 specialorthogonal automorphism,27 Rembieliirski,J., 164,287 specialorthogonal group,27 representation,adjoint, 238 specialunitary automorphism,55 representation,Y ahIen, 254 specialunitary group, 55 reversion,L35,162 sphere,40 reversionof H,64 spheres,conformal, 248 Reziprozitiitsgezetz, 282 sphere,unit,40 right linear space,67 Spin group, 146 rigid motion, 173,281 spinoraction,257 ring, 5 spinor space,133,1,54 rotation,27, 143 Springer,A, 167,286 rotationof R3,62 Spurgeon,C., 277, 288 rotation of R4, 65 square,twisted, 99 squareof vector,22 Samelson,H,283,287 stabiliser.21 scalar.2 standardatlas, 111

f, Index 295 standardc hart,1 11,2 15 ten, 100,1 02,155 standardc omplexs ymplecticp lane,5 0 tensora lgebra,1 24 standardr eal symplecticp lane,47 tensorp roduct, 81 SteenrodN, .E.,2 64,2 76,288 theoreticalp hysics,1 65 stereographic projection, 4I, 249 theta triad,26'7 Stiefelm anifold, 2M, 262 Trts, 1., 282, 288 Stiefelm anifold, complex,2 65 topologicalg r ottp,2 25 Study, E., ix, 256, 281, 282, 288 topological group map,2 25 Study'sp rinciple of trality,282 topologicalm anifold, 215 subalgebra9, topologicals pace,1 91 subgroup, top olo gical, 225 topologicals ubgroup,2 25 submanifold.2 12 topology, 191 submanifold,s mooth,2 12 topology,i nduced,1 93 submersion2, 14, 222 topology,p roduct, 196 submersive2, 14 topology,q uotient, 196 submodule,5 topology,s ubspace1, 93 subspacea, ffine,6 transformation,M iibius, 250 subspaceli,n ear,3 transitivea ctionso n spheres2, 56 subspaceto pology, 193 transitive grorp action, 2L, 226 sum,d irect, 3 translation,2 54 sum of squares3, 6 transpose4, superalgebra8, 6 transposition,4 , 24 superfield,1 34 Trautman, 4., 166,288 surjection,2 triad, theta,267 surjective,2 1.rialrty,256 surjectivec riterion, 213 triality, Freudenthal'sp rinciple of, 278 swap,141, 54 triality, Study'sp rinciple of, 282 symplectic,e tymologyo f, 79 triality automorphism,2 67 symmetricc orrelation,2 4, 52, 74, 95 triangle, Cayley,t84 symplecticg roup, complex,5 0 triangle,H amilton, 183 symplectic group, rcal, 47 triangle inequality,1 92 symplecticm ap,47,50 triple product, scalar,6 1,182 symplecticp lane,c omplex,5 0 Tucker,R W., 166,285 symplectic plane, real, 47 twisted square,9 9 symplecticp lane,s tandardc omplex,5 0 type of orthonormal subset,1 26 symplecticp lane,s tandardr ea1,4 7 symplecticq uaternionicg roup, 78 unit quaternion,6 0 symmetrics econdd ifferential,2 10 unitary group, 55, 104 symplectics pace,c omplex,5 0 unitary linear map, 55 symplectics pace,r eal, 46 universalC lifford algebr4 130

tangency,202 Vahlen,K .Th., 167,2 54, 288 tangentb undle, 21.92, 20 Vahlen representation,254 tangentb undle map,22l van der Waerden8, .L., 1'76,288 tangent bundle space,2I9 vector,2, 140,1 67 tangent map,2l9 tangentp rojection,2 20 Wall, C.T.C.,x , 288 tangents pace2. I2, 219,2 21 Waterman,P .L., 167,2 88 langent vector,22l Weyl,H ., 79,243,288 target, 1 Weyl spinors,1 66 target of a map, 194 Witt decomposition3, 7, 109 Tavlor series2" 09 Witt index, 37, 109

pnh..Bibliorhe{-I F a s s a u I I i