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{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 fol- lowing/') 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.
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