Selected Title s i n Thi s Serie s

44 Max-Alber t Knus , Alexande r Merkurjev , Marku s Rost , an d Jean-Pierr e Tignol, Th e boo k o f involutions , 199 8 43 Lui s A . Caffarell i an d Xavie r Cabre , Full y nonlinea r ellipti c equations , 199 5 42 Victo r Guillemi n an d Shlom o Sternberg , Variation s o n a theme b y Kepler , 199 0 41 Alfre d Tarsk i an d Steve n Givant , A formalization o f set theory without variables , 198 7 40 R . H . Bing , Th e geometri c topolog y o f 3-manifolds , 198 3 39 N . Jacobson , Structur e an d representation s o f Jordan algebras , 196 8 38 O . Ore , Theor y o f graphs, 196 2 37 N . Jacobson , Structur e o f rings, 195 6 36 W . H . Gottschal k an d G . A . Hedlund , Topologica l dynamics , 195 5 35 A . C . Schaeffe r an d D . C . Spencer , Coefficien t region s fo r Schlich t functions , 195 0 34 J . L . Walsh , Th e locatio n o f critica l point s o f analyti c an d harmoni c functions , 195 0 33 J . F . Ritt , Differentia l algebra , 195 0 32 R . L . Wilder , Topolog y o f manifolds , 194 9 31 E . Hill e an d R . S . Phillips , Functiona l analysi s an d semigroups , 195 7 30 T . Rado , Lengt h an d area , 194 8 29 A . Weil , Foundation s o f algebraic geometry , 194 6 28 G . T . Whyburn , Analyti c topology , 194 2 27 S . Lefschetz , Algebrai c topology , 194 2 26 N . Levinson , Ga p an d densit y theorems , 194 0 25 Garret t Birkhoff , Lattic e theory , 194 0 24 A . A . Albert , Structur e o f algebras , 193 9 23 G . Szego , Orthogona l polynomials , 193 9 22 C . N . Moore , Summabl e serie s an d convergenc e factors , 193 8 21 J . M . Thomas , Differentia l systems , 193 7 20 J . L . Walsh , Interpolatio n an d approximatio n b y rational function s i n the comple x domain, 193 5 19 R . E . A . C . Pale y an d N . Wiener , Fourie r transform s i n the comple x domain , 193 4 18 M . Morse , Th e calculu s o f variations i n the large , 193 4 17 J . M . Wedderburn , Lecture s o n matrices , 193 4 16 G . A . Bliss , Algebrai c functions , 193 3 15 M . H . Stone , Linea r transformation s i n Hilber t spac e an d thei r application s t o analysis , 1932 14 J . F . Ritt , Differentia l equation s fro m th e algebrai c standpoint , 193 2 13 R . L . Moore , Foundation s o f point se t theory , 193 2 12 S . Lefschetz , Topology , 193 0 11 D . Jackson , Th e theor y o f approximation, 193 0 10 A . B . Coble , Algebrai c geometr y an d thet a functions , 192 9 9 G . D . Birkhoff , Dynamica l systems , 192 7 8 L . P . Eisenhart , Non-Riemannia n geometry , 192 2 7 E . T . Bell , Algebrai c arithmetic , 192 7 6 G . C . Evans , Th e logarithmi c potential , discontinuou s Dirichle t an d Neuman n problems , 1927 5.1 G . C . Evans , Functional s an d thei r applications ; selecte d topics , includin g integra l equations, 191 8 5.2 O . Veblen , Analysi s situs , 192 2 (See the AM S catalo g fo r earlie r titles ) This page intentionally left blank Editorial Boar d Joan S . Birma n Susan J . Friedlander , Chai r Stephen Lichtenbau m

1991 Mathematics Subject Classification. Primar y 11E39 , 11E57 , 11E72 ; Secondary 11E88 , 16K20 , 16W10 , 17A75 , 17C40 , 20G15. This monograp h yield s a comprehensiv e expositio n o f th e theor y o f centra l simpl e algebra s with involution , i n relatio n wit h linea r algebrai c groups . I t aim s t o provid e th e algebra-theoreti c foundations fo r much o f the recent work on linear algebraic groups over arbitrary fields . Involution s are viewe d a s twiste d form s o f similarit y classe s o f hermitia n o r bilinea r forms , leadin g t o ne w developments o n th e mode l o f the algebrai c theor y o f quadrati c forms . Beside s classica l groups , phenomena relate d t o trialit y ar e als o discussed , a s wel l a s group s o f type F4 o r Gi arisin g fro m exceptional Jorda n o r compositio n algebras . Severa l result s an d notion s appea r her e fo r th e first time, notabl y th e discriminan t algebr a o f a n algebr a wit h unitar y involutio n an d th e algebra - theoretic counterpar t t o linea r group s o f type D4. For researc h mathematician s an d graduat e student s workin g i n centra l simpl e algebras , alge - braic groups , nonabelia n Galoi s cohomolog y o r Jorda n algebras . Cover desig n b y Herber t Rost .

Library o f Congres s Cataloging-in-Publicatio n Dat a The boo k o f involution s / Max-Alber t Knu s .. . [e t al.]. p. cm . — (Colloquiu m publication s / America n Mathematica l Society , ISS N 0065-925 8 ; v. 44 ) Includes bibliographica l reference s an d index . ISBN 0-8218-0904- 0 (alk . paper ) 1. Linea r algebrai c groups . 2 . Hermitia n forms . 3 . Homolog y theory . 4 . Galoi s theory . I. Knus , Max-Albert . II . Series : Colloquiu m publication s (America n Mathematica l Society ) ; v. 44 . QA179.B66 199 8 98-2220 2 512/.5-dc21 CI P

Copying an d reprinting . Individua l reader s o f this publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , o r multiple reproduction o f any material i n this publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o [email protected] . © 199 8 by the America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 3 0 2 0 1 0 0 9 9 9 8 http://dx.doi.org/10.1090/coll/044 America n Mathematica l Societ y Colloquiu m Publication s Volum e 44

Th e Boo k of Involution s

Max-Alber t Knu s Alexande r Merkurje v Marku s Ros t Jean-Pierr e Tigno l

With a prefac e by J. Tit s Contents

Preface x i Introduction xii i Conventions an d Notation s xvi i Chapter I . Involution s an d Hermitia n Form s 1 § 1. Centra l Simpl e Algebra s 3 l.A. Fundamenta l theorem s 3 l.B. One-side d ideal s i n central simpl e algebra s 5 l.C. Severi-Braue r varietie s 9 §2. Involution s 1 3 2.A. Involution s o f the firs t kin d 1 3 2.B. Involution s o f the secon d kin d 2 0 2.C. Example s 2 3 2.D. Li e and Jorda n structure s 2 7 §3. Existenc e o f Involutions 3 1 3.A. Existenc e o f involutions o f the firs t kin d 3 2 3.B. Existenc e o f involutions o f the secon d kin d 3 6 §4. Hermitia n Form s 4 1 4.A. Adjoin t involution s 4 2 4.B. Extensio n o f involutions an d transfe r 4 5 §5. Quadrati c Form s 5 3 5.A. Standar d identification s 5 3 5.B. Quadrati c pair s 5 5 Exercises 6 3 Notes 6 7 Chapter II . Invariant s o f Involutions 7 1 §6. Th e Inde x 7 1 6.A. Isotropi c ideal s 7 2 6.B. Hyperboli c involution s 7 4 6.C. Odd-degre e extension s 7 9 §7. Th e Discriminan t 8 0 7. A. Th e discriminan t o f orthogonal involution s 8 0 7.B. Th e discriminan t o f quadratic pair s 8 3 §8. Th e Cliffor d Algebr a 8 7 8.A. Th e spli t cas e 8 7 8.B. Definitio n o f the Cliffor d algebr a 9 1 8.C. Li e algebra structures 9 5

V vi CONTENT S

8.D. Th e cente r o f the Cliffor d algebr a 9 9 8.E. Th e Cliffor d algebr a o f a hyperbolic quadrati c pai r 10 6 §9. Th e Cliffor d Bimodul e 10 7 9.A. Th e spli t cas e 10 7 9.B. Definitio n o f the Cliffor d bimodul e 10 8 9.C. Th e fundamenta l relation s 11 3 §10. Th e Discriminan t Algebr a 11 4 10.A. The A-power s o f a central simpl e algebr a 11 5 10.B. The canonica l involutio n 11 7 10.C. The canonica l quadrati c pai r 11 9 10.D. Induced involution s o n A-power s 12 3 10.E. Definitio n o f the discriminan t algebr a 12 7 10.F. Th e Braue r clas s o f the discriminan t algebr a 13 0 §11. Trac e For m Invariant s 13 2 11.A. Involutions o f the first kin d 13 3 ll.B. Involution s o f the secon d kin d 13 8 Exercises 14 5 Notes 14 8

Chapter III . Similitude s 15 3 §12. Genera l Propertie s 15 3 12.A. The spli t cas e 15 3 12.B. Similitude s o f algebras with involutio n 15 8 12.C. Proper similitude s 16 3 12.D. Functorial propertie s 16 8 §13. Quadrati c Pair s 17 2 13.A. Relation wit h th e Cliffor d structure s 17 2 13.B. Cliffor d group s 17 7 13.C. Multipliers o f similitudes 19 0 § 14. Unitar y Involution s 19 3 14.A. Odd degre e 19 3 14.B. Even degre e 19 4 14.C. Relation wit h th e discriminan t algebr a 19 5 Exercises 19 9 Notes 20 3

Chapter IV . Algebra s o f Degree Fou r 20 5 §15. Exceptiona l Isomorphism s 20 5 15.A. Bi = d 20 7 15.B. A\ = D 2 21 0 15.C. B 2 = C 2 21 6 15.D. A 3 = Ds 22 0 §16. Biquaternio n Algebra s 23 3 16.A. Albert form s 23 5 16.B. Albert form s an d symplecti c involution s 23 7 16.C. Albert form s an d orthogona l involution s 24 5 §17. Whitehea d Group s 25 3 17.A. SKi o f biquaternion algebra s 25 3 17.B. Algebras with involutio n 26 6 CONTENTS vi i

Exercises 27 0 Notes 27 4

Chapter V . Algebra s o f Degree Three 27 9 §18. Etal e an d Galoi s Algebra s 27 9 18.A. Etale algebra s 28 0 18.B. Galoi s algebras 28 7 18.C. Cubic etale algebra s 29 6 §19. Centra l Simpl e Algebra s o f Degree Three 30 2 19.A. Cyclic algebra s 30 2 19.B. Classificatio n o f involutions o f the secon d kin d 30 4 19.C. Etale subalgebra s 30 7 Exercises 31 9 Notes 32 1

Chapter VI . Algebrai c Group s 32 3 §20. Hop f Algebra s and Grou p Scheme s 32 4 20.A. Group scheme s 32 5 §21. Th e Li e Algebra an d Smoothnes s 33 4 21.A. The Li e algebra o f a group schem e 33 4 §22. Facto r Group s 33 9 22.A. Group schem e homomorphisms 33 9 §23. Automorphis m Group s o f Algebras 34 4 23.A. Involutions 34 5 23.B. Quadratic pair s 35 0 §24. Roo t System s 35 2 24.A. Classification o f irreducible roo t system s 35 4 §25. Spli t Semisimpl e Group s 35 5 25.A. Simple spli t group s o f type A, B, C, D, F , an d G 35 7 25.B. Automorphisms o f split semisimpl e group s 36 0 §26. Semisimpl e Group s ove r a n Arbitrar y Fiel d 36 0 26.A. Basic classificatio n result s 36 4 26.B. Algebraic group s o f small dimensio n 37 3 § 27. Tit s Algebra s o f Semisimple Group s 37 5 27.A. Definition o f the Tits algebra s 37 6 27.B. Simpl y connecte d classica l group s 37 8 27.C. Quasisplit group s 37 9 Exercises 38 0 Notes 38 1

Chapter VII . Galoi s Cohomoiog y 38 3 §28. Cohomoiog y o f Profinite Group s 38 3 28.A. Cohomoiogy set s 38 3 28.B. Cohomoiog y sequence s 38 5 28.C. Twisting 38 7 28.D. Torsors 38 8 §29. Galoi s Cohomoiog y o f Algebraic Group s 39 1 29.A. Hilbert's Theore m 9 0 and Shapiro' s lemm a 39 2 29.B. Classificatio n o f algebras 39 5 viii CONTENT S

29.C. Algebras with a distinguished subalgebr a 39 8 29.D. Algebras with involutio n 39 9 29.E. Quadrati c space s 40 6 29.F. Quadrati c pair s 40 8 §30. Galoi s Cohomolog y o f Roots o f Unity 41 3 30.A. Cyclic algebra s 41 4 30.B. Twisted coefficient s 41 6 30.C. Cohomologica l invariant s o f algebras o f degree three ... . 42 0 §31. Cohomologica l Invariant s 42 3 31.A. Connecting homomorphism s 42 3 3l.B. Cohomologica l invariant s o f algebraic group s 42 9 Exercises 44 2 Notes 44 6

Chapter VIII . Compositio n an d Trialit y 45 1 §32. Nonassociativ e Algebra s 45 1 §33. Compositio n Algebra s 45 4 33.A. Multiplicative quadrati c form s 45 4 33.B. Unital compositio n algebra s 45 6 33.C. Hurwitz algebra s 45 8 33.D. Composition algebra s without identit y 46 2 §34. Symmetri c Composition s 46 3 34.A. Para-Hurwitz algebra s 46 4 34.B. Petersson algebra s 46 6 34.C. Cubic separabl e alternativ e algebra s 46 9 34.D. Alternative algebra s with unitar y involution s 47 7 34.E. Cohomologica l invariant s o f symmetric composition s ... . 48 0 §35. Cliffor d Algebra s an d Trialit y 48 1 35.A. The Cliffor d algebr a 48 1 35.B. Similitude s an d trialit y 48 3 35.C. The grou p Spi n an d trialit y 48 5 §36. Twiste d Composition s 48 9 36.A. Multipliers o f similitudes o f twisted composition s 49 3 36.B. Cycli c compositions 49 5 36.C. Twisted Hurwit z composition s 49 9 36.D. Twisted composition s o f type A' 2 50 4 36.E. Th e dimensio n 2 case 50 6 Exercises 50 7 Notes 50 9

Chapter IX . Cubi c Jordan Algebra s 51 3 §37. Jorda n Algebra s 51 3 37.A. Jordan algebra s o f quadratic form s 51 4 37.B. Jordan algebra s o f classical type 51 5 37.C. Freudenthal algebra s 51 6 §38. Cubi c Jordan Algebra s 51 9 38.A. The Springe r decompositio n 52 2 §39. Th e Tits Constructio n 52 4 39.A. Symmetric composition s an d Tit s construction s 52 8 CONTENTS i x

39.B. Automorphisms o f Tits construction s 52 9 §40. Cohomologica l Invariant s 53 3 40.A. Invariants o f twisted composition s 53 9 §41. Exceptiona l Simpl e Li e Algebras 53 9 Exercises 54 0 Notes 54 2 Chapter X . Trialitaria n Centra l Simpl e Algebra s 54 7 §42. Algebra s o f Degree 8 54 7 42.A. Trialitarian triple s 54 7 42.B. Decomposabl e involution s 55 0 §43. Trialitaria n Algebra s 55 2 43.A. A definition an d som e properties 55 2 43.B. Quaternioni c trialitaria n algebra s 55 5 43.C. Trialitarian algebra s o f type 2 D^ 55 8 §44. Classificatio n o f Algebras an d Group s o f Type D4 56 0 44.A. Groups o f trialitarian typ e D4 56 1 44.B. The Cliffor d invarian t 56 3 §45. Li e Algebras and Trialit y 56 5 45.A. Local triality 56 7 45.B. Derivation s o f twisted composition s 56 9 45.C. Li e algebras an d trialitarian algebra s 56 9 Exercise 57 1 Notes 57 1 Bibliography 57 3 Index 58 5 Notation 58 9 This page intentionally left blank Preface

Quatre de s meilleur s algebriste s d'aujourd'hu i (j'aimerai s dire , comm e jadis ,

xi xii PREFAC E

fasse ic i figure d e parent pauvre) , les algebres de composition, ..., regoiven t autan t d'attention. On l' a compris , c e Livr e es t tou t a l a foi s u n livr e d e lectur e passionnan t e t un ouvrag e d e referenc e d'un e extrem e richesse . J e sui s reconnaissant au x auteur s de I'honneu r qu'il s m'on t fai t e n m e demandan t d e l e prefacer , e t plu s encor e d e m'avoir permi s d e l e decouvrir e t d'apprendr e a m'en servir .

Jacques Tit s Introduction

For u s an involutio n i s an anti-automorphis m o f order tw o o f an algebra . Th e most elementary example is the transpose fo r matrix algebras. A more complicate d example o f a n algebr a ove r Q admittin g a n involutio n i s th e multiplicatio n alge - bra o f a Riemann surfac e (se e the note s a t th e en d o f Chapter I fo r mor e details) . The centra l proble m here , to giv e necessary an d sufficien t condition s o n a divisio n algebra ove r Q t o b e a multiplicatio n algebra , wa s completel y solve d b y Alber t (1934/35). T o achiev e this , Alber t develope d a theor y o f centra l simpl e algebra s with involution , base d o n the theory o f simple algebras initiated a fe w years earlie r by Brauer, Noether , an d als o Albert an d Hasse , and gav e a complete classificatio n over Q . Thi s i s the historica l origi n o f our subject , howeve r ou r motivatio n ha s a different source . Th e basi c object s ar e stil l centra l simpl e algebras , i.e. , "forms " of matri x algebras . A s observe d b y Wei l (1960) , centra l simpl e algebra s wit h in - volution occu r i n relatio n t o classica l algebrai c simpl e adjoin t groups : connecte d components o f automorphism group s o f central simple algebras with involution ar e such groups (wit h the exception o f a quaternion algebr a with an orthogonal involu - tion, wher e the connecte d componen t o f the automorphis m grou p i s a torus), and , in thei r turn , suc h group s ar e connecte d component s o f automorphis m group s o f central simpl e algebra s with involution . Even i f thi s i s mainl y a boo k o n algebras , th e correspondenc e betwee n alge - bras an d group s i s a constant leitmotiv . Propertie s o f the algebra s ar e reflecte d i n properties o f the group s an d o f relate d structures , suc h a s Dynki n diagrams , an d vice versa . Fo r exampl e w e associat e certai n algebra s t o algebra s wit h involutio n in a functorial way , such a s the Cliffor d algebr a (fo r orthogona l involutions ) o r th e A-powers and the discriminant algebr a (fo r unitary involutions) . Thes e algebras are exactly the "Tit s algebras," define d b y Tits (1971 ) in terms o f irreducible represen - tations o f th e groups . Anothe r exampl e i s algebrai c triality , whic h i s historicall y related with groups o f type D4 (E. Cartan) an d whos e "algebra " counterpar t is , so far a s w e know, systematicall y approache d her e fo r the first time . In the first chapte r w e recall basic properties o f central simpl e algebras and in - volutions. A s a rule for the whole book, without howeve r going to the utmost limit , we try to allow base fields of characteristic 2 as well as those of other characteristic . Involutions are divided up into orthogonal, symplectic and unitary types. A central idea o f this chapter i s to interpret involution s in terms o f hermitian form s ove r skew fields. Quadrati c pairs , introduced a t th e end o f the chapter , giv e a correspondin g interpretation fo r quadrati c form s i n characteristic 2 . In Chapter II we define several invariants o f involutions; the index is defined fo r every type o f involution. Fo r quadratic pair s additiona l invariant s ar e the discrim - inant, th e (even ) Cliffor d algebr a an d th e Cliffor d module ; fo r unitar y involution s we introduce th e discriminan t algebra . Th e definitio n o f the discriminan t algebr a

xiii XIV INTRODUCTION

is prepared fo r b y the construction o f the A-power s o f a central simple algebra. Th e last par t o f this chapte r i s devoted t o trace form s o n algebras , whic h represen t a n important tool for recent results discussed in later parts o f the book. Ou r method o f definition i s based o n scalar extension : afte r specifyin g th e definition s "rationally " (i.e., ove r a n arbitrar y bas e field), the mai n propertie s ar e prove n b y working ove r a splittin g field. Thi s i s i n contras t t o Galoi s descent , wher e construction s ove r a separable closur e ar e show n t o b e invarian t unde r th e Galoi s grou p an d therefor e are defined ove r the base field. A main sourc e o f inspiration fo r Chapters I and II is the pape r [291 ] o f Tits o n "Forme s quadratiques, groupe s orthogonau x e t algebre s de Clifford. " In Chapte r II I w e investigate the automorphis m group s o f central simpl e alge - bras with involutions. Inne r automorphism s ar e induced b y elements which w e call similitudes. Thes e automorphis m group s ar e twisted form s o f the classica l projec - tive orthogonal , symplecti c an d unitar y groups . Afte r provin g result s whic h hol d for al l types o f involutions, w e focus o n orthogonal an d unitar y involutions , wher e additional informatio n ca n b e derive d fro m th e invariant s define d i n Chapte r II . The nex t tw o chapters ar e devote d t o algebra s o f lo w degree . Ther e exis t certai n isomorphisms amon g classica l groups , know n a s exceptiona l isomorphisms . Fro m the algebr a point o f view, this i s explained i n the first par t o f Chapter I V b y prop- erties o f the Cliffor d algebr a o f orthogonal involution s o n algebras o f degree 3, 4, 5 and 6 . I n the secon d par t w e focus o n tensor product s o f two quaternion algebras , which we call biquaternion algebras . Thes e algebras have many interesting proper - ties, whic h coul d b e th e subjec t o f a monograp h o f it s own . Thi s ide a wa s a t th e origin o f our project . Algebras with unitary involutions are also of interest fo r odd degrees, the lowest case being degree 3. Fro m the group point o f view algebras with unitary involution s of degree 3 are o f type A 2. Chapte r V gives a new presentation o f results o f Albert and a complet e classificatio n o f thes e algebras . I n preparatio n fo r this , w e recal l general results o n etale an d Galoi s algebras . The ai m o f Chapte r V I i s t o giv e th e classificatio n o f semisimpl e algebrai c groups ove r arbitrar y fields. W e us e th e functoria l approac h t o algebrai c groups , although w e quote without proo f som e basic results o n algebrai c group s ove r alge - braically close d fields. I n the central section w e describe i n detail Weil's correspon - dence [310 ] betwee n centra l simpl e algebra s wit h involutio n an d classica l groups . Exceptional isomorphism s ar e reviewe d agai n i n terms o f this correspondence . I n the las t sectio n w e define Tit s algebras o f semisimple group s and giv e explicit con - structions o f them i n classical cases . The them e o f Chapter VI I i s Galoi s cohomology . W e introduce th e formalis m and describ e many examples . Previou s results ar e reinterpreted i n this setting an d cohomological invariant s ar e discussed . Mos t o f the technique s develope d her e ar e also needed fo r the followin g chapters .

The last thre e chapters are dedicated to the exceptional group s o f type G 2, -F 4 and t o D4, which , i n vie w o f triality , i s als o exceptional . I n th e Wei l correspon - dence, octonion algebras play the algebra role for G 2 an d exceptional simple Jordan algebras the algebr a rol e fo r F4. Octonion algebra s ar e a n importan t clas s o f compositio n algebra s an d Chap - ter VII I give s a n extensiv e discussio n o f composition algebras . O f specia l interes t from th e grou p poin t o f view ar e "symmetric " compositions . I n dimensio n 8 these are o f two types, corresponding t o algebraic groups o f type A 2 o r type G 2. Trialit y INTRODUCTION xv is define d throug h th e Cliffor d algebr a o f symmetri c 8-dimensiona l compositions . As a step towards exceptional simple Jordan algebras , w e introduce twisted compo - sitions, which are defined ove r cubic etale algebras. Thi s generalizes a constructio n of Springer . Th e correspondin g grou p o f automorphism s i n th e spli t cas e i s th e semidirect produc t Spin 8 x#3. In Chapter I X we describe differen t construction s o f exceptional simple Jorda n algebras, du e t o Freudenthal , Springe r an d Tit s (th e algebr a side ) an d giv e inter - pretations fro m th e algebrai c grou p side . Th e Springe r constructio n arise s fro m twisted compositions , define d i n Chapte r VIII , an d basi c ingredient s o f Tit s con - structions are algebras o f degree 3 with unitary involutions , studied i n Chapter III . We conclude this chapter b y definin g cohomologica l invariant s fo r exceptional sim - ple Jordan algebras . The las t chapte r deal s wit h trialitaria n action s o n simpl e adjoin t group s o f type D4. T o complet e Weil' s progra m fo r oute r form s o f D 4 ( a cas e no t treate d by Weil) , w e introduc e a ne w notion , whic h w e cal l a trialitaria n algebra . Th e underlying structur e i s a centra l simpl e algebr a wit h a n orthogona l involution , o f degree 8 ove r a cubi c etal e algebra . Th e trialitaria n conditio n relate s the algebr a to it s Cliffor d algebra . Trialitaria n algebra s als o occu r i n th e constructio n o f Li e algebras o f type D4. Som e indications i n this direction are given in the last section . Exercises an d note s ca n b e foun d a t th e en d o f eac h chapter . Omitte d proof s sometimes occu r a s exercises . Moreove r w e include d a s exercise s som e result s w e like, but whic h we did not wish to develop fully. I n the notes we wanted to give com- plements an d t o loo k a t som e result s fro m a historica l perspective . W e have trie d our bes t t o b e useful ; w e cannot, however , giv e strong guarantee s o f completenes s or eve n fairness . This boo k i s the achievemen t o f a joint (an d ver y exciting ) effor t o f fou r ver y different people . W e are aware that th e result i s still quite heterogeneous; however , we flatter ourselve s that the differences i n style may be viewed as a positive feature . Our work started out as an attempt to understand Tits' definition o f the Cliffor d algebra o f a generalized quadratic form , an d ende d up including many other topic s to whic h Tit s mad e fundamenta l contributions , suc h a s linea r algebrai c groups , exceptional algebras , triality , .. . No t onl y wa s Jacque s Tit s a constan t sourc e o f inspiration throug h hi s work , bu t h e als o had a direc t persona l influence , notabl y through hi s threa t — earl y i n th e inceptio n o f ou r projec t — t o spea k evi l o f our wor k i f i t di d no t includ e th e characteristi c 2 case. Finall y h e als o agree d t o bestow hi s blessings o n our boo k sou s forme d e preface. Fo r al l that w e thank hi m wholeheartedly. This boo k coul d no t hav e bee n writte n withou t th e hel p an d th e encourage - ment o f man y friends . The y ar e to o numerou s t o b e liste d her e individually , bu t we hope they wil l recognize themselves and find her e our warmest thanks . Richar d Elman deserve s a specia l mentio n fo r hi s commen t tha t th e mos t usefu l boo k i s not th e on e to whic h nothin g ca n b e added , bu t th e on e whic h i s published. Thi s no-nonsense statemen t helpe d u s se t limit s to ou r endeavor . W e were fortunate t o get usefu l advic e o n variou s point s o f th e expositio n fro m Ottma r Loos , Antoni o Paques, Parimala, Miche l Racine, David Saltman, Jean-Pierre Serr e and Sridharan . We thank al l o f the m fo r lendin g helpin g hand s a t th e righ t time . A numbe r o f people wer e nice enough to read an d commen t o n drafts o f parts o f this book: Ev a Bayer-Fluckiger, Vladimi r Chernousov , Ingri d Dejaiffe , Albert o Elduque , Darrel l Haile, Lu c Haine , Pa t Morandi , Holge r Petersson , Ahme d Serhir , Ton y Springer , xvi INTRODUCTIO N

Paul Swet s an d Olive r Villa . W e kno w al l o f the m ha d bette r thing s t o do , an d we are grateful. Ski p Garibaldi and Adrian Wadsworth actuall y summoned enoug h grim self-discipline to read a draft o f the whole book, detecting many shortcomings , making shrew d comment s o n th e organizatio n o f the boo k an d polishin g ou r bro - ken English . Eac h deserve s a medal . However , ou r capacit y fo r makin g mistake s certainly exceed s ou r friends ' sagacity . W e shall gratefull y welcom e an y commen t or correction . Tignol had the privilege to give a series of lectures on "Centra l simple algebras, involutions an d quadrati c forms " i n April 199 3 at th e National Taiwa n University . He wants to thank Ming-chan g Kan g an d th e Nationa l Researc h Counci l o f Chin a for thi s opportunit y t o tes t hig h dose s o f involution s o n a ver y patien t audience , and Eng-Tjio e Ta n fo r makin g hi s stay i n Taiwan a most pleasan t experience . Th e lecture note s fro m thi s cras h cours e serve d a s a blueprin t fo r th e firs t chapter s o f this book . Our projec t immensel y benefite d b y reciproca l visit s amon g th e authors . W e should lik e t o mentio n wit h particula r gratitud e Merkurjev' s sta y i n Louvain-la - Neuve i n 1993 , with suppor t fro m th e Fond s d e Developpemen t Scientifiqu e an d the Institu t d e Mathematiqu e Pur e e t Applique e o f th e Universit e catholiqu e d e Louvain, Rost' s stay s i n Zuric h a t th e Mathematic s Researc h Institut e (FIM ) i n winter 1995-9 6 an d a t th e Mathematic s Departmen t o f th e ET H Zuric h fo r th e winter semeste r o f 1996-97 , an d Tignol' s sta y a t th e Mathematic s Departmen t o f the ET H Zuric h fo r th e winte r semeste r o f 1995-96 , bot h wit h suppor t fro m th e Eidgenossische Technisch e Hochschule . Moreover , Merkurje v gratefull y acknowl - edges suppor t fro m th e Alexande r vo n Humbold t foundatio n an d th e hospitalit y of the Bielefel d universit y fo r th e yea r 1995-96 , an d Tigno l i s grateful t o th e Na - tional Fun d fo r Scientifi c Researc h o f Belgiu m an d t o th e Europea n Commissio n for partia l suppor t unde r th e "credi t au x chercheurs " an d th e TM R programme s respectively (contrac t ERB-FMRX-CT97-0107) . The fou r author s enthusiasticall y than k Herber t Ros t (Markus ' father ) fo r th e design o f the cover page, in particular fo r hi s wonderful an d colorful renditio n o f the Dynkin diagra m D4. The y als o giv e specia l prais e t o Serge i Gelfand , Directo r o f Acquisitions o f the American Mathematical Society, for his helpfulness an d patienc e in taking car e o f all their wishe s fo r th e publication . Conventions an d Notation s

Maps. Th e imag e o f a n elemen t x unde r a ma p / i s generally denote d /(#) ; the notatio n x? i s also use d however , notabl y fo r homomorphism s o f lef t modules . In that case , we also use the right-hand rule for mapping composition; fo r the image ofxGl unde r th e composit e map X — > Y— > Z w e set eithe r g o f(x) or x^9 an d the composit e i s thus eithe r g o / o r fg. As a general rule , modul e homomorphism s ar e written o n the opposit e sid e o f the scalars . (Righ t module s ar e usuall y preferred. ) Thus , i f M i s a module ove r a ring R, i t i s also a module (o n the opposite side) ove r End#(M), an d the i?-modul e structure define s a natural homomorphism :

R-^EndEndRiM){M). Note therefor e tha t i f S C End#(M ) i s a subring , an d i f w e endo w M wit h it s natural 5-modul e structure , the n Ends(M ) i s the opposite o f the centralize r o f S in Endfl(M) :

P Ends(M) = (C EndR{M)S)° . Of course, i f R i s commutative, ever y right R- module MR ma y also be regarded as a left R- module RM, an d every endomorphism o f MR als o is an endomorphism o f RM. Note howeve r tha t wit h th e conventio n above , th e canonica l ma p End#(M# )— > Endft(flM) i s an anti-isomorphism . The characteristi c polynomia l an d it s coefficients . Le t F denot e a n ar - bitrary field. Th e characteristi c polynomia l o f a matri x m G Mn{F) (o r a n endo - morphism m o f an n-dimensiona l F-vecto r space ) i s denoted

n n 1 n 2 n (0.1) P m(X) =X - s 1(m)X - + s 2(m)X - - • • • + (-l) 5n(m).

The trace an d determinan t o f m ar e denoted tr(ra ) an d det(m ) :

tr(ra) = si(ra) , det(m ) = s n(m). We recal l the followin g relation s betwee n coefficient s o f the characteristi c polyno - mial:

f 2 2 (0.2) Proposition . Form, m G Mn(F), we have si(ra) — s\(m ) — 2s2(m) and

f / Si(m)si(m ) — si(mm) = s 2{m + m!) - s 2{m) - s 2(m ).

Proof: I t suffice s t o prov e thes e relation s fo r generi c matrice s m = (xij)i

an( an algebrai c closur e o f Q(xij | 1 < i,j < n)) , w e hav e Si(ra ) — J2i

The trace an d nor m o f £ are denoted T L/F{£) an d N L/F(£) (o r simply T(£), N(£)):

TL/F(e) = s1(e), N L/F(e) = Sn (e). We als o denot e

(0.3) S L/F(£) = S(£) = s 2(t). The characteristic polynomial i s also used to define a generic polynomial fo r centra l simple algebras , calle d th e reduced characteristic polynomial: se e (1.6) . General - izations to certain nonassociativ e algebra s ar e give n i n §32 . Bilinear forms . A bilinear for m b: V xV -^ F on a, finite dimensional vecto r space V ove r a n arbitrar y field F i s calle d symmetric i f b(x, y) — b(y, x) fo r al l x, y G V, skew-symmetric i f b(x,y) = —b(y,x) fo r al l x, y £ V an d alternating if 6(x,x) = 0 fo r al l x G V. Thus , th e notion s o f skew-symmetri c an d alternatin g (resp. symmetric) for m coincid e i f char F ^ 2 (resp. charF = 2) . Alternatin g form s are skew-symmetric i n ever y characteristic . If 6 is a symmetric o r alternating bilinea r for m o n a (finit e dimensional ) vecto r space V , the induce d ma p

6: V-> V * =Kom F(V,F) is defined b y b(x)(y) — b(x,y) fo r x, 2 / G V. Th e bilinea r for m 6 is nonsingular (o r regular, or nondegenerate) i f 6 is bijective. (I t suffice s t o require that b be injective , i.e., that th e onl y vecto r x G V suc h tha t b(x,y) = 0 fo r al l ? / G V i s x = 0 , sinc e we ar e dealin g wit h finite dimensiona l vecto r space s ove r fields.) Alternately , b is nonsingular i f an d onl y i f the determinan t o f it s Gra m matri x wit h respec t t o a n arbitrary basi s o f V i s nonzero:

det(6(ei,eJ-))1

(ai,...,an)((a;i,...,xn),(2/i,...,2/n)) = aix^yi H ha nxn?/n.

We also defin e th e n-fold Pfister bilinear form ((a\, ..., a n)) b y

((ai,...,an)) = (1,-ai) ® •••

If b: V x V — > F i s a symmetri c bilinea r form , w e denot e b y <#> : V— > F th e associated quadrati c map , define d b y qb{x) = b(x,x) fo r xeV . Quadratic forms . I f q : V — > F i s a quadrati c ma p o n a finite dimensiona l vector spac e ove r an arbitrary field F, th e associated symmetri c bilinear for m b q is called th e polar o f g ; it i s defined b y

bq(x, y) = q(x + y) - q{x) - q(y) fo r x, y G V, hence b q(x,x) = 2q(x) fo r al l x € V . Thus , the quadratic ma p q^ associate d t o b q is qb q — 2g . Similarly , fo r ever y symmetric bilinea r for m 6 on V , w e have b qh = 26 . 1 Let V - = {x G V | b q(x,y) = 0 for y G V}. Th e quadrati c ma p q i s calle d nonsingular (o r regular, o r nondegenerate) i f either F- 1 = {0 } or di m V-1 = 1 and ^(V-1-) 7 ^ {0}. Th e latter cas e occurs only i f charF = 2 and V i s odd-dimensional . Equivalently, a quadrati c for m o f dimensio n n i s nonsingula r i f an d onl y i f i t i s equivalent ove r a n algebrai c closur e t o Y17=i x2i-i#2i (i f n i s even ) o r t o XQ + Ylili x 2i-iX2i (i f n i s odd). The determinant an d th e discriminant o f a nonsingula r quadrati c for m g o f dimension n ove r a field F ar e defined a s follows : le t M b e a matrix representin g q in the sens e tha t q{X) = X-M-X l

t where X = (#i,... , xn) an d denote s the transpose o f matrices; the condition that q is nonsingular implie s that M + M l i s invertible i f n i s eve n o r charF ^ 2 , an d has rank n — 1 if n i s odd and char F = 2 . Th e matrix M i s uniquely determined b y q up to the addition o f a matrix o f the form N — N*; therefore, M + M l i s uniquely determined b y q. If char F ^ 2 we set

x2 x x2 det 9 = det(|(M 4 - M*)) • F G F /F and

discg = (-l) n(n~1)/2detg G F x/Fx2. Thus, th e determinant (resp . the discriminant) o f a quadratic for m i s the determi - nant (resp . the discriminant ) o f its polar for m divide d b y 2 n. If char F = 2 and n i s odd w e set (0.4) de t ^ = disc q = q{y) • Fx2 G F x/Fx2 where y G Fn i s a nonzero vecto r suc h that ( M - f M* ) • y = 0 . Suc h a vector y i s uniquely determine d u p to a scalar factor , sinc e M + M * has rank n — 1, hence th e definition abov e does not depen d o n the choic e o f y. If char F = 2 and n i s even w e set

1 det9 = s 2((M + M')- ^) + P(^) e F/p(F) and

disc 9 = Vlin^ll + de t g G F/p(F) xx CONVENTION S AN D NOTATION S where m = n/2 an d p(F) = {x + x 2 \ x G F}. (Mor e generally , fo r fields o f characteristic p ^ 0 , p i s define d a s p(x) = x - f xp, x G F.) Th e followin g lemm a shows that th e definitio n o f det q does not depen d o n the choic e o f M:

(0.5) Lemma . Suppose cha r F = 2 . Let M,N G Mn(F) andW = M + M*. IfW is invertible, then

r 1 2 s2(W-\M + N + TV') ) = s 2(W - Af) + ^(W^N) + (s^W^N)) . Proof: Th e secon d relatio n i n (0.2 ) yield s

1 s2{W- M + W^iN + N*)) = r_1 1 -1 ^(W^M) + s 2(W (JV + N )) + s^W^M)*! (W ^ + #*) ) + 5i(W- 1MW~1(A^ + 7V t)). In orde r t o prov e the lemma , w e show below : 1 t l 2 (0.6) s 2(W~ (N + N )) = (s!(W- N)) 1 1 (0.7) s 1(W~ M)s1 (W-\N + N )) = 0 (0.8) si^-^W-^-Z V + iV*) ) =si(W" 1^). Since a matrix and its transpose have the same characteristic polynomial, the traces of W~ lN an d (W^Nf = ^W' 1 ar e the same , henc e siiW^N*) = si^W"1) = ^(W^iV) . Therefore, s i (W^iV + iV*) ) = 0 , and (0.7 ) follows . Similarly, w e have sxiW^MW^N1) = s^NW^M'W'1) = s^W^M^^N), hence the lef t sid e o f (0.8 ) i s l l 1 Sl{W- MW- N) + siiW^AftW^N) = ^(r^M + Af'JW ^). Since M 4 - M* = W , (0.8 ) follows . The secon d relatio n i n (0.2 ) show s that th e lef t sid e o f (0.6 ) i s 1 1 t 1 1 £ 1 s2(W~ N) + s 2(W- N ) + 5i(VT- iV)51(W 7V ) + ^(T^AW" ^). Since W~ lN an d W^W^JVyW ( = W~ lNl) hav e the same characteristic poly - 1 x 1 nomial, w e hav e s i(W~ N) — Si{W~ N ^ fo r i = 1,2 , henc e th e first tw o term s cancel an d th e thir d i s equal t o si(W~ 1N)2. I n orde r t o prov e (0.6) , i t therefor e suffices t o sho w 1 1 t s1(W~ NW- N ) = 0. Since W = M + M\ w e have W"1 = W~ lMW~l + W^M'W1, henc e and (0.6 ) follow s i f w e sho w that th e tw o term s o n the righ t sid e ar e equal . Sinc e Wl = W w e have (W^MW^NW^N 1)1 = NW^^W^M^'1, henc e

= siiW^MtW^NW^N*). • CONVENTIONS AN D NOTATION S xx i

Quadratic forms are called equivalent i f they can be transformed int o each other by invert ible linear changes of variables. Th e various quadratic forms representing a quadratic map with respect to various bases are thus equivalent. I t i s easily verifie d that th e determinant de t q (hence also the discriminant dis c q) is an invariant o f the equivalence class o f the quadratic for m q\ the determinant an d the discriminant ar e therefore als o defined fo r quadratic maps. Th e discriminant o f a quadratic for m (o r map) o f even dimension i n characteristic 2 is also known a s the pseudodiscriminant or the Arf invariant o f the form. Se e §8.D for the relation between the discriminan t and th e eve n Cliffor d algebra .

Let ai , ... , a n £ F . I f charF ^ 2 w e denot e b y (ai,.. . ,an) th e diagona l quadratic for m 2 (<*i,..., an) = OLIX\ H + a nx n which i s the quadrati c for m associate d t o th e bilinea r for m (ai,... , a n). W e als o define th e n-fold Pfister quadratic form ((ai,... , an)) b y

«ai,..., a„» = (1 , -ax) • • •<8 > (1, -an) where ® = ®F is the tenso r produc t ove r F. I f charF = 2 , the quadrati c form s [a 1,0^2] and [ai ] ar e define d b y 2 [ai,a2] = &iXf + XiX2 + a 2Xl an d [a\] = a\X , and the n-fold Pfister quadratic form ((ai,... , an]] b y

((ai,...,an]] = ((ai,...,a n_i» 0 [l,a n]. (See Baeza [28 , p. 5 ] or Knus [157 , p. 50 ] for the definitio n o f the tensor produc t o f a bilinea r for m an d a quadratic form. ) Fo r instance ,

((aua2]} = {x\ + x\x2 + a2xl) + a x(xl + x 3x4 +a 2xl). This page intentionally left blank Bibliography

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Index absolute Galoi s group , 27 9 Cayley-Dickson process , 45 9 absolutely simpl e group , 36 4 central algebra , 45 1 additive group , 32 6 central isogeny , 34 2 adjoint anti-automorphism , 1 central simpl e L-algebra , 55 2 adjoint group , 36 4 central simpl e algebra , 3 adjoint involution , 2 , 24 , 4 2 centralizer, 5 adjoint representation , 34 3 centroid, 45 1 adjoint semisimpl e group , 35 6 character, 32 7 admissible pair , 52 5 characteristic polynomial , x v afrine grou p scheme , 32 5 Clifford algebra , 87 , 9 2 Albert algebra , 51 7 Clifford bimodule , 11 0 Albert form , 23 5 Clifford grou p o f a quadratic pair , 178 , 35 1 Albert quadrati c space , 23 5 Clifford invariant , 481 , 563 Albert's theorem , 236 , 30 8 closed embedding , 32 7 algebraic group , 33 8 closed subgroup , 32 7 algebraic grou p scheme , 32 6 cohomological invarian t o f algebrai c group , algebraic torus , 33 3 429 Allen invariant , 57 0 cohomologous cocycles , 38 4 alternating bilinea r form , xv i commutative grou p scheme , 32 5 alternating hermitia n form , 4 2 comodule, 34 3 alternative algebra , 45 1 comorphism, 32 5 annihilator o f a n ideal , 8 composition algebra , 45 6 1 Arason invariant , 43 6 composition algebr a o f type Aquaternion algebra , 46 4 isotropic algebr a wit h involution , 7 3 para-unit, 46 6 isotropic ideal , 72 , 7 3 Peirce decomposition , 51 6 isotropic involution , 7 3 Petersson algebra , 46 6 pfaffian, 2 0 Jordan algebra , 51 3 pfaffian adjoint , 27 5 Jordan algebr a o f a quadratic form , 51 4 pfaffian characteristi c polynomial , 1 9 Jordan product , 2 7 pfaffian norm , 1 9 pfaffian trace , 1 9 kernel, 328 , 38 5 Pfister bilinea r form , xv i Kummer sequence , 41 3 Pfister for m o f an involution , 30 4 Pfister quadrati c form , xi x left-invariant derivation , 33 4 pointed quadrati c space , 52 0 Lie algebr a o f a group scheme , 33 4 polar o f a cubi c form , 47 4 Lie algebra o f Jordan type , 56 9 Lie algebra o f type D4 , 56 9 polar o f a quadratic form , xvi i Lie invarian t function , 45 2 principal homogeneou s set , 288 , 38 8 Lie product, 2 7 product algebra , 41 6 local triality, 56 7 product o f group schemes , 32 8 loop i n algebrai c group , 43 2 profinite group , 38 3 projective genera l linea r group , 34 5 maximal subtorus , 35 6 projective orthogona l group , 35 0 minimal dominan t weight , 35 4 projective similitude , 15 9 minimal Tit s algebra , 37 7 projective symplecti c group , 34 7 Moore determinant , 23 9 projective unitar y group , 34 6 multiplication algebra , 45 1 proper isometry , 154 , 16 4 multiplicative group , 32 6 proper similitude , 154 , 157 , 164 , 16 7 multiplicative quadrati c form , 45 4 pseudodiscriminant, xi x multiplicative type , 33 2 multiplicity o f a weight, 34 4 quadratic map , 48 9 multiplier, 488 , 49 0 quadratic norm , 48 9 multiplier map , 18 4 quadratic pair , 5 5 multiplier o f a similitude , 153 , 15 8 quadratic trace , 51 7 quasisplit group , 37 9 nondegenerate bilinea r form , xv i quaternion algebra , 2 5 nondegenerate quadrati c space , xvi i quaternion basis , 2 5 nonsingular bilinea r form , xvi , 1 nonsingular cubi c form , 47 4 radical o f a quadratic form , 34 8 nonsingular hermitia n form , 23 , 4 2 reduced characteristi c polynomial , 5 nonsingular quadrati c space , xvi i reduced dimension , 6 norm o f a vecto r space , 3 7 reduced Freudentha l algebra , 517 , 52 0 norm o f a n algebra , 36 , 55 4 reduced norm , 5 normal subgroup , 32 7 reduced roo t system , 35 2 normalizer, 32 7 reduced trace , 5 reduced unitar y Whitehea d group , 26 7 Okubo compositio n algebra , 475 , 48 0 reduced Whitehea d group , 25 3 1-cocycle, 361, 384 reflection, 35 2 opposite algebra , 3 regular bilinea r form , xv i order o f a finite grou p scheme , 33 3 regular form , 34 8 orientation, 20 7 regular loca l ring , 33 7 orientation o f a quadratic space , 17 1 regular quadrati c space , xvi i oriented quadrati c space , 17 1 regular space , 34 8 orthogonal group , 348 , 35 0 representation, 34 2 588 INDEX restriction map , 38 5 *-action, 37 6 restriction scheme , 32 7 strictly power-associativ e algebra , 45 1 root, 353 , 35 6 sub-Grassmannian variety , 1 2 root lattice , 35 3 surjective homomorphism , 32 7 root system , 356 , 36 4 symbol algebra , 2 7 roots o f unity , 32 8 symmetric bilinea r form , xv i Rost invariant , 43 5 symmetric compositio n algebra , 46 4

symmetric compositio n o f type M 2, 48 0 sandwich map , 3 2 symplectic basis , 1 6 Schur index , 4 symplectic group , 34 7 second involution , 55 0 symplectic involution , 1 6 second Tit s construction , 52 5 system o f simpl e roots , 35 3 second Tit s process , 52 6 semilinear action , 27 9 T' (secon d involution) , 55 0 semisimple algebrai c group , 35 5 Tits algebra , 37 7 separability idempotent , 28 5 Tits clas s o f a simpl y connecte d semisimpl e separable algebra , 344 , 45 3 group, 42 6 separable homomorphism , 34 1 Tits clas s o f an algebr a wit h unitar y involu - Severi-Brauer variety , 1 0 tion, 42 7 Shapiro's Lemma , 39 4 Tits construction , 52 5 signature o f a form , 13 6 Tits process , 52 5 signature o f an involution , 13 7 torsor, 288 , 38 8 similar spaces , 16 8 trace o f an element , x v similitude, 488 , 49 0 transfer o f a bilinear form , 52 4 similitude o f a bilinea r space , 15 3 transpose o f a linea r map , 1 similitude o f a quadratic pair , 15 9 trialitarian algebra , 552 , 55 3 similitude o f an algebr a with involution , 158 trialitarian algebr a o f type G2, 55 3 simple group , 35 7 trialitarian algebr a o f type l D±, 55 8 simply connecte d group , 36 4 trialitarian algebr a o f type 6 Z>4, 55 8 simply connecte d semisimpl e group , 35 6 trialitarian algebr a o f type 3 D4, 55 8 simply transitiv e action , 38 8 trialitarian algebr a o f type 2 D4, 55 8 skew field , 3 trialitarian isomorphism , 55 3 skew-hermitian form , 4 2 trialitarian triple , 54 8 skew-symmetric bilinea r form , xv i triality, 451 , 547 Skolem-Noether theorem , 5 trivial group , 32 6 smooth algebrai c group , 33 8 twisted p-form , 39 2 smooth grou p scheme , 33 8 twisted Cayle y composition , 49 0 S TV-condition, 34 4 twisted compositio n o f type G2 , 49 9 1 solvable algebrai c group , 35 5 twisted compositio n o f type A2, 50 5 special Cliffor d group , 177 twisted compositio n o f type 2 ^2, 50 6 special Jorda n algebra , 51 3 twisted compositio n ove r L , 48 9 special linea r group , 32 8 twisted form , 36 1 special unitar y group , 34 6 twisted Hurwit z composition , 49 9 spin group , 18 7 twisting, 38 7 spinor group , 349 , 35 1 2-cocycle, 38 4 spinor norm , 35 1 type o f a roo t system , 35 6 spinor norms , 18 7 split algebra , 4 unital algebra , 45 1 split Cayle y algebra , 46 2 unitary group , 34 6 split Freudentha l algebra , 51 7 unitary involution , 2 1 split semisimpl e group , 35 6 unitary Whitehea d group , 26 7 split torus , 33 3 vector representation , 17 7 split trialitaria n algebra , 55 3 split twiste d composition , 49 0 Wedderburn's theorem , 3 , 30 3 splitting field , 4 weight, 34 4 Springer construction , 52 3 weight lattice , 35 3 Springer decomposition , 52 3 Weyl group , 35 3 stabilizer, 32 7 standard identification , 5 5 Zorn matrices , 50 7 Notation

()# (Freudentha l adjoint) , 470 , 51 9 Ad (adjoin t representation) , 34 3 x (Freudentha l product) , 470 , 51 9 ad (inne r derivation) , 34 4 • (Jordan product) , 51 3 Ar (r-invarian t elements) , 38 3 _L, xvi i Ax (Tit s algebra) , 37 7 = F (tenso r product) , xi x AlgF, 28 0 (C,7l)A, 50 0 a\, 54 9 (L,a) (cycli c algebra) , 302 , 41 4 as, 48 1

(U,q)K, 499 aa,T, 38 6 (V,$) (roo t system) , 35 3 Alt(A,cr) (alternatin g element s i n A), 1 4 op {VIQ) (quadrati c space) , 7 4 y4 (opposit e algebr a o f A), 3 (V,q,() (oriente d quadrati c space) , 17 1 Ared, 33 1 x x Det (Z,a)p (quaternio n algebra) , 19 0 (A x A ) , 52 9 (^J/) (quadrati c pair) , 5 5 ,4-7brsr (A-torsors) , 38 8 Aut(A,cr), 34 5 (a, £>) F (quaternio n algebra) , 2 5 ia^)c,K (symbo l algebra) , 2 7 Aut(J,A), 52 9 Aut(J/A), 52 9 ((ai,..., a n)) (Pfiste r bilinea r form) , xv i Aut^, 36 2 ((ai,. .. ,an]], xi x Autalg(A), 32 8 (ai,.. . ,an), xi x (/, 0,28 2 AutF(A, cr,/), 15 9 [ai], xi x AutF(J,A), 52 9 [ai,a2], xi x AutF(J/A), 52 9 Aut (v), 32 8 [a, 6)F (quaternio n algebr a i n char . 2) , 2 5 G AutK(A,a), 34 5 {a,6}z, 45 2 AutH(AH), 32 8 A(p,w) (twiste d p-form s o f w), 362 , 39 2 B(A,

589 590 NOTATION

C(A, a, f) (eve n Cliffor d algebr a o f A), 9 2 discg, xvi i C{G) (cente r o f G) , 35 7 disccr (discriminan t o f a), 8 1 c(T) (Cliffor d invariant) , 56 3 Dyn(3>) (Dynki n diagram) , 35 3 C(V,q) (Cliffor d algebr a o f (V,q)), 8 7 E(a), 55 7 C0(V,q) (eve n Cliffor d algebr a o f (V,q)), 8 7 1C, 47 2 E{£) (Alle n invariant) , 57 0 C3 (cycli c group), 30 2 EQ (roo t system) , 35 5 E7 (roo t system) , 35 5 Gn (roo t system) , 35 4 C , 20 7 ^8 (roo t system) , 35 5 n e C+, 51 5 ( ij)i

A(X), 29 1 Gi x G 2, 32 8 6° (connectin g map) , 38 5 G2, 37 3 61 (connectin g map) , 38 6 g2(B,r) (cohomolog y class) , 42 0 A(g) (Dickso n invarian t o f g), 15 7 G2", 464 Der(A,M) (derivatio n module) , 33 4 G2, 37 3 Der(r), 56 9 G2 (roo t system) , 35 5 Der(J), 56 9 93(J), 53 7 Der(J/L), 56 9 GL (restrictio n scheme) , 32 6 Der(y, L,Q,/3) , 56 9 GA{R), 32 5 det(m) (determinan t o f ra), x v Ga (additiv e grou p functor) , 32 6 det b (determinant o f b) , xv i T (absolut e Galoi s group) , 27 9 det q, xvi i r+(V,qr), 34 9 deter (determinan t o f a), 8 1 T(A,cr,/) (Cliffor d group) , 35 1 df (differentia l o f F), 33 5 r(i4,<7,/) (Cliffor d group) , 17 8 dimG (dimensio n o f G), 33 7 T(G,L), 49 9 disc b (discriminant o f b), xv i r+(V,qf) (specia l Cliffor d group) , 17 7 NOTATION 591

1 rA, 49 0 f- (idea l orthogona l t o i), 7 2 7£, 28 1 Isom(X), 17 0 GD (Cartie r dual) , 33 3 Iso(A,a), 15 9

G-GalF (Galoi s algebras) , 29 0 Iso(V,6) (isometrie s o f (V,6)) , 15 3 GL(V) (genera l linea r group) , 32 6 Iso(A, a) (isometrie s o f (A, a)), 34 5 GLi(A), 32 6 J(A,X) (firs t Tit s construction) , 52 6 GLn, 32 6 J(B,T,U,V) (secon d Tit s construction) , 52 4 Gm (multiplicativ e grou p functor) , 32 6 J(L,V) (Springe r construction) , 52 3 G^ K (nor m on e elements) , 40 2 q) fl(A,cr), 2 8 J(V, (Jorda n algebr a o f a quadrati c GO(A,a,/), 35 0 form), 51 4 GO(A,ff), 15 9 J°, 51 4 GO(A,cr), 34 7 K*K', 41 6 GO(A,(T,/), 15 9 ker(/) (kerne l of/), 32 8 GO(V,6), 15 4 GO(V,g), 15 5 L°, 28 4 GO+(V,g), 15 7 A (roo t lattice) , 35 3 GO+(A,a), 16 4 A-)- (con e o f dominant weights) , 35 3

GO+(A,*")> 15 9 LK, 28 0

GSp(V,fc), 15 4 £x (lef t multiplicatio n wit h x), 45 5 GSp(A,cr), 34 7 G*, 32 7 M(A) (multiplicatio n algebra) , 45 1 M[K] (twiste d module) , 41 6 G-TorsT (G-torsors) , 29 0 GU(A,a), 15 9 Mn(C) (matri x algebra) , 51 6 GU(B,r), 19 3 Mr,s(D) {r x s-matrices) , 7 GU(B,r), 34 6 fi(t) (multiplie r oft) , 48 3 GU(V,/i), 15 5 M» (a, 6), 45 2 /xn (root s o f unity), 32 8 ?t(A, o- ) (Jorda n algebr a o f Sym(A, a)), 2 8 Mn[K] (twiste d group) , 41 8 Wn(C,a), 51 6 ^(F), 43 1 ft3(C,a)°, 51 7 /i (multiplie r map) , 18 2 if°(r,A), 38 3 \i (multiplie r map) , 18 4 /f2(r, A ) (cohomolog y set) , 38 4

HiHH2, 32 8 N(i) (nor m off) , xv i #2(r, A ) (cohomolog y group) , 38 4 N(x,y,z) (cubi c form) , 47 4 Hl(F,A) (cohomolog y groups) , 39 1 NA (generi c norm), 45 2 #const (constan t grou p scheme) , 33 2 Nc (nor m o f x), 45 7 -f/diag (diagonalizabl e grou p scheme) , 33 2 nil(A) (nilpoten t elements) , 33 1

Het, 33 2 Nj (nor m o f J), 51 6 nH (n-torsio n o f if), 41 3 NK/F(A) (nor m (corestriction ) o f A), 3 7 Hom^ (G, if) (grou p schem e homom.) , 32 5 N(K/F) (grou p o f norms), 11 4

Hurwm (Hurwit z algebras) , 46 0 NL/F(l) (nor m oil), xv i Hurw(para-Hurwitz algebras) , 46 4 Nrd^(a) (reduce d nor m o f a), 5 Nc/, 32 7 %{G) (Rost invariant) , 43 5 1° (annihilato r o f i), 8 0(A,a), 347 IF (fundamenta l idea l o f WF), 14 1 0(A,cr,/), 160 InF (power s o f the fundamenta l ideal) , 14 1 0(A,a,/), 350 im(/) (imag e o f F), 32 7 0(V,

0+(V,6), 15 4 R(C), 45 5 0+(V,g), 15 7 rdimM (reduce d dimensio n o f M), 6 0+(V,g), 34 8 res (restrictio n map) , 38 5

0(V,6), 15 4 pv (hyperplan e reflection) , 15 7 O/fii, 48 0 P{E® A,cra(X) (reduce d characteristi c polynomial o f a) , 5 Sj(a), 516 , 51 9 S'iCi U(J3) (reduce d unitar y Whitehea d Prpa S (X) (pfaffia n characteristi c polynomial o f s), 1 9 group), 26 7 Pa (pfaffia n adjoint) , 24 5 SKi(A) (reduce d Whitehea d group) , 25 3 PSim(A,cr), 15 9 Skew(A, cr ) (skew-symmetri c element s PSim(V, 6 ) (projectiv e similitude s o f (V , 6)), in A) , 1 4 153 Skew(A,cr)° (trac e zer o elements) , 2 8 SLi(A), 25 3 Pe, 24 6 SL(£), 19 4 Q+(A,(T,£), 411 SL/FW, xv i Q(A,a), 411 SL(V), 32 8

Q0(A,aJ), 411 SLi(A), 32 8 qh, xvii Sn (spino r norm) , 18 6 Qn, 171 Sn(A,cr,/), 18 7 <#, 171 Sp(A,cr), 15 9 Q°, 306 Sp(A,cr), 34 7 Quot(S) (fiel d o f fraction s o f 5) , 34 1 Sp(V,6), 15 4 Q/Z(i)(F), 43 1 Sp(V,/i), 34 7 NOTATION

Spin(A,<7,/), 18 7 x-^ (lef t modul e homomorphism) , x v Spin(A,cr,/), 35 1 X(A), 35 4 Spin(V,g), 18 7 Xi, 35 7 Spin(V,g), 34 9 x *c r (a ® 6) (righ t modul e structure) , Spin8, 48 6 Spin±(y,g), 35 9 Z(A) (centroid) , 45 1 Z(A,tr,/), 18 7 SrdA(a), 5 l SSym(A,a)x, 40 6 Z (T,A) (1-cocycles) , 38 4 2 SSym(B,r)x, 53 1 Z (r,A) (2-cocycles) , 38 4 SSym(B,T)x, 40 1 SU(B,r), 19 4 SU(B,r), 34 6 SU(V,/i), 15 5 Sv, 32 7 Sym(A,cr) (symmetri c element s i n A) , 1 4 Sym(A,cr)', 40 0 Sym(B,r)°, 30 6 Symd(A,cr) (symmetrize d element s i n A), 14

T{£) (trac e off), xv i T(A, a) (trialitaria n algebra) , 55 8 t(B,r) (Tit s class) , 42 7 T(£) (trialitaria n algebr a o f a Li e algebra) , 570 T(x,y) (bilinea r trac e form) , 28 1 t+, 48 3 t~, 48 3 TA (generi c trace), 45 2 r', 55 0

Tc(x) (trac e o f z), 45 7 0+, 48 5 0-, 48 5 0a, 38 7 Tj(a), 516 , 51 9 TL/F(£) (trac e o f ^) , xv i

(TL/F)*(6), 52 4 tr(ra) (trac e o f m), x v

TrdA(a) (reduce d trac e o f a), 5 Tr, 53 4 TTfT/(x,j/), 53 4

U(B,r), 34 6 UKi(B) (unitar y Whitehea d group) , 26 7 U(A,a), 15 9 U(£,r), 19 3 U(V,/i), 15 5 y-1, xvi i V* (dua l spac e o f V) , xv i VG (G-invarian t subspace) , 34 3

W($) (Wey l group) , 35 3 WF (Wit t rin g o f F), 14 1 x*, 470 , 51 9 X(f), 28 1 X(L), 28 0 x *y, 47 2

Xc, 44 4