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America n Mathematica l Societ y

Colloquiu m Publication s Volum e 39

Structur e an d Representation s of Jorda n Algebra s

Natha n Jacobso n

America n Mathematica l Societ y Providence , Rhod e Islan d

http://dx.doi.org/10.1090/coll/039

2000 Mathematics Subject Classification. P r i m a r y 16-02 , 17-XX .

Library o f Congres s Cataloging-in-Publicat io n Dat a

Jacobson, Nathan , 1910 -Structure an d representation s o f Jorda n algebras .

p. cm . — (America n Mathematica l Societ y Colloquiu m publications , ISS N 0065-925 8 ; v. 39 ) Providence, America n Mathematica l Society , 1968 . Includes bibliography . ISBN 978-0-8218-4640- 7 (alk . paper ) 1. Jordan algebras . I . Title . II . Colloquiu m publication s (America n Mathematica l Society ) ;

v. 39 .

QA1 .A5225 vol . 3 9 512'.8 6801943 9

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Copyright 196 8 b y th e America n Mathematica l Society . Reprinted b y th e America n Mathematica l Society , 1991 , 200 8

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PREFACE The purpos e o f thi s boo k i s t o giv e a comprehensive accoun t o f th e structur e

and representatio n theor y o f Jorda n algebra s ove r a field o f characteristi c no t two. I t may b e appropriat e a t thi s poin t t o indicat e th e limit s w e hav e se t our -selves an d wha t lie s immediately beyon d these limits . I n the first place, w e not e that a substantial par t of the theory carries over to algebras over a commutativ e ring with 1 containing a n element \ suc h that \ + \ = 1 . A reade r who has nee d of a result of this generality wil l have no difficulty i n ascertaining whether or not the corresponding result in the field case carries over.

A more serious and significant extension of the theory, which can now be made, is th e passag e fro m a linea r theor y t o a quadrati c one . I n th e cas e o f a specia l Jordan algebr a thi s amount s t o th e replacemen t o f th e Jorda n produc t a.b = \{ab +ba), whic h i s bilinear , b y th e produc t aba 9 whic h i s linea r i n b and quadratic in a. A s the Jordan theory has developed it has become increasingly a quadrati c theor y base d o n th e produc t {aba} =bU a =2(fc. a).a — b.a- 2, where a.b is th e give n bilinea r produc t an d a- 2 = a.a. I n a specia l Jorda n al -gebra {aba} coincide s with aba. An important step in the transition to a quadratic theory wa s taken b y th e autho r i n a paper whic h appeare d in 196 6 ([38] i n the Bibliography), i n whic h w e gav e a n Artinian-lik e structur e theor y fo r Jorda n algebras founde d o n axiom s o n quadrati c ideals . Thes e ar e define d b y mean s of th e compositio n {aba}. Thi s theor y i s considere d i n detai l i n Chapte r IV . The author's structure theory has been extended by McCrimmon in [6 ] and [14 ] to quadratic Jordan algebras over an arbitrary commutative ring with 1 . A sketch of McCrimmon' s theor y ca n b e foun d i n ou r note s o n "Furthe r Result s an d Open Questions " a t th e en d o f thi s book .

A par t o f th e Jorda n theor y ca n b e founde d als o o n th e notio n o f inverses . This ha s bee n don e i n th e boo k Jordan Algebras b y Brau n an d Koeche r an d more recentl y b y Springe r i n a n unpublishe d pape r whic h cover s als o th e cas e of algebras of characteristi c two. The connection betwee n inverses and the quad-ratic compositio n aba i n associativ e algebra s i s give n b y Hua' s identit y (p . 2). The drawbac k o f a theor y base d o n inverse s i s tha t on e mus t b e i n a situatio n in whic h on e ha s a n ampl e suppl y o f invertibl e elements . Th e theor y seem s t o work bes t fo r finite-dimensional algebra s wit h 1 ove r a n infinit e field. I n thi s case th e invertibl e element s for m a Zarisk i ope n subse t o f th e algebra . O n th e other hand , th e theor y doe s no t appl y ver y wel l to infinite-dimensiona l algebra s or i n arithmeti c setting s i n whic h invertibl e element s ma y b e scarce . Fo r thi s reason w e hav e preferre d t o bas e th e theor y on a.b an d {aba}.

V

A numbe r o f importan t application s o f Jorda n algebra s hav e bee n found . The Jorda n theor y ha d it s birt h i n a n attemp t b y P . Jorda n an d subsequentl y by Jordan , vo n Neuman n an d Wigne r t o formulat e th e foundatio n o f quantu m mechanics i n term s o f th e produc t A . B = \{AB + BA) rathe r tha n th e asso -ciative produc t AB. A ver y importan t are a o f application s o f the Jorda n theory, especially o f exceptiona l Jorda n algebras , i s t o exceptiona l Li e group s an d al -gebras an d relate d geometries . Thi s i s discusse d i n par t i n Chapte r IX . Indica -tions o f additiona l result s an d th e origina l paper s containin g thes e ar e give n in th e "Furthe r Result s etc. " Another extremel y interestin g an d promisin g are a of application s i s t o rea l an d comple x analysis , particularl y t o homogeneou s cones, Siege l half-spaces an d automorphic functions . Th e most complete accoun t of thes e extremel y interestin g result s presentl y availabl e i s i n Koecher' s Uni -versity o f Minnesot a mimeographe d note s [4] . Thi s aspec t o f th e theor y ha s been completel y omitte d i n cu r discussion .

In th e cours e o f writin g thi s boo k I hav e ha d invaluabl e hel p fro m Kevi n McCrimmon—first, i n carefull y readin g an d criticisin g severa l version s o f th e manuscript, second, in communicating improved proofs of a number of important results and finally, in carefully readin g the proofs. I am indebted also to Marshal l Osborn fo r readin g th e proof s an d t o Miche l Racin e fo r hel p i n compilin g th e Bibliography. Also I am greatly indebted to Jacques Tits who took tim e off from his ow n importan t researche s o n algebrai c group s t o derive , vi a th e theor y o f algebraic groups , th e elegan t construction s o f exceptiona l Jorda n algebra s which we have given in Chapter IX . I wish to record my sincere appreciation fo r all o f thes e contributions .

The intentio n o f writin g thi s boo k ha d it s inceptio n wit h a n invitatio n b y the Amertean Mathematica l Societ y t o give Colloquium Lecture s at the 1955 summer meeting. Th e subjec t ha s change d enormousl y i n th e intervenin g years . Sinc e there remai n man y natura l ope n question s an d possibilitie s fo r application s i t is likely t o chang e substantiall y als o i n the next decade.

Key West, January 22 , 196 8

VI

TABLE O F CONTENT S

Chapter I . FOUNDATION S 1

1. Specia l Jordan algebra s an d Jorda n algebra s 3 2. Fre e specia l Jordan algebras . Cohn' s theore m 6 3. Homomorphi c image s o f special Jordan algebra s 1 0 4. Som e constructions o f Jorda n algebra s 1 2 5. Alternativ e algebra s an d Jorda n algebra s 1 5 6. Varietie s o f algebras. Free algebra s 2 3 7. Basi c Jordan identities . Jorda n tripl e produc t 3 3 8. Strongl y associativ e subalgebras . Jorda n tripl e

product identitie s 3 7 9. Macdonald' s theorem. Shirshov' s theorem 4 0

10. ^-identities . Exceptional characte r o f §(D)3 4 9 11. Inverse s and zer o divisor s 5 1 12. Homotop y an d isotop y 5 6

Chapter IT . ELEMENT S O F REPRESENTATIO N THEOR Y 6 3

1. Associativ e specialization s an d universa l envelopes 6 5

2. Specia l universa l envelope s fo r Jorda n algebra s with 1 an d fo r direc t sum s 7 2

3. Example s 7 4 4. Associativ e algebra s wit h involution s an d Jorda n

algebras 7 6 5. Bimodule s fo r algebra s in a variet y rT(I) 7 9 6. Birepresentation s for an algebra in a variet y ^ ( / ) 8 2 7. Multiplicatio n specialization s an d universa l

envelopes 8 6 8. Extensio n o f algebras and facto r set s 9 1 9. Jorda n bimodule s an d universa l multiplicatio n

envelopes 9 5 10. Associativ e specialization s an d multiplicativ e

specializations 9 9 11. Universa l multiplicatio n envelope s fo r Jorda n

algebras wit h identit y element s 10 2

vii

12. Som e basi c criteria 10 6 13. Example s an d applications 11 0

Chapter III. PEIRC E DECOMPOSITION S AN D JORDA N MATRI X

ALGEBRAS 117

1. Peirc e decomposition s 11 8 2. Jorda n matri x algebra s 12 5 3. Coordinatizatio n theorem s 13 2 4. Perfectio n o f certain classe s o f associative algebra s

with involution s 13 8 5. Unita l bimodule s fo r Jordan matri x algebras 14 4 6. Som e remark s o n the universa l envelope s o f Jordan

matrix algebra s 14 7 7. Liftin g o f idempotent s an d Jordan matri x

algebras 14 8

Chapter IV . JORDA N ALGEBRA S WIT H MINIMU M CONDITION S O N

QUADRATIC IDEAL S 15 2

1. Quadrati c ideal s 15 3 2. Axiom s and first structure theore m 15 7 3. Determinatio n o f a class of alternative algebra s wit h

involution 16 2 4. Simpl e Jorda n algebra s o f capacity tw o 17 1 5. Secon d structur e theore m 17 8 6. Specia l universa l envelopes . Isomorphism s an d

derivations o f specia l simpl e Jorda n algebra s satisfying th e axiom s 18 3

7. Alternativ e set o f axioms 18 7

Chapter V. STRUCTUR E THEOR Y FO R FINITE-DIMENSIONA L JORDA N

ALGEBRAS 18 9

1. Ideal s an d associato r ideal s 19 0 2. Solvabl e ideals 19 2 3. Finite-dimensiona l ni l algebras 19 5 4. Reduce d Jorda n algebra s 19 7 5. Structur e o f finite-dimensional semisimpl e Jorda n

algebras 20 0 6. Reduce d simple Jorda n algebra s 20 2 7. Finite-dimensiona l centra l simple Jordan algebras 20 6

viii

Chapter VI. GENERI C MINIMUM POLYNOMIALS , TRACES AN D NORM S . . . 21 3

1. Differentiatio n of rational expressions 21 4 2 Differentia l calculu s of rational mapping s 21 5 3 Generi c minimum polynomial of a strictly power

associative algebr a 22 1 4. Som e important example s 22 9 5. Multiplicativ e propertie s o f the generi c nor m 23 4 6. Separabl e algebra s 23 8 7. Nor m similarit y of Jordan algebras . The grou p of

norm preserving linear transformation s 24 1 8. Determinatio n o f M(3 ) fo r 3 specia l centra l simpl e

of degree ^3 24 7 9. Th e Lie algebras 9W(3) , 2R 0(3) an d D e r3 fo r 3

separable 25 3

Chapter VII. REPRESENTATIO N THEOR Y FO R SEPARABL E JORDA N ALGEBRA S 25 9

1. Structur e o f Clifford algebra s 26 0 2. Structur e of meson algebras 26 4 3. Universa l envelope s fo r finite-dimensional specia l

central simpl e Jorda n algebra s o f degre e ^ 3 27 2 4. Bimodule s fo r compositio n algebra s an d reduce d

Jordan algebra s 27 7 5. Separabilit y of 3 an d U(%) 28 6 6. Th e theorem of Albert-Penico-Taft 28 6 7. Derivation s int o bimodule s 29 2 8. Uniquenes s o f the Albert-Penico-Taft decom -

position 30 3

Chapter VIII. CONNECTION S WIT H LI E ALGEBRA S 30 7

1. Associato r structur e o f a Jorda n algebra . Abstrac t Lie tripl e system s 30 7

2. Som e result s o n completel y reducibl e Li e algebras of linea r transformation s wit h application s to Jordan algebras 31 2

3. Operato r commutativity 32 0 4. Derivation s o f semisimpl e Jorda n algebra s o f

characteristic 0 int o bimodule s 32 3 5. Th e Tits-Koeche r constructio n o f a Li e algebr a fro m

a Jorda n algebr a 32 4 6. Subalgebra s and ideal s o f R = &(3) 32 9

ix

7. Applicatio n t o a proof of the Albert-Penico Theorem 33 3

8. Associativ e bilinea r forms 33 6 9. Examples . Functorial constructions 33 9

10. Associato r nilpotent Jorda n algebra s 34 3 11. Carta n subalgebra s o f Jorda n algebras . Associato r

regular element s 34 9 12. Applicatio n t o generi c trace s 35 2 13. Conjugac y o f Carta n subalgebra s 35 3

Chapter IX . EXCEPTIONA L JORDA N ALGEBRA S 35 6

1. Preliminarie s o n identitie s an d subalgebra s 35 7 2. Element s o f ran k one . Uniquenes s o f th e coefficien t

algebra 36 4 3. Au t 3 /$ e 37 0 4. Condition s for isomorphis m 37 8 5. Mor e o n Au t 3 38 2 6. Invarian t factor theorem for split 3 38 9 7. Moufan g projective planes 39 3 8. Elation s 39 7 9. Harmonicit y 40 2

10. Fundamenta l theore m o f projectiv e geometr y fo r th e planes $ ( 3) 40 4

11. Connection s with exceptional Li e algebras 40 7 12. Exceptiona l Jorda n divisio n algebra s 41 2

FURTHER RESULT S AN D OPE N QUESTION S 42 3

BIBLIOGRAPHY 44 3

SUBJECT INDE X 45 1

X

FURTHER RESULTS AND OPEN QUESTIONS

CHAPTER I

There ar e a numbe r o f importan t ope n question s concernin g identitie s an d inverses i n Jorda n algebras . W e procee d t o indicat e som e o f these .

First, a s i n th e text , le t FJ ir) denot e th e fre e Jorda n algebr a wit h 1 an d r (free) generator s x l9x29-~9xr (p . 40 ) an d le t FSJ ir) b e th e fre e specia l Jorda n algebra wit h 1 and r (free) generator s u l9u29•••,", (p . 7) . Le t Ji (r) be the kerne l of the homomorphism o f FJ (r) ont o FSJ ir) sendin g 1 -• 1 , x^-m,-, i = 1,2 , - , r . It i s immediate tha t $t{r) i s a T-idea l i n FJ (r), tha t is , an idea l mappe d int o itsel f by ever y homomorphis m o f FJ (r) int o itself . W e kno w tha t St ir) ^ 0 if r ^ 3 . A Jordan algebr a wit h 1 an d r generator s i s a homomorphi c imag e o f a specia l Jordan algebr a i f an d onl y i f i t satisfie s al l th e identitie s i n Jt (r). Th e result s o f Glennie indicated in the text (p. 51) provide nonzer o elements o f 5fc(3). However , a complete determination of R(r) fo r r ^ 3 has not yet been given. I t is not known either i f 5t (r), r ^ 3 , i s finitely generate d a s a T-ideal .

Let &{{x l9-"9xr}}' b e th e fre e nonassociative algebr a generated b y x l9-~9x,9

X9 th e idea l i n this algebr a generate d b y al l th e element s o f th e for m ab — ba, (a2b)a — a2(ba)9 a 9 6eO{{xJ} . Assum e tha t <t> i s infinit e an d let /e*{{X|} } be homogeneou s o f degre e n . W e clai m tha t iff i s a n identit y fo r al l finite-di-mensional Jorda n algebras , the n / i s a n identit y fo r al l Jorda n algebras . (Thi s result ha s bee n communicate d t o u s b y Koecher. ) W e hav e t o sho w tha t th e hypothesis implie s that /e 2. Sinc e O is infinite , X i s a homogeneous ideal , that is, if/ ! + / 2 + •• • +fke% wher e deg/ j =j9 the n ever y fjeX. Le t X n+i b e th e ideal generate d b y X an d al l monomial s i n th e x' s o f degre e ^ n + I . The n X„+i i s also a homogeneous idea l an d the homogeneou s par t o f degre e n in this ideal i s containe d i n I . I t i s clea r tha t 0{{x l}}/Ir t+1 i s a Jorda n algebr a an d since ther e ar e onl y a finite numbe r o f distinc t monomial s i n th e x t o f degre e ^ n , ^>{{xi}}IX„ +1 i s finite dimensional . Hence / i s a n identit y fo r thi s Jorda n algebra. The n feXH+l9 whic h implie s t h a t / e X .

Let {3 a | aeA} b e a family o f Jorda n algebra s wit h 1 . W e defin e a. free com-position o f th e 3 « t o b e a Jorda n algebr a ̂ f j wit h 1 togethe r wit h a famil y {^a |aeA} wher e *j a i s a homomorphis m o f 3 « int o ̂ J sending 1 - * 1 such that , if 3 i s any Jorda n algebr a wit h 1 and £ « is a homomorphis m o f 3« i°t o 3 wit h 1 -* 1, the n ther e exist s a uniqu e homomorphis m A o f ̂ J int o 3 satisfyin g 1*= 1 and C « = ^ » a e A . f t i s immediat e tha t an y tw o fre e composition s ar e

423

424 FURTHER RESULTS AND OPEN QUESTIONS

equivalent i n the obvious categorica l sense . In particular, the algebra ̂ } is deter-mined u p to isomorphism. Also < P i s generated by the subalgebras 32"- It is e a s y to prov e th e existence o f a free compositio n fo r any family o f Jordan algebras . For example , let 3t = FJ ir)jR, 3 2 = FJ (s)/£ wher e R an d 2 ar e ideals. Identif y FJir) an d FJ is) wit h th e subalgebra s o f FJ ir+s) generate d b y x u-~9xr an d xr+l9"-9xr+s respectivel y and let 3JI be the ideal in FJ{r+s) generate d by R and fi. Then it is easily seen that S$ = FJ (r+5)/2ft an d the homomorphisms x + 51 -> x 4-SOI, x e FJ(r) an d >> + £-> y 4- 501, ye FJis) constitut e a fre e compositio n o f 3i an d 32- Thi s procedur e i s easily generalize d (se e Cohn's Universal Algebra, p . 142 for a general argument) . W e shall cal l th e free compositio n {ty,n a} th e free pro-duct o f th e 3 * i f n a i s a monomorphis m fo r ever y a e A . I n thi s case , if A = {l,2,---,w} , then w e writ e S$ = 3 i * 3 2 * •" *3n- I t i s easy t o se e that the free produc t o f associativ e algebra s (sam e definition s a s fo r Jorda n algebras ) exists. Thi s implie s tha t th e fre e produc t o f specia l Jorda n algebra s exist . O n the othe r hand , fre e product s ma y no t exis t fo r exceptiona l Jorda n algebras . For example , le t 3i b e a finite-dimensional simpl e exceptiona l Jorda n algebra . Then i t i s know n tha t i f 3 i s any Jordan algebr a wit h 1 containing 3i a s sub-algebra wit h 1 , the n 3 = 3 i ®^ > wher e ( £ is a n associativ e Jorda n algebr a with 1 (Jacobson [18]) . It follows that 3 satisfie s every multilinear identity which holds for 3 i. No w let 32 = FJ {2) and let ̂ 3 be a free composition of 3i an d 32 • Then * P satisfies al l the multilinear identitie s satisfie d b y 3i • Hence $ canno t contain a subalgebra isomorphi c to 32 (& ex - 4 , p. 363). Thus $ i s not the free product o f 3i an d 32-

Let 3 i an d 3 2 b e associativ e Jorda n algebra s (wit h 1 ) an d 3 3 = W ( 1 )

with generato r z . Sinc e th e 3» a r e specia l th e fre e produc t 3 i * 3 2 * 3 3 exists . Also th e fre e associativ e produc t ^ ' o f th e associativ e algebra s 3 * exists. Le t 3 i * ' 3 2 * ' 3 3 b e the subalgebra o f the special Jorda n algebr a S$'+ generated by the image s o f th e 3 i i n ̂ P '• W e hav e th e canonica l homomorphis m a o f 3 i *3 2 * 35 ont o 3 i * ' 3 2 * ' 33 . Le t R b e the kernel o f a and let 3 b e the sub -space o f 3i *32 *33 spanne d by products o f elements of 3i an d 32 and a single 2 (that is , degree 1 in z). If 3 i = 3 2 = FJ (1\ the n Macdonald' s theorem state s that 5 l n 3 = 0- Th e sam e resul t ha s bee n prove d b y McCrimmo n i n [9 ] i f 3 i = 3 2 i s isomorphic t o the group algebr a o f an infinite cycli c group . I n both of thes e case s th e universa l multiplicatio n envelop e o f 3 * is commutativ e (cf . §2.3 an d §2.13) . W e conjecture tha t thi s conditio n o n the 3u l = 1>2 , is suf-ficient t o insur e tha t & n 3 = 0. * Thi s woul d constitut e a generalizatio n o f Macdonald's an d McCrimmon' s theorems . Anothe r consequenc e o f th e con -jectured resul t woul d be that the free produc t o f two associative Jorda n algebras having commutativ e universa l envelope s i s special . Thi s would constitute a gen-eralization o f Shirshov' s theorem .

* This conjecture was arrived at in a conversation with McCrimmon.

FURTHER RESULT S AND OPEN QUESTIONS 425

McCrimmon's analogu e o f Macdonald' s theore m ca n b e use d t o establis h identities in x, y9 x- 1 , y'" 1

9 z which are of degree one or zer o i n z b y verifyin g these fo r specia l Jorda n algebras . Example s o f thi s sor t ar e th e identitie s {xyx}.{x'~ 1 y'~1x _ 1 } = 1 , {xyx}' 2.{x'~1y~1x'~1} = {xyx} , whic h follo w di -rectly fro m Theore m 1.13(6) , p . 52. Another exampl e is

{)>{(x•~1 + / ~ 1 ) { x z x } ( r - , +y'- l)}y] = {( x + y) z(x + y)).

On th e othe r hand , McCrimmon' s theore m i s no t adequat e fo r establishin g more genera l rationa l identitie s i n x, y whic h involv e inverse s o f polynomial s in x an d y an d mor e genera l rationa l expressions , e.g . x*" 1 +0>'~1 — x)'" 1

(which occur s in Hua's identity , ex. 3, p. 54). T o handl e suc h identitie s we con-sider th e fre e Jorda n algebr a FJ in+2) generate d b y x 9y9xl9--9xn. Le t p. = P i(x,y9x!,-•-,x,) b e a n elemen t o f th e subalgebr a generate d b y l 9x,y9

*!,—,x£ , / = 0,1 , . . ,n . Let / = /(P 0,---,Pn) b e the ideal i n FJ{n+2) generate d by th e In elements P i^1.xi — 1 9 Pi^1.x:2 — x^. No w let 3 b e a Jordan algebr a with 1 , a 9 b elements o f 3- Assume P 0(a,b) i s invertible i n 3 wit h a x a s its in-verse, Pi(a, b9ax) i s invertible with inverse a2, and, in general, Pf_ x(a9 b,al9 • • •, flf_ x) is invertibl e wit h invers e a t. The n th e homomorphis m o f FJ in+2) int o 3 s u c h that 1 -• 1, x -> a, y^b, x t -+ af maps / int o 0 , so we have the homomorphism of FJ {n+2) 11 int o 3 sendin g 1->1 , x + I-+a, y + I-+b, x i + l^a {. I t i s clear als o tha t P i.i + / i s invertible i n FJ {n+2)jI wit h invers e x , + /. Th e ele-ments o f / ma y be regarded a s rational identitie s i n the two elements x , y. A s an example , le t n = 3 , P 0 = x , P j = y , P 2 = x — x2 . The n Hua' s identit y i s equivalent t o the statement tha t th e following element s ar e contained i n /:

(*i ~ * 3) • (* - {xyx}) - 1 , (x , - x 3)'2. (x - {xyx}) - (x x - x 3).

We conjectur e tha t fo r an y choice o f th e P t th e subalgebr a o f 3 generate d by 1 , a, b 9 al9---9an (a s indicated) is special (assuming that P i-1(a9b9al9---9ai-i) is invertibl e wit h invers e a (). Thi s woul d constitut e a generalization o f th e the -orem of Shirshov-Cohn. If 3 i s algebrai c (that is, every element of 3 i s algebraic) then the subalgebra generate d by 1, a, b 9 al9~-9an i s generated b y l,a,fa . Henc e this i s specia l b y the Shirshov-Coh n theorem . I n general, i f the subalgebra gen-erated b y l 9a9b9al9--'9a„ i s special , s o tha t thi s ca n be identifie d wit h a sub-algebra o f ?l +, 3 1 associative, the n a t- and P l_1(a,b1 ,a1 , •••,<!,•-1 ) ar e inverse s in the associative algebra 91 . Then the verification o f the fact that certain elements of 3 ar e mapped int o 0 unde r th e homomorphis m w e defined ca n be reduce d to th e verification o f certain rationa l identitie s i n associativ e algebras . I n par -ticular, i t is clear that the validity of Hua's identity for algebraic Jordan algebras, hence fo r finite-dimensional Jordan algebras , i s a consequence o f Hua's identit y for associativ e algebras . T n general, i f the conjecture w e have mad e i s true, then


any rationa l identit y i n tw o element s o f a Jorda n algebr a coul d b e establishe d by provin g correspondin g rationa l identitie s i n tw o element s o f a n associativ e algebra.

One ca n als o attemp t t o reduc e identitie s whic h ar e rationa l i n tw o element s and integra l an d o f th e first degre e i n a thir d elemen t t o associativ e rationa l identities i n a simila r manner . Her e le t FJ (n+3) b e th e fre e Jorda n algebr a generated b y l 9x9y9z9xl9--9xn9 Pfcx 9y9z9xl9-~9x^ a n elemen t o f th e subal -gebra generate d b y l,x,y,z 9xU'~,xi9 I th e idea l generate d b y P,-! .*, — 1 , Pi-l.xi-

2 — x t. Le t 0>{u 9v9w9ul9•••,«„} b e th e fre e associativ e algebr a generate d by 1 and the indicated elements, I s th e ideal generated b y Pi-iu t — 1 , uiPi.1 — 1 , where P t = P i{u9v9w,ul9-~,un). The n w e hav e a homomorphis m o f FJ (n+3)

into th e Jorda n algebr a (0{---}// s)+ suc h tha t 1->1 , x-*u + I s9 y-+v + I S9

z-+w + I 59 Xi-Ûi + Is. W e conjectur e tha t i f feFJ{n+3) i s o f degre e 1 i n z and i s mappe d int o 0 unde r th e homomorphis m o f FJ° l+3) int o (<&{•-•} I Id* > then fel. I f thi s wer e th e case , the n w e woul d hav e a reductio n o f identitie s rational i n tw o element s an d integra l o f th e first degre e i n a thir d elemen t t o associative rationa l identities .

There ar e a numbe r o f interestin g question s o n th e embeddabilit y o f Jorda n integral domain s i n divisio n rings . W e recal l first a well-known resul t of Ore' s that an y associativ e rin g withou t zero-divisor s ^ 0 whic h ha s the (left ) commo n multiple propert y ca n b e embedde d i n a divisio n rin g (Jacobson , Theory of Rings, p . 11 8 o r Cohn' s Universal Algebra, p . 275) . Th e commo n multipl e property fo r a n associativ e rin g 2 1 is tha t i f a an d b ar e nonzer o element s o f % then SHa n 2l b ^ 0 for th e left ideal s %a, %b. W e hav e calle d a n elemen t o f a Jordan algebra 3 a zero divisor in 3 if the mapping U a is not injective in 3 > that is, ther e exist s a b j£ 0 suc h tha t bU a = 0 . W e shal l no w sa y tha t th e element s a, b e 3 have a common multiple i f Û anÛb 5 * 0 fo r th e quadrati c ideal s 31/*, 3^6 - I n v *ew ° f Ore' s resul t on e i s tempte d t o conjectur e tha t i f 3 i s a Jordan algebr a i n whic h an y tw o nonzer o element s hav e a commo n multipl e then 3 ca n be embedded in a Jordan division algebra .

It i s a well-know n result , du e independentl y t o Malce v an d B . H . Neumann , that th e fre e associativ e algebr a 0{x 1,x2,~-,xn} ca n b e embedde d i n a divisio n algebra (Cohn' s Universal Algebra, p . 276) . I t follow s fro m thi s tha t th e fre e Jordan algebr a FJ (2) ca n b e embedde d i n a divisio n algebra . Henc e thi s i s a n integral domain . I s th e fre e Jorda n algebr a FJ (n\ n ^ 3 , a n integra l domai n and ca n thi s b e embedde d i n a divisio n algebra ? I f th e latte r statemen t i s tru e then i t woul d provid e example s o f Jorda n divisio n algebra s whic h ar e infinit e dimensional ove r thei r centers . O n th e othe r hand , i t i s als o possibl e tha t FJ(n) contain s nonzer o zero-divisor s an d possibl y nonzer o nilpotent elements if n ^ 3 . Thi s i s th e situatio n fo r fre e alternativ e algebra s (se e Hum m an d Klein -feld [7]) .


CHAPTER I I

The notio n o f bimodul e an d universa l multiplicatio n envelop e fo r associativ e and Li e algebra s ar e th e startin g point s o f th e cohomolog y theorie s fo r thes e classes o f algebras . Th e theor y i n th e first case wa s initiate d b y Hochschil d an d in the secon d cas e b y Chevalley an d Eilenberg . W e recal l the basi c foundationa l results i n thes e theorie s (se e Cartan-Eilenber g [1 ] fo r th e details) . Le t 2 1 be a n associative algebr a wit h 1 , 50 i a unita l bimodul e fo r 21 . Th e unita l universa l multiplication envelop e fo r 2 1 i s C/i(2l ) = 21 ® 21° an d 2 1 i s a bimodul e (th e regular bimodule ) fo r 21 . On e define s th e nt h cohomology group of 2 1 with co-efficients in 501 for n = 0,1,2 , b y H"(2l,50t ) = Ext£ l(9r)(2l,50t). Thes e ar e vecto r spaces ove r th e bas e field <J> , //°(2t,50i) i s isomorphi c t o th e subspac e o f 501 of elements u suc h tha t ua = au 9 ae% tf l(2l,50t) s Der(2I,50l)/Inder(2l,50t) , where Inder(2t,50t ) i s th e subspac e o f derivation s o f 2 1 int o 50 1 o f th e for m a -* [a,u], a e 21, u e50i. // 2(2l,50t) i s isomorphi c t o the vector space o f equiv -alence classe s o f nul l ( = singular ) extension s O-»50t->(£->2l-> O wher e th e vector space compositions o n such equivalence classes are the Bae r compositions. Next le t £ b e a Li e algebra , U(£ ) th e universa l multiplicatio n envelop e ( = Birk -hoff-Witt algebra ) an d regar d O a s a trivia l bimodul e fo r £ s o tha t a / = 0 = /a , aeO, Ze£ . I f 50 1 i s a n £-bimodule , on e define s H\2M) = Ext^ fi)(O,50l), n = 0,1,2 , •••. The n #°(£,50i ) i s isomorphi c t o th e subspac e o f 50 1 of u suc h that ul = 0 an d H\2M) £ Der(£,50Z)/lnder(£,50l ) wher e Inder(£,50t ) i s th e set o f derivation s o f th e for m l-*lu (=—«/ ) , Zefl , ue 501 . Also , a s i n th e associative case // 2(fl,50i) i s isomorphic to the vector space o f equivalence classe s of nul l extensions o f £ b y 501.

In bot h th e associativ e an d Li e cases , on e ha s simpl e standar d resolution s of the bimodules 2 1 and 0 respectivel y whic h lead to determinations of//" ( — ,501) by cochains , cocycle s an d coboundaries . Also i n bot h case s on e ha s importan t interpretations o f th e vanishin g o f H n fo r n = 1,2,3 .

A definitio n o f cohomology space s for n ^ 2 in any variety of algebra s has been given b y Gerstenhabe r i n [2 ] a s follows . Le t 2 1 e 1^(1), the variet y of algebra s defined b y a set o f identities J , 501 an /-bimodule fo r 21 . Define a singular exten-sion of length two of 21 by 501 to be a null extension of 21 by 501 as defined on p. 91. If n > 2 , defin e a singular extension of length n to b e a n exact sequenc e o f bi -modules O->50l-»50l n_1 - > •50t 2-*(£->O togethe r wit h a singula r extensio n 0->(£-*(£-*2I—>0. Thes e giv e th e exac t sequenc e

0^501^501^! - > • • -+50t 2-+(£->21-0.

Morphisms, equivalences , additio n an d scala r multiplicatio n o f equivalenc e classes of singular extensions can be defined. Then one defines the nth cohomology group H "(21,501) for n ^ 2 o f 2 1 e iT(I) wit h coefficients i n 501 as the vector space of equivalenc e classe s o f singula r extension s o f lengt h n of 2 1 by 501. These defi -nitions ar e equivalent t o th e classica l one s in the associative an d Li e cases.

428 FURTHE R RESULT S AND OPEN QUESTION S

Gerstenhaber's definition s o f H n(3l,50l) fo r n ^ 2 hav e bee n recentl y supple -mented b y Glassma n i n [1 ] t o giv e definition s o f //°(3l,50i ) an d i/'(31,501) . W e shall now indicat e these . Le t 50^ and 50t2 be 7-bimodules fo r 3 1 e ^(J), rj a homo-morphism o f 50tj int o 50l2. The n an y derivatio n D o f 3 1 into SJlj determines th e derivation Drj of 31 into 5012 so we have the linear mapping fj: D-+Dn o f Der( 31,50 )̂ into Der(3I,50l 2). I n thi s wa y on e obtain s a functor fro m th e category o f 3I-bi -modules t o th e categor y o f vecto r space s s o tha t 9 K -> Der(3t,50t), */-**/ . A n inner derivation functor J: 501- » .7(31,501), /;-•/;|.7(3l,50t ) i s define d t o b e a subfunctor o f th e precedin g whic h respect s epimorphisms . I t ca n b e show n tha t the inne r derivatio n functor s ar e i n 1- 1 correspondenc e wit h submodule s o f Der(31,1/(31)) considere d i n a natura l wa y a s lef t £/(3l)-module . Relativ e t o th e choice o f J on e define s ifj(3l,9W ) = Der(3I,50l)/J(3I,aft) . No w assum e th e inner derivation functo r J i s finitely generated i n th e sens e tha t th e lef t U(3l)-modul e J((7(3l)) i s finitely generated . Le t {d iid2,~'->dk} b e a se t o f generator s fo r thi s module an d le t X i9 1 ^ i ^ k, b e a fre e 3I-bimodul e wit h generato r x t corre -sponding to the generator 1 of (7(31) . Le t d t b e th e derivatio n o f 3 1 into X { cor -responding t o d t an d le t Y be th e submodul e o f £ * © A\ generate d b y th e ele -ments flZ{</„ fle«. Pu t <£{,, } = (Z©JT,)/ y an d defin e 7/° lR}(3l,50t) = Hom^^G^jjJOl) fo r an y bimodul e 501 of 31 . For suitabl e choice s o f J an d {d j in the associativ e an d Li e cases the definitions o f H° an d H l give n b y Glassma n are equivalen t t o th e usua l ones . Also i f 0->50t ' ->50i-»50T ->0 i s a shor t exac t sequence o f 3l-bimodules , the n on e ca n defin e connectin g homomorphism s t o obtain th e usua l lon g exac t sequenc e o f cohomolog y spaces .

All th e result s just indicate d ca n b e develope d fo r algebra s ove r commutativ e rings. Th e notio n o f bimodules , universa l multiplicatio n envelope s an d exten -sions hav e bee n considere d b y Knopfmache r i n [1 ] an d [2 ] i n thi s generality .

CHAPTER 11 1

The Corollar y (p . 138 ) t o th e Coordinatization Theore m ha s bee n generalize d recently b y McCrimmo n i n [1 1 J. I f e x an d e 2 ar e nonzer o orthogona l idempo -tents i n a Jorda n algebr a 3 > the n thes e ar e sai d t o b e interconnected i f ^ 3 ; ? . ^ , / # y = l , 2 , Z u = %U^ r Thi s i s equivalen t to : 3iy-« i = 3« . 3/i = 3^« 4 • I* is cI e a r t n a t connectednes s implies interconnectedness. McCrimmo n has proved that i f 3 i s a Jordan algebra wit h 1 = Z l e, , wher e th e e { are nonzero interconnected orthogona l idempotent s an d n ^ 4 , the n 3 I s special (an d henc e any Jorda n algebr a containing 3 a s subalgebra wit h the sam e identity i s special) . This theore m ha d bee n conjecture d b y the author an d wa s suggeste d b y Martin -dale's theorem . I t i s eas y t o se e tha t an y tw o nonzer o idempotent s o f a simpl e Jordan algebr a ar e interconnected . A s ha s bee n show n b y McCrimmon , a con -sequence o f thi s an d a generalizatio n o f th e foregoin g theore m t o algebra s no t necessarily containin g 1 i s th e followin g strikin g re&ult : I f 3 i s a simpl e Jorda n


algebra (no t necessaril y containin g 1 ) whic h contain s thre e nonzer o orthogona l idempotents whos e su m i s no t a n identit y elemen t fo r 3, the n 3 i s special .

CHAPTR R IV

The structur e theor y o f thi s chapte r reduce s th e stud y o f th e Jorda n algebra s which are nondegenerate an d satisfy th e minimu m conditions o n quadratic ideal s to tha t o f Jorda n divisio n algebras . Thu s th e situatio n i s quit e simila r t o tha t which obtain s fo r Artinian semisimpl e algebras . At th e presen t tim e th e relatio n between Jorda n an d associativ e divisio n algebra s remain s t o b e clarified. A nat -ural questio n her e i s th e following : I s ever y specia l Jorda n divisio n algebr a o f one o f th e followin g types : (1 ) a Jorda n algebr a o f a symmetri c bilinea r form , (2) A+ wher e A is an associative division algebra, (3) £)(A, J) where A is a division algebra wit h involutio n J ? A t th e presen t n o example s o f exceptiona l Jorda n division algebra s whic h ar e infinit e dimensiona l ove r thei r center s ar e known .

There i s n o satisfactor y analysi s a s ye t o f degenerat e Jorda n algebra s satis -fying appropriat e minimu m condition s o n quadrati c ideals . I n particular , i t would b e interestin g t o hav e a definition o f a radica l fo r a Jordan algebr a satis -fying th e minimu m condition s whic h i s analogou s t o on e o f th e definition s o f the radica l o f a n Artinian associativ e algebras .

It woul d b e interestin g t o exten d t o Jorda n algebra s othe r aspect s o f th e as -sociative structur e theory . On e interestin g directio n woul d b e th e developmen t of a theor y o f Jorda n P/-algebras , tha t is , Jorda n algebra s satisfyin g a poly -nomial identity . Sinc e ther e exis t nontrivia l identitie s whic h ar e satisfie d b y al l special Jorda n algebras , th e so-calle d s-identitie s (p . 49 ) i t i s natura l t o defin e a P/-Jorda n algebr a t o b e one whic h satisfies an identity p which is not an s-iden-tity. Fo r example , th e Jorda n algebr a o f a nondegenerat e symmetri c bilinea r form satisfie s th e identit y p = [ [x 1 ,x 2 ,x 3 ] , 2 ,x 4 ,x 5 ] (ex . 6 , p . 364) , an d i t i s easily see n tha t thi s i s no t a n s-identity . Ar e thes e an d simpl e Jorda n algebra s which ar e finite dimensiona l ove r thei r center s th e onl y simpl e Pi-Jorda n al -gebras? The sam e questio n ca n b e aske d fo r Jorda n divisio n algebras .

The structur e theor y o f thi s chapte r ha s bee n extende d b y McCrimmo n ([6 ] and [14] ) to algebras over an arbitrary commutative rin g with 1 . As we indicated in Chapte r I , i n considerin g specia l Jorda n algebras , t o obtai n a genera l theor y one ha s t o replac e th e bilinea r compositio n a.b b y th e compositio n bU a whic h is linea r i n b an d quadrati c i n a, and , i f th e existenc e o f 1 is not assumed, then one mus t conside r als o th e unar y compositio n a' 2. T o simplif y th e discussio n we stic k t o th e cas e o f algebra s wit h 1 . I t i s convenient t o formulat e th e axiom s in term s o f th e mapping s U a. Fo r th e sak e o f compariso n wit h th e Jorda n al -gebras considered i n this book, w e give first an analogous definitio n o f the usual ones i n operato r form . Accordingly, w e define a Jordan algebra with 1 over a field Q> of characteristic not two to be a triple (3, R, 1), where 3 i s a vector space

430 FURTHER RESULT S AND OPEN QUESTIONS

over O , 1 a particula r elemen t o f 3 , an d R a mappin g of 3 int o H o m ^ ^ S ) satisfying th e followin g conditions :

1. R i s <D-linear . 2. Rt = 1 . 3. [* . ,*•*.]= 0 . 4. R a = L a i f xL a = aR x.

It is clear that this definition i s equivalent to the usua l one and that if P is an ex-tension field of O then 3P > 1 and the extension R o f R t o a linear mappin g o f 3p satisfies condition s 1-4 . Henc e w e obtain the Jordan algebra (3 P , R, 1). The fore-going definitio n ca n als o b e give n fo r lef t unita l <D-module s ove r a n arbitrar y commutative rin g wit h 1 , provide d tha t O contain s a n elemen t \ suc h tha t i + i = 1 . I n thi s case als o 3p> R an d 1 would satisf y th e sam e axiom s fo r any commutative rin g extension P o f O (with th e sam e 1) . A homomorphism n of a Jordan algebr a (3,-R , 1) into a Jordan algebr a (3 ' , /? \ l ') i s a O-homomorphis m of 3 int o 3' suc h that 1 * = 1 ' and (bRa)

n = b nR'a«. We no w giv e McCrimmon' s definitio n o f a quadratic Jordan algebra with 1

over an arbitrary commutative ring <b with 1 as a tripl e (3, U, 1) where 3 i s a unital lef t O-module , 1 a distinguished element of 3> and U is a mapping a-+U a

of 3 int o Hom <I)(3,3) satisfyin g th e followin g axioms : 1. £ 7 is O-quadratic , tha t is , U aa = a 2l/fl, a e O , ae% an d U atb = U a+b- U a

— U b i s O-bilinea r i n a an d b. 2. U x = 1 . 3. £/ f l<W = VbVa. 4. 1^1/ * = £/*K M wher e xV atb = 6U fl>Jc.

If P is a commutative ring extension wit h 1 of O, then U has a unique extensio n to a quadratic mapping o f 3 P = P ® *3 int o HomP(3p,3p)- Her e ^ Pi e P > aie 3 then U 1Piai = Ip?C/ f l i+ L^jPtP/U^j. W e requir e als o

5. Condition s 3 an d 4 hol d i n ever y 3 P-Axiom 5 i s equivalen t t o intrinsi c condition s o n 3 > namely , th e validit y o f

certain linearization s o f 3 and 4 . I t i s easy t o se e tha t i f O is a field wit h mor e than three elements, then these linearizations are consequences o f 3 and 4. Henc e 5 i s superfluou s i n thi s case .

We pu t a' 2 = W a, a o b = (a + b)' 2 -a'2- b 2 = \U mJb9 V a = U aA. The n aob = boa an d aoa = 2a' 2. Als o V x = l/ 1§1 = 2 . W e writ e {afrc } = fc(/ ar, so {aba} = 26l/ fla sinc e l/ fllI = 2l/ a. Also , {abc} = cK a>5.

As a specia l cas e o f th e notio n o f homomorphis m fo r genera l algebra s wit h finitary compositions (cf . Cohn' s Universal Algebra, p . 49), one defines a homo-morphism r\ of (3 , t f , l ) int o th e quadrati c Jorda n algebr a (3\U\V) t o b e a O-homomorphism o f 3 int o 3 ' suc h tha t 1 * = 1 ' an d (bU aY = b nUa«. Th e class o f quadrati c Jordan algebras with 1 over O is a category whos e morphism s are homomorphisms . I f O is a field of characteristi c no t tw o an d 3 i s a Jordan algebra wit h 1 over O then 3 define s a quadratic Jordan algebra with 1 , (3, V, 1)

FURTHER RESULTS AND OPEN QUESTIONS 431

in whic h 1 = 1 , t/ a = 2R* - R a.2. All the condition s excep t 4 ar e clear an d thi s has bee n prove d o n p . 328 . Conversely , le t (%U 91) b e a quadrati c Jorda n al -gebra wit h 1 over a field o f characteristi c no t two . The n i t ca n b e shown that , if w e put R a = \V a, w e obtain a Jordan algebr a wit h 1 , (%R, 1) . On e can sho w also tha t th e tw o construction s ar e inverse s an d homomorphism s o f (3 , U 91) coincide wit h homomorphism s of (%R,1). On e obtain s i n thi s wa y a categor y isomorphism o f th e categor y o f quadrati c Jorda n algebra s wit h 1 ove r 0 wit h the categor y o f Jorda n algebra s wit h 1 over 0 . Th e sam e resul t hold s fo r an y ring O which contains an element \ suc h that i + i = 1 , i f one defines a Jorda n algebra ove r 0 a s indicate d abov e (tha t is , a s i n th e field case) .

Let S H be a n associativ e algebr a wit h 1 over a commutativ e rin g 0 (wit h 1) . Define xU a = axa . The n xUab = axb + bxa, xV ab = abx + xba. I t is immediate that 1- 5 hold , s o 9 1 with 1 and th e indicate d U i s a quadrati c Jorda n algebr a with 1 , % iq) = (%U,l). I f (3,C/,1 ) i s a quadrati c Jorda n algebr a wit h 1 , a subalgebra R i s a 0-submodule containin g 1 and closed unde r bU a. Subalgebra s of algebra s 2l (q), 9 1 associativ e wit h 1 , ar e calle d special quadratic Jordan algebras with 1 . I f (91 , J) i s a n associativ e algebr a wit h involutio n an d 1 the n §(21, J), th e se t o f J-symmetri c elements , i s a subalgebr a o f 9l (fl). Le t 9 3 b e a vector space over a field of any characteristic,/a quadrati c form. Then it is readily verified tha t th e subspac e 3 = 0 1 + 9 5 of th e Cliffor d algebr a C(93,/ ) i s a sub-algebra o f C(93,/) (q). W e call this the quadratic Jordan algebra off.

We shal l se e i n a momen t tha t th e standar d exceptiona l Jorda n algebra s als o havp analogues i n quadrati c Jorda n algebra s (see als o the note s o n Chapter IX) .

It seem s likel y tha t muc h o f th e usua l Jorda n theor y ca n b e carrie d ove r t o quadratic Jorda n algebras . W e procee d t o indicat e tha t thi s i s indee d th e cas e with th e structur e theor y a s develope d i n thi s chapter . I n th e note s o n th e re -maining chapters, extensions o f som e othe r parts of the theory wil l b e indicated.

We conside r first th e basi c concept s o f th e structur e theory . On e define s a n inner ( = quadratic ) ideal 9 5 o f (3 , U, 1) t o b e a 0-submodul e o f 3 suc h tha t %Ub £ 23 , b e 93. An outer ideal 2 3 is a 0-submodule suc h tha t bU a e 23, b e 23, a e 3- A n ideal i s a subset whic h is bot h an inner and an oute r ideal . I f 93 is an ideal i t i s easily see n that i f beS B an d a,ce3> the n bU atC9 b' 2, b o c, bV aceSB. It follow s tha t th e 0-modul e 3/2 3 i s a quadrati c Jorda n algebr a wit h 1 in th e obvious way . Th e kerne l o f a homomorphis m i s a n ideal . I f be!$ the n 3 ^ i s an inne r idea l calle d th e principal inner ideal generate d b y b. I f O contains i , then an y oute r idea l i s an ideal . Otherwise , thi s may no t b e the case , a s the fol -lowing example s show : (1 ) Le t 3 h e the specia l quadrati c Jorda n algebra §(Z„ ) n x n symmetri c matrice s ove r th e rin g Z o f integers , an d le t 9 3 be th e subse t of integral matrices with even diagonal elements . Then S B is an outer ideal whic h is no t a n ideal . (2 ) Le t O be a field o f characteristi c two , P a n extensio n field such tha t th e subfiel d 0(P 2) ove r Q> generate d b y th e square s o f th e element s of P doe s no t coincid e wit h P . Le t §(P n) b e th e specia l quadrati c 0-algebr a o

432 FURTHER RESULT S AND OPEN QUESTION S

n x n symmetri c matrice s ove r P . Le t 33 be the subset consistin g of the matrice s with diagona l entrie s i n <J>(P 2). Then 2 3 is an oute r idea l whic h i s not an ideal .

The oute r idea l 3 ' ° f (3» *A 0 generate d b y 1 is a subalgebr a whic h w e shall call th e core o f 3- Any subalgebra o f 3 containin g 3 ' wil l b e called a n ample subalgebra o f 3-

In a n associativ e algebr a wit h ] th e relation s aba = a an d ab 2a = 1 impl y ab = 1 = ba. Thi s suggest s the definition: ae{%U,l) i s invertible wit h b as an inverse i f blla = a, b' 2 Ua = 1. This is equivalent to the usua l notions in a Jordan algebra an d on e has a close analogue of Theorem 1.1 3 (on inverses) . Fo r example , the invers e i s unique an d UaUb = 1 = U hUa for b the inverse a'" 1. A quadrati c Jordan algebr a wit h 1 is a division algebra i f every nonzero element of the algebra is invertible .

If w i s an invertible element of (3, U,l)} then w e put l( t t ) = u'"\ U {au) = UuUa.

Then (3 , U("\ 1 (M)) i s a quadratic Jorda n algebr a calle d the u-isotope o f (3, U, 1). As i n the case of Jordan algebra , isotop y i s an equivalence relation .

An elemen t e of a quadrati c Jorda n algebr a (3 , U, 1) is idempotent i f e'2 = e and th e idempotent s e,f ar e orthogonal i f fU e = eof = eUf = 0. Le t {e.j / = 1,.••,/! } b e orthogonal idempotent s suc h tha t Z " ^ = 1 . Put EH = Ue.9

Eu = U e.te for / ^ y. Then 1 = £ , ^ £ , 7 an d the Eu ar e orthogonal projections . Hence 3 = Z © 3 u - Thi s IS ca^e(^ t ne Peirce decomposition o f 3 relativ e to the ev

The 3it are inner ideals . There are many importan t relation s connecting the Peirce components 3,7 analogous t o those w e derived fo r Jordan algebras . On e defines connectedness and strong connectedness for orthogonal idempotents in a quadratic Jordan algebr a wit h 1 as for Jordan algebras . Th e principal result s carr y over : transitivity hold s an d on e can pas s fro m connecte d idempotent s t o strongl y connected idempotent s in an isotope as in Lemma 5 of §3.1 (p. 123).

We conside r nex t th e definition o f quadratic Jorda n matri x algebras . Fo r th e sake o f simplicity w e stick t o the case of standard involutions ; the more genera l case of canonical involution s is reducible to this via isotopy by a diagonal matrix . First, let (D, j) b e an associative algebra with involution an d 1 . For any n = 1,2,3 , •••we defin e th e standard quadrati c Jorda n matri x algebr a £>(D„) to be the se t of n x n matrice s ove r D whic h ar e symmetric unde r th e standard involutio n X -*X l ( d = dJ) wher e AU B = BAB and 1 and the dMnodule structur e ar e as usual. I n studyin g thes e algebra s i t is convenient t o modify ou r usual definitio n of fl[(/], a e D b y puttin g a\if\ = ae ih a e ^ ( D j ) , a[i/ ] = ae^^-ae^ a e D . Using these on e can develop formulas fo r the U operator analogou s to (18 ,)-(23/) of p . 126 . These an d the formulas whic h on e obtains fro m th e Jordan matri x algebra §(D 3) , wher e (£>J ) i s alternative ove r a field of characteristi c 5 * 2, lead to th e definition o f $(£>3) for (D J) alternative. Here one begins with an alternative algebra wit h involutio n (DJ ) and 1 satisfyin g th e conditio n tha t ddeN($)), the nucleus of D, for every d e D. Then it can be shown that D' = § ( D J) n iV(D ) is a O-submodul e o f D such tha t dd'deT)' fo r all d e D, d ' e D ' . On e defines§(D 3)

FURTHER RESULT S AND OPE N QUESTIONS 433

to be the set of matrices which are symmetric under the standard involution i n D3

and which have diagonal elements in T>'. (Not e tha t i f J) is associative thi s is the same as our former definition. ) Equivalently , w e let §(D3) b e the 4>-module direct sum o f thre e copie s o f X ) and thre e copie s o f D ' denote d respectivel y a s D [ y ] , i <j, 3y[it] , i,j = 1,2,3 . I f / > y, w e put d\jj~\ = d[ji\, wher e d[ji] i s the image in D [ J J] o f de X> . It is easy to see that there is a unique quadratic mapping a -* Ua

of §(© 3) = ©[12 ] © £[23 ] © Tj[13 ] © £'[11 ] © £'[22 ] © £'[33 ] int o Hom0(fy(T)3),<Q(T)3)) such that the following formulas hold:

(1) b'[ii]U aVQ = a'b'a'lii],

(2) a V'1UaliJi = da 'aiJJ\

(3) b[ij]U a0n = aba\ij\,

(4) K L » M < / K D y ] } = a'bc'[if]>

(5) {aVtylVW*]} = (a'bc + FbTtii],

(6) {aViMUWk]} = a'bc{ikl

(7) {«'[«'/]6'[«]c[y] } = a'b'c[ia

(8) {amb'UjWQ} = ab'c{ikl

(9) {a[«j]bDi]c[/k] } = o(bc)[ik] ,

(10) {a[y]*D*M*fl } = ( f l(^) + «(6c) ) [//],

where {abc} = 6(7 fl c and it is understoo d that all the U formulas not covered b y these an d a\Ji\ = d\ij'] ar e 0 (e.g. a'[ii\U b,ljn = 0 if / 5^ /). W e remark tha t th e condition tha t ddeN($)) implie s tha t dd an d d + de/X)' fo r deT). Henc e th e right-hand side s o f (I)-(IO) ar e contained in£)(T) 3). I t is a formidable task whic h has been carried out by McCrimmon to show that(£(D3), I/ , 1), where 1 = Z ? 1[»] satisfies th e axiom s fo r a quadrati c Jordan . W e shal l no w defin e a standard quadratic Jordan matrix algebra t o be either an algebra of the form(§(X)n), U, 1), where (T),7) is associative with involution and 1, or one of the algebras (§(3)3), U 91) just defined .

Let 3 b e suc h a standar d quadrati c Jorda n matri x algebr a an d le t X) 0 b e th e O-submodule o f I) generated b y the elements dd, deT). The n it is easily seen that if we identify d'\ii\ wit h d'eu, d[ij~\ with deu + de jh i < j , the n the core 3' = §(£>„)' > n ^ 3 i s th e subse t o f matrice s havin g diagona l element s i n 33 0. Th e element s ei = ! [" ] a n d ![(/] > ' <J» a r e containe d i n 3'- The e£ are orthogonal idempotent s such tha t Ze t- = 1 and e> t an d e j9 i ^ j 9 ar e strongl y connecte d b y l[i/J . A s fo r ordinary Jorda n algebras , on e ha s a Stron g Coordinatization Theore m whic h i s a structur e theore m fo r quadrati c Jordan algebras (3, U, 1 ) such that 1 = Z " ei9

where th e e t ar e nonzer o strongl y connecte d orthogona l idempotent s an d n *z 3


The resul t state s tha t unde r thes e condition s on e ha s a homomorphis m rj o f (3, U, 1 ) int o a standar d quadrati c Jorda n matri x algebr a £)(£)„) suc h tha t 3 n

is an ample subalgebra of §(D n) and the kernel of rj is an ideal consisting of absolute zero divisors z (that is , U z = 0 ) such that 2z = 0 .

We ca n no w stat e McCrimmon' s extension s o f ou r structur e theorems . Th e class o f algebra s t o whic h thes e appl y ar e th e quadrati c Jorda n algebra s wit h h (3 > U* 1) whic h contai n n o absolut e zero-divisor s 5*0 , an d whic h satisf y th e minimum condition s for inner ( = quadratic ) ideals (p . 157) . Such an algebra is a direct su m o f simpl e quadrati c Jorda n algebra s satisfyin g th e sam e conditions . A determinatio n o f thes e ca n b e give n b y th e followin g theorem : A quadrati c Jordan algebr a (% (J, 1) i s simpl e wit h minimu m condition s fo r inne r ideal s i f and onl y i f i t i s o f on e o f th e followin g types : I . a quadrati c Jorda n divisio n algebra, TI. an ample subalgebra of a quadratic Jordan algebra of a nondegenerate quadratic form , III . a n ampl e subalgebr a o f a quadrati c Jorda n matri x algebr a £>(03,/y) wher e CD,;" ) i s a n octonio n algebr a wit h standar d involution , IV . a n ample subalgebr a o f a quadratic Jordan algebra £>($! , J), wher e (%,J) i s a simple Artinian associativ e algebra with involution.

We not e finally tha t McCrimmo n ha s succeede d als o i n extendin g th e theor y to quadratic Jordan algebras without 1 by giving axioms in terms of the U operator and a unary composition a -* a2, whic h insure that such a system can be imbedde d in a quadratic Jordan algebra with 1 .

CHAPTER V

Braun an d Koeche r [1 ] hav e define d th e radica l o f finite-dimensional Jordan algebra 3/3 * wit h 1 * 6 b e th e intersectio n o f th e radical s o f al l th e symmetri c bilinear form s / o n 3 int o extensio n fields o f O suc h tha t / (z , 1) = 0 fo r al l nilpotent z . I t i s clea r tha t rad 3 a s define d i n thi s chapte r i s containe d i n th e Braun-Koecher radica l ft. Conversely , le t a$md% s o a ha s a nonzer o imag e a i n some simpl e homomorphic imag e 3 o f 3- Le t P b e the center of 3 s o P is a finite-dimensional extensio n field o f O . Le t i b e th e generi c trac e bilinea r for m on 3/ P s o i i s nondegenerat e (b y Chapte r VI , p . 24 0 o r Chapte r VIII , p . 353) , i ha s values in P and /(£,! ) = 0 for z nilpotent. Sinc e a 5* 0, a i s no t containe d in the radica l o f f. Defin e / b y f(x9y) = i(x,y). The n i t i s immediat e tha t / i s a form of the type considered by Braun and Koecher and a is not contained in the radical of / . Henc e a $ ft. Thu s ft c rad 3 and ft = rad3 .

CHAPTER V I

The author ha s prove d i n [32 ] tha t th e generi c nor m n(x) , x = Z " €itti9 of any finite-dimensional simpl e Jorda n algebr a i s a n irreducibl e polynomia l i n €>[^l5 •••,£„]. Th e metho d o f proo f o f thi s resul t i s a n extensio n o f on e use d b y Dieudonne i n [2 ] t o prov e the sam e resul t for simpl e associative algebras . Sinc e


any simple associative algebra 9 1 over a field of characteristic not tw o determine s the simpl e Jorda n algebr a 9l + wit h th e sam e generi c norm , th e theore m fo r associative algebra s o f characteristi c no t tw o follow s fro m th e Jorda n algebr a result. The Jordan theorem gives the following consequence of Theorem 3 (iv): If ̂ 1 is a simple Jorda n algebra and the remaining hypotheses are as in Theorem 3 (iv) then 2(£i , ••-,£,,) is a power of the generic norm. I t is easy to extend thi s resul t t o the semisimple case.

The classica l theore m o n compositio n o f quadrati c form s (se e Theore m 4.5 , Jacobson [22] , and the references given there) has been generalized b y Schafer in [14]. Hi s results are that if 91 is an algebra which possesses a nondegenerate homo-geneous for m Q(x) o f degre e n permittin g th e compositio n Q(xy) = Q(x)Q(y) and the characteristic i s 0 or p > n, then 9 1 is alternative. Moreover , i f 91 is finite dimensional, the n 9 1 is separable . Nondegenerac y o f Q in Schafer' s sens e mean s that i f Q(x l,x2,-~,x„) i s th e symmetri c multilinea r for m associate d wit h Q a s usual s o tha t Q(x 9 --,x) = Q(x), the n th e onl y z suc h tha t Q(z,x 2, •~,x n) = 0 for al l Xi is z = 0 . Schafer' s result , alon g wit h Theore m 3 , ca n b e use d t o giv e a complet e determinatio n o f th e nondegenerat e Q whic h permi t compositio n o n a finite-dimensional algebra . Schafe r conjecture d tha t hi s conclusion s ar e vali d without th e assumptio n o f finiteness o f dimensionality . I n anothe r pape r [13 ] Schafer considered the problem of characterizing Jordan algebras by commutativity and th e existenc e o f a nondegenerate for m Q satisfyin g th e Jorda n compositio n Q({xyx}) = Q(x) 2Q(y). H e wa s abl e t o trea t thi s proble m onl y fo r deg g = 2 and 3.

An extensiv e generalizatio n o f thes e result s o f Schafer' s ha s bee n give n b y McCrimmon i n [2] . H e consider s finite-dimensional algebra s 91/ 0 an d define s a for m (homogeneou s polynomia l functio n o f degre e > 0 ) Q o n 9 1 t o admit composition i f there exists two rational mappings E annd F of 91 into Hom4)(9l, 91) such that: 1. £(1) = 1 = F(l) , 2 . A" E = <xa L, A\ F = fla R where a, ft are nonzero elements of O and aL and aR are respectively the left and right multiplications by a, 3. Q(aE b) = Q(a)e{b), Q(aF b) = Q{a)j{b) fo r som e rationa l function s e an d / in 91 (whenever the various functions are defined). A form Q is called nondegenerate if th e correspondin g trace form t(a,b) = — A^A^logg i s a nondegenerat e symmetric bilinea r form in a an d b. I f Q admit s compositio n an d th e charac -teristic is 0 or a prime p exceeding the degree of Q, then this condition is equivalent to Schafer' s condition s fo r nondegeneracy . Wit h thes e definitions , McCrimmo n has prove d tha t i f 9 1 is a finite-dimensional algebra wit h 1 over a n infinit e field which possesse s a nondegenerat e for m Q admittin g composition , the n 9 1 i s a separable noncommutative Jorda n algebr a (definitio n i n ex. 8 , p . 33 ) an d 9l + i s a separabl e (commutative) Jordan algebra . The converse i s also valid as is shown by McCrimmon in [2] and Osborn in [7]. McCrimmon's theorem implies Schafer's theorem on composition in the usual sense with the improvement that the condition on the characteristic can be replaced by the assumption that the number of elements


in the bas e field exceeds the degree o f Q. McCrimmon' s theorem yield s als o th e generalization o f Schafer' s result s o n Jorda n compositio n t o form s o f arbitrar y degree.

In [8 ] McCrimmo n generalize d th e foregoin g result s t o algebra s fo r whic h finiteness of dimensionality i s not assumed and has obtained an affirmative answe r to Schafer' s conjectur e note d above . I n [10 ] th e sam e autho r ha s give n a n ex -tension of the theory of the generic minimum polynomial to strictly power associa-tive algebra s wit h 1 which ma y b e infinit e dimensiona l bu t ar e "genericall y al -gebraic" in a certain sense.

The basic tool in these papers of McCrimmon's is that of the differential calculu s of rationa l mappings . Thi s metho d wa s use d systematicall y first b y Koeche r i n applying th e structur e theor y o f Jorda n algebra s ove r th e rea l field to determin e the homogeneou s domain s o f positivit y an d mor e genera l "co-domains " (se e Koecher [4 ] and also Vinberg [1]) .

The following theore m has been proved b y Meyberg i n [3] : Let 3 an d 3 ' ̂ > e

finite-dimensional central simpl e Jordan algebras with the same underlying vecto r space o f characteristi c ^ 3 (and 5 * 2) an d assum e tha t 3 an d 3' hav e th e sam e structure grou p (equivalently , i n vie w o f Theorem s 6. 6 an d 6.7 , i f |4> | i s larg e enough, the same groups o f norm similarities) , then 3 an d 3' ar e isotopes.

CHAPTER VI I

It would b e interesting t o obtai n proo f o f the main result s o f thi s chapter (the separability o f C/(3 ) fo r 3 separable , th e Albert-Penico-Taft theorem , Harris ' theorem) withou t usin g th e classificatio n o f simpl e Jorda n algebras . Fo r charac -teristic 0 , suc h proof s ar e give n i n Chapter VIII . Perhaps i t ma y b e possibl e t o transfer th e result s o f thi s cas e t o characteristi c p b y a metho d o f reductio n modulo p.

It i s unknow n at th e presen t tim e whethe r o r no t anythin g lik e Theore m 6.1 6 is valid i n the characteristi c p case . I n particular, the followin g questio n appear s to b e unsettled: Let 3 = Si ® 5 R wher e Si is separable and 9 t i s the radical. Le t SB be a separabl e subalgebra . The n doe s ther e exis t a n automorphis m rj such tha t 931* £ ft? (Theore m 6.1 5 give s an affirmative answe r in a special case. )

CHAPTER VII I

We shall indicate th e grou p theoretic backgroun d fo r som e o f th e Li e algebras considered in this chapter and the extensions of these notions to quadratic Jordan algebras (cf. Koeche r [8] and McCrimmo n [6]) .

We consider first the structur e group T(3) . W e defined thi s to b e the group o f isotopies o f 3 ont o 3 an d w e saw in §1.1 2 that T(3 ) i s the se t o f bijectiv e linea r mappings a in 3 fo r which there exists a linear mapping /? such that U XOL = /?(/** , x e 3 - The n 1 * i s invertibl e an d p = <x(U x*yl. Henc e th e conditio n is :


Ux* = U^a l U x* or, with a slight change of notation, we can define T(3)t o be the set of bijective linea r mappings W of 3 suc h that

(1) U xW = W*U XW9 W* = U iwW'\ x e 3 .

It follows tha t W e T(3) i f and onl y i f W is a bijective linea r mappin g in 3 suc h that ther e exist s a bijectiv e linea r mappin g W* in 3 satisfyin g U xW = W*UXW. Then necessarily W* — U lw W~l.

These consideration s carr y ove r t o quadratic Jorda n algebra s (3 , U91) with 1 over an arbitrary commutative ring d> wit h 1 . We defirfe the structure group T(3) to b e the group o f bijectiv e O-endomorphism s W of 3 fo r which ther e exist s a bijectiv e O-endormorphis m W* such tha t U xW = W*U X W, x e 3- I t i s clea r from th e axiom 3 for quadratic Jorda n algebra s wit h 1 that i f a i s invertible , then (7 aer(3) . Also , U IW= W*W i s bijective , s o \W i s invertibl e an d W* = U lwW~l e T(3). I t is immediate that W -> W* is an anti-automorphism in T(3) and W** = W. If a is invertible and WeTQ), the n l/ flH, = W*U aW is in -vertible. Henc e a W is invertible i n 3 an d (aW)'1 = (aW)U~w = a'~\W*)~ l. If WeTQ) an d 1W = 1 then W* = W l an d C/XW = WU xW. Thi s state s tha t W is a homomorphism of (3, U, 1) so W is an automorphism. Henc e the group of automorphisms Aut3 £ T(3 ) and Aut3 is the subgroup of T(3) fixing the element 1. The subgroup r\(3) of F(3) generate d b y the invertible U a is called th e inner structure group. Th e elements of 1^(3) n Aut 3 ar e called inner automorphisms.

Let D b e the algebra ove r <t > with basi s (1,0> where t 2 = 1 (cf. the proof of Theorem 2.1(7) on p. 67). Consider (3 ,̂ U, 1) and let (5(3) be the set O of endomor-phisms A o f 3 suc h tha t 1 + At e IXSD). Her e ^ * s ^e D-endomorphis m i n 3$ extendin g th e given A. The condition tha t W = 1 + At e T ^) i s tha t (1 ) holds fo r x e 3- Since W~ l = 1 — At, thi s give s th e condition tha t /4e©(3 ) i f and only if

(2) UX*A=UI.IAU X + LU X9A], X G 3 .

It follows fro m this , o r directly fro m th e definition, tha t ©(3 ) is a Li e algebra (subalgebra o f Hom 0(3,3)~ ) whic h i s restricte d i f p<b = 0 fo r a prim e p (cf. Serre, Lie Algebras and Lie Groups, Ne w York, 1965 , p. L.A. 1.3). We call ©(3) the structure Lie algebra o f (3, U, 1). It is easily seen that A e ©(3) i f and onl y if there exist s a n ^'eHom^Q , 3) suc h tha t

(3) U x,xA = A'U x+UxA, x e 3 .

Then A' is uniquely determined and in fact, A' = U 1A — A. One can deduce from the axiom s fo r quadrati c Jorda n algebra s wit h 1 the identit y U xVab + V hJUx

=* Ux,xva,b which shows that Vajb e ©(3)- Since UuiA = V x 1A we see that A' e ©(3). Hence A -+A' i s an anti-automorphism of the Lie algebra ©(3) and e: A ->A s — A' is an automorphism satisfyin g e 2 = 1 . Also we have V ab = — V ha.


If Xe(5(3)and Wer(3),the n W'\l + AtWerQJ. Henc e W~ lAWe(5Cs). Thus T(3) acts on (5(3 ) b y means o f A -> W~1AW an d this is an automorphis m of (5(3) - O n e c a n establis h easil y fro m th e definition s tha t (W~ 1AW)' = W*A'(W' 1)*.

The definin g conditio n (3 ) fo r Ae (5(3) give s U xyA + U ytJcA = A'U xy + U xyA which implies that AVX,Z + V xA, = 7 X,2A, + V Xt2A. Hence [T fl>b, A]=VaAib + F fl>w. This implies that the subset fi(3) o f sums of the operators V ab is an ideal in ©(3) -We shal l cal l thi s idea l th e inner structure Lie algebra o f (3, U, 1).

We define a derivation D of (3, U, 1) to be a O-endomorphism suc h that ID = 0 . and [U X,D] = U XjXD, x e 3 . I t i s clear from (3 ) that D e (5(3 ) an d D ' = - D o r D = D. Th e se t o f derivation s De r 3 i s th e subalgebr a o f (5(3 ) o f mapping s satisfying I D = 0 . This i s clear from (2). The elements of £(3) n Der 3 ar e called inner derivations. Amon g these one has the strictly inner derivations V ab — V ba. The inne r derivations an d strictl y inner derivations constitut e ideal s Inde r 3 an d Strinder3 in Der3 .

We consider nex t th e extensio n o f th e Tits-Koeche r constructio n o f th e Li e algebra 51(3 ) t o quadrati c Jorda n algebras . Since V ab = 2a A b if 3 i s a Jordan algebra an d V atb i s define d a s i n (3 , C7,1) with U a = 2R} a — Ra.2, on e i s lea d t o define R(3) = 3 © 3 © #(3)> where 3 i s a copy of 3 an d fi(3) is the inner structure Lie algebra . W e defin e a produc t [ , ] i n 51(3) a s i n (26 ) o n p . 32 5 with th e modification tha t a A 6i s replace d by V ab(ct Koeche r [8]). Asmex. 1,p . 329, w e can i ntroduce the same bilinear product [ , ] in g(3 ) = 3 © 3 © ®(3)> © ( 3 ) t h e

structure Li e algebra. Th e verificatio n give n in the text carrie s over to sho w tha t g(3) an d 51(3) are Lie algebras. W e have the automorphism e in 5 (3 ) : a + b + A -» b + a +A (cf . p . 327) and s2 = 1 .

We now retur n t o Jorda n algebra s wit h 1 (over a field o f characteristi c 5 * 2, though everythin g i s vali d fo r a commutativ e rin g O containing i). Alon g wit h the Jordan algebra 3 w e consider the associated quadratic Jordan algebra (3, U 91), where U a = 2R 2

a - R fl.2. Then Va>b = 2aAb = 2(R ttmb - [R flRj). A linear mapping D is a derivation of the Jordan algebra if and only if it is a derivation of (3, U, 1). Since ©(3 ) include s th e R a = \V a^ i t i s immediat e tha t (5(3 ) = K(3 ) © Der 3 and fi(3) = R(3 ) © [R(3) , *<3>]-"

We shal l no w prov e a recen t theore m o f Koecher' s whic h i s a n importan t addition to the theory of 51(3) given in the text. This states that an y derivatio n D of 51(3 ) i n t o 5(3 ) c a n b e extende d t o a n inne r derivatio n D o f g(3 ) an d every derivation o f g(3 ) i s inner . To be consistent wit h the text , w e use the definitions given in the text rather than the modifications just indicate d fo r quadrati c Jorda n algebras. W e not e first that i f A i s a linea r transformation i n 3 suc h tha t ther e exists a linea r transformatio n A suc h tha t [aAfc,A ] = a A Lb + aAbA, the n A e ©(3) an d A = A. Thi s follow s b y retracin g the step s o f th e argumen t give n above t o sho w tha t S(3 ) i s a n idea l i n ©(3) - No w le t D b e a derivatio n o f th e ideal 51(3 ) i n t o 5(3) - Sinc e [ a + 5 + L,R, ] = a — b w e ca n subtrac t a n inne r


derivation fro m D to obtai n a derivation mappin g R t int o ©(3) - Henc e w e may assume R lDe®Q)- s i n c e [>>Ki]_ = « for " 3 , w e hav e [aD 9Rt] + [>,I^D ] = AD , and ^ since [ a D ^ J e S + 3 an d K i D e © ^ ) , s o [ a ^ D ^ j e S , w e hav e aD e3 + 3 - Henc e w e can write aD = aD t + aD 2, wher e D t an d D2 ar e linea r mappings o f 3 int o 3 an d 3 respectively . The n [fl,/*i ] = a give s aD j + aD 2

= aDij- aD 2 + aiR^D). SmcQ_aR{D e% w e obtain D2 = 0 , so 3D c 3 . Similarly , 3D < = 3 . The n [ 3 3 ] D c [ 3 3 ] , an d sinc e [ 3 3 ] = £ , w e se e that _also £D s f i . Let Dx b e the restriction of D to 3 an d write the restriction of D to 3 a s x -* xD2 , x e 3 - Le t D 3 b e the restrictio n o f D t o £ . I f a e 3 an d L e £ , the n [a,L ] = aL and aLD t = [aD l5 L ] + [a , LD 3] = aD xL + aLD 3. Henc e LD 3 = [ L , ^ ] . If 6 6 3, the n [a A fc, DJ = ( a A b)D 3 = [a , S] D = \aD u B] + [a, M) 2] = aD lAb + aA bD 2. B y the resul t note d a t the beginning , w e have Dx e ©(3 ) and D 2 =D V I t follow s tha t th e derivatio n D coincide s wit h th e restriction s t o ft(3) o f th e inne r derivatio n X-+[X,D X~\ determine d b y D te(5Q). I t follow s easily that every derivation of 5(3) i s inner and that Derg(3) = 5(3 ) = Der5t(3) .

We consider next the automorphisms of 5(3) and 51(3)- We have already defined the automorphism e: a + b + L^>b + a +Lof 5(3 ) andft(3) . Le t W eT(3) th e structure grou p an d A €©(3). The n W~ lAWe(SQ). Th e definitio n o f T(3 ) implies tha t W~\a A b)W = aW A b(W*y\ Henc e A-+W~ XAW i s a n auto -morphism of ©(3) which maps £(3) into itself. Also, a + B + A-+aW + b(W*)~ l

4- W ~ XA W is an automorphism OL W of 5(3) whic h maps R(3) into itself. It can be shown tha t th e OL W ca n b e characterize d a s th e automorphism s o f 5(3 ) (51(3) ) which ma p 3 int o itself. If a e% the n (ad of = 0 i n 5(3 ) (cf . ex . 4 , p . 329) . I t follows tha t T a = exp(a d a) wher e Zexp(a d a)) = 1 + [Xa ] + i [ [Aa]a ] i s a n automorphism i n 5(3 ) mappin g 51(3 ) int o itself . Th e automorphis m e , a^ fo r W e T(3), Ta for a e 3 generat e a subgroup Aut' 5(3) of the group of automorphisms of 5(3) - Relation s betwee n thes e generator s hav e bee n considered b y Koecher. There exis t Jorda n algebra s 3 suc h tha t Aut ' 5(3 ) # Au t 5(3) .

For finite-dimensional Jordan algebras , Koecher ha s defined i n a recent paper [10] a group of rational mappings which is closely related to the foregoing groups . This i s th e grou p E(3 ) generate d b y th e structur e grou p T(3) , th e translation s ta: x -> x + a and the mapping j: x -> — x'"1 define d on the set of invertible elements. In general , th e mapping s o f S ar e rationa l define d o n Zarisk i open subset s o f 3 and the composition is the usual composition o (cf. §6.2) . The Hua identity can be written in operator form as j o ta o j o ta. -1 o j o ta = U a. This shows that the inner structure group I \ (3) i s contained i n the subgroup S t(3) generate d by j an d the translations. I f T denote s th e subgrou p o f translations , the n i t follow s fro m Hua's identity that S = T o TojoTojo T (with obvious meaning). Koecher has given a beautiful characterizatio n o f th e grou p S b y a differential equatio n suc h that a rationa l mappin g i s containe d i n 3 i f and only i f i t satisfie s th e equation. Using this , he ha s proved tha t S i s a n algebraic group (not linear) . Also h e ha s studied the question of the uniqueness of the representation of an element according


to the decomposition of S as T o Toj o To j o T. One obtains a linear representation of S ont o Aut'3(j) sending W -• OLW, t a -• ia, j - » e.

CHAPTER I X

A numbe r o f characterization s o f finite-dimensional exceptional centra l simpl e Jordan algebras based on forms have been given. We have already noted Schafer' s characterization in [13] by means of cubic forms satisfying the Jordan composition (see th e note s o n Chapter VI) . Another importan t characterizatio n i s one du e t o Springer in [1] . In this, one assumes that 3 i s a finite-dimensional algebra with 1 over a field of characteristi c ^ 2 , 3 equipped wit h a quadratic form Q with non-degenerate associated bilinear form (x,y) suc h that (1) Q(x2) = Q(x) 2 i f (x, 1) = 0 , (2) (xy,z) = (x,yz), (3 ) Q(l ) = 3/2 . Then Springe r ha s shown that 3 i s a central simple Jordan algebra o f degree three with Q(x) = %t(x2)(x'2 = x2) and conversely.

Another characterization base d on cubic forms i s one which was first suggested by Freudentha l (cf . [6] ) an d was established b y Springe r in [5 ] fo r finite-dimen-sional algebras over fields of characteristic ^ 2,3 . Recently, this has been extended by McCrimmon in [15 ] to quadrati c Jorda n algebra s ove r arbitrar y fields. Th e Springer-McCrimmon axioms ar e th e following . Le t 3 b e a finite-dimensional vector spac e ove r a n arbitrar y field O equipped wit h a cubi c for m n and a dis -tinguished element / e 3 satisfyin g th e following conditions :

1. n ( l ) = l . 2. t(x,y) = — A^A^lognis a nondegenerat e symmetri c bilinea r form . 3. I f x* i s defined b y t(x*,y) = A y

xn, the n x # # = n(x)x. Now define x x y = (x + y)* — x* — y* an d

(1) yU x = t(x,y)x-x* x y.

Then McCrimmo n ha s shown , usin g the technique s o f the differentia l calculu s o f rational mappings , that (%U,1) i s a quadratic Jorda n algebr a wit h 1 . (Actually, the hypothesi s o f finiteness o f dimensionalit y i s no t essential . On e requires onl y a formulation whic h permits the application of the differential calculus. ) Le t (X>J) be a composition algebr a ove r a n arbitrar y field 0 , § (T>3,J y) the spac e o f 3 x 3 matrices over D whic h are symmetric under the canonical involution X -> y~ lXly, y = diag {Vi,72*73)5 7i 5^ 0 in O. Let n(X) b e defined as usual (equation (50), p. 232) and le t 1 be the usua l identit y matrix . The n i t can b e show n tha t th e condition s 1-3 ar e satisfied , s o i f U i s define d b y (1) , the n (§(D 3,Jy), U,l) i s a quadrati c Jordan algebr a wit h 1 . Th e structur e the n define d i s a n isotop e o f a standar d quadratic Jorda n matri x algebr a a s define d i n th e note s o n Chapte r I V b y a n octonion algebra .

Similarly, on e ca n exten d Tits ' construction s t o quadrati c Jorda n algebras . We indicat e onl y th e first o f these . Henc e le t % be a centra l simpl e associativ e algebra (with 1 ) of degree three. Then the generic norm n on 31 is a cubic form and


t(a, b) = - A * Afc lo g n i s a nondegenerate symmetri c bilinea r for m o n 31 . Let 3 = 2l o ® 21 1 ® 2l 2

a direc t sum of three copies of 91, a -* a,- a linear isomorphism of S 3I onto 91, . Let / ibea nonzer o element o f Q> and defin e

n(x) = n(a) + M^O ) + /*~ ln(c)

for x = a 0 + b± + c 2, a,b,ce s2I. The n i t ca n b e show n tha t thi s an d 1 = 1 0

satisfy th e condition 1- 3 s o (1 ) define s a quadratic Jordan algebra wit h 1 . I f the characteristic i s not two, this coincides with the quadratic Jordan algebra define d by Tits.

There i s a n extensiv e literatur e o n exceptiona l algebrai c groups , Li e algebra s and geometrie s connecte d wit h exceptiona l Jorda n algebras . W e indicat e thi s briefly. Tae-I l Su h i n [1 ] ha s prove d tha t an y isomorphis m betwee n th e littl e projective groups o f Moufan g projectiv e planes i s induced b y an isomorphism o r correlation between the planes. His proof is based on a classification of involutions contained in the little projective group. Another method o f obtaining Suh' s result and an extension of this to the middle projective group has been given by Veldkamp in [2] . An extensiv e stud y of elliptic and hyperbolic Moufan g planes , that is , the geometry o f Moufan g plane s relativ e t o certai n type s o f polarities , ha s bee n made b y Springe r an d Veldkam p i n [1] . Recently , Springe r an d Veldkamp i n [2] an d Veldkam p i n [3] , [4 ] an d [5 ] hav e develope d a geometr y o f spli t Jordan algebras . Thi s i s agai n base d o n th e element s o f ran k one .

Additional result s o n th e group s o f automorphism s an d o f nor m preservin g transformations o f reduce d exceptiona l simpl e Jorda n algebra s ar e give n i n Jacobson [26] and [27], These are algebraic groups of types F4 and E6 respectively. Soda in [1] has studied the groups Aut3 \&, where 3 i s reduced simple exceptional and St i s a cubic subfield . Thes e ar e exceptional simpl e algebrai c group s o f typ e D4, whic h had been studied previously i n a geometric fashion b y Tits [3] . Soda's results are based in part on some given by Springer in [9].

In a series of papers [6]-[14], Freudentha l has studied certain real forms of the Lie algebras E n an d E 8 an d relate d these to symplectic and metasymplectic geom-etries. Certai n form s o f Li e algebra s o f type s D 4 an d E 6 hav e bee n studie d respectively by Allen in [1 ] and by Ferra r in [1].

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SUBJECT INDE X

Albert-Jacobson theorem, Albert-Jacobson-McCrimmon

theorem, Albert-Paige theorem, Albert-Penico-Taft factor , Albert-Penico-Taft theorem, Albert's theorem

on nil Jordan algebras, on semisimple Jordan algebras, on simple Jordan algebras, on solvable Jordan algebras,

algebra, alternative, alternative division, Cayley, Clifford, composition, degree of an, exterior, free /-Lie, Malcev, meson, nil, of octonions, opposite, simple, strongly special, unramified,

algebra of octonions, algebra with involutions,

center of , central simple, centroid of , degree of , derivation of , homomorphism of an, ideal of , multiplication algebra of , simple Artinian, perfect, subalgebra of ,

algebraic Jordan algebras, alternative algebra, alternative bimodule, alternative division algebra, associates, associative,

algebra with involution, specialization,

associator, ideal, nilpotent element, regular element,

369

198 50

303 287 417,421 196 201 204 195

15 169 17 75

162 222 74 26 4

33 115 195 17 13 60

101 229

17 208 208 208 207 209 144 13

128 207 178 76

128 54 15 80

169 318 199 13 63 15

190 346 351

basic Jordan identities, bimodule, bimodule with involution,

associative, alternative,

birepresentation, regular,

capacity, Cartan subalgebra, Cayley algebra, Cayley bimodule with involution,

center, center of algebra with involution, central,

simple algebra with involution, centroid,

of algebra with involution, chain rule, Clifford algebra, Clifford group,

even (or second), Cohn's theorem, collineation, composition algebra, Coordinization Theorem, correlation,

degree, of an algebra,

degree of the algebra with involution,

derivation, of an algebra with involution,

derivation into bimodule, derived elements, differential, direct limit, directional derivative,

elation, elementary semisimple Jordan

algebra, elements of rank one, Euler's differential equation, exceptional Jordan algebra, exchange automorphism, extension, exterior algebra,

factor set, First Structure Theorem, formally real, free associative algebra,

33 80

145 145 145 84 90

158 350 17

, 279 18

, 208 206 208 206 207 215 75

371 372

8 379 162 137 397

209 223

209 35

144 292 84

217,218 69

214,216

397

318 364 219

6,11 101 91 74

93 161 205 25

451

452 SUBJECT INDE X

free /-algebra, free Jordan algebra, free monad, free monoid, free nonassociative algebra, free special Jordan algebra,

generic element, generic minimum polynomial, generic norm, generic trace, Glennie's theorem,

Harris' theorem, homogeneous rational mapping, homomorphism of an algebra

with involution, homotope, homotopy, Hua's identity,

ideal of algebra with involution, inner automorphism,

of associative algebra with involution,

inner derivation, into bimodule,

inner norm preserving group, inner structure group, invariant, inverse, invertible, involution,

canonical, of orthogonal type, of symplectic type, of the first kind, of the second kind, standard,

isotope, isotopic, isotopy, Jacobson and Richart theorem, Jacobson's theorem, Jordan algebra,

elementary semisimple, exceptional, free, free special, of a symmetric bilinear form, noncommutative, nondegenerate, purely inseparable, reduced reflexive, regular, special, split,

Jordan, bimodule, division algebra, element, homomorphism,

26 40 24 23,24 25 7

222 223 223 223 51

293,410 219

13 56 57 2,54

128 60

248 35

292 246 59

220 52 52

125 341 285 208 208

19,125 56 57 57 2

369,401 6

318 6,11

40 7

14 33

155 344 197 76 55 3,4,6

204

80 53 7 2

integral domain, matrix algebra, product, triple product,

Jordan, von Neumann and Wigner theorem,

left free cyclic bimodule, Lie,

algebra, element, invariant, product, semi-invariant,

Lie triple system, derivation of ,

inner, of linear transformations, ideal in a, standard Lie envelope o f a, universal Lie envelope of a,

linear property, 1ineari7atinn little projective group,

Macdonald's theorem, main involution, Malcev algebra, Martinaale theorem, meson algebra, monad (monoid), Moufang's identities, multiplication algebra,

of algebra with involution, multiplication specialization,

nil algebra, nilpotent, nonassociative algebra, noncommutative Jordan algebra, nondegenerate Jordan algebra, norm,

class, preserving group, semisimilarity, similarity,

nucleus l i U v l V U i J ^

null extension, octonions, opposite algebra, operator commutativity, orthogonal idempotents,

connected, strongly,

Osborn's theorem,

Peirce decomposition, formulas (PD), of alternative algebras,

Penico sequence, Penico solvable, perfect associative algebra with

involution,

53 127

3 36

205

279

4 9

220 4

220

308 309 35

311 310 310 78 28

398

41 65 33

141 115 23 16

206 207 63,86

195 195

1 33

155

379 241 394 241

18 91

16 13

320

122 122 176

118,120 121 165 192 332

76

SUBJECT INDEX 453

Pfaffian, 23 1 polynomial functions, 21 5 polynomial mapping, 21 8 power associative, 3 6 power associativity, 5 purely inseparable Jordan algebra, 34 4 primitive idempotents,

absolutely, completely,

projective transformation,

quadratic ideal, maximal, minimal, minimal conditions for, principal,

quasi-inverse,

radical, rational function, rational mappings, reduced,

Jordan algebra, orthogonal groups,

reducing set of idempotents, reflexive Jordan algebra, regular bimodule,

with involution, regular birepresentation, regular Jordan algebra, reversal operation,

Second Structure Theorem, semiderivation, semi-invariant, separable, semisimple, Schafer's theorem, Shirshov's theorem, Shirshov-Cohn theorem,

197 158 397

153 157 154 157 154 55

192 216 217 197 197 372 197 76 81

278 90 55 8

179 185 220 238 192 346 47 48

5-identity, simple algebra, simple components, simple Artinian algebra with

involution, solvable ideals, special Jordan algebra, special universal envelope,

construction of a, squared, unital, unital squared,

special universal functor, spin group, spinorial norm, split Jordan algebra, split null extension, Springer's theorem,

strictly power associative, strongly associative algebra, strongly special algebra, structure group, subalgebra of algebra with

involution,

TafVs theorem, type of an involution,

49 60

162

178 192

3,4,6 65 68

100 73

104 69

372 372 204 80

378, 381,406 222 38

101 59

128

336 209

universal /-multiplication envelope, 88 universal multiplication, 9 5

functor, 9 0 universal unital multiplication

envelope, 10 3 unramined algebra, 22 9

variety, 2 5

zero divisor, 5 3 absolute, 15 4

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