A Generalization of an Algebra of Chevalley

Journal of Algebra 233, 398᎐408 Ž2000. doi:10.1006rjabr.2000.8434, available online at http:rrwww.idealibrary.com on

A Generalization of an Algebra of Chevalley R. D. Schafer Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Communicated by Efim Zelmano¨ Received March 15, 2000

C. Chevalley, in The Algebraic Theory of Spinors w1x, defined by means of a Clifford algebra a certain 24-dimensional commutative algebra ᑝ which he used to prove the principle of triality and to define the ŽCayley. algebra ᑩ of octonions. Reversing the construction, he used ᑩ and its involution x ª x s tŽ x.1 y x

Ž 1.

to obtain ᑝ as the algebra of triples Ž a, b, c . of octonions with multiplication

Ž a, b, c . ( Ž x, y, z . s bz q yc, cx q z a, ay q xb

ž

/

Ž 2.

w1, p. 125, Ž3.x. Chevalley’s algebra resonates in more recent research w2, 3x. ᑝ is related to the 27-dimensional exceptional simple Jordan algebra ᑢ Ž ᑩ 3 . ŽAlbert algebra. w5, p. 102x as follows: ᑢ Ž ᑩ 3 . consists of those 3 = 3 matrices

␰1 Xs c b



c ␰2

b a

a

␰3

0

with elements in ᑩ which are self-adjoint with respect to conjugate transposition, and multiplication is given by X ⭈ Y s XY q YX 1 2

Žhere the usual is deleted in order to give Ž2., but the Jordan algebras are isomorphic under X ª 21 X .. Consider the 24-dimensional subspace of 398 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.

399

AN ALGEBRA OF CHEVALLEY

ᑢ Ž ᑩ 3 . consisting of matrices 0 X0 s c b



c 0 a

b a 0

0

y x 0

0

Ž 3.

and define the product of X 0 in Ž3. and 0 Y0 s z y



z 0 x

to be the truncated product 0



ya q bx

Ž X 0 ⭈ Y0 . 0 s ay q xb xc q az

cx q za

0

cy q zb .

0 bz q yc

0

Ž 4.

Then the resulting 24-dimensional algebra is clearly isomorphic by Ž2. and Ž4. to Chevalley’s algebra ᑝ under the mapping X 0 ª Ž a, b, c .. The algebra ᑩ of octonions is an example of a quadratic algebra ᑫ with 1 over a field F, where x 2 y t Ž x . x q nŽ x . 1 s 0

for all x in ᑫ

Ž 5.

with t Ž x . and nŽ x . in F, and Ž1. is an involution of ᑫ Žequivalently, t Ž xy . s t Ž yx . w4, p. 203x.. ŽBy abuse of language we may include ᑫ s F1 as a quadratic algebra with t Ž ␣ 1. s 2 ␣ , nŽ ␣ 1. s ␣ 2 and the identity map as involution.. Assume throughout this paper that F is a field of characteristic / 2, 3. We generalize Chevalley’s algebra by taking any Žpossibly infinite-dimensional . quadratic algebra ᑫ with 1 over F having Ž1. as an involution, and letting ᑛ be the set of all triples Ž a, b, c . of elements in ᑫ with multiplication Ž2.. Equivalently, ᑛ is the set of matrices Ž3. with elements in ᑫ, together with the truncated product Ž4.. We shall call any such commutative algebra ᑛ an algebra of type C. In this paper we shall characterize algebras of type C by internal properties of the algebras. One such property is that the mapping T, defined by

Ž a, b, c . ª Ž c, a, b . s Ž a, b, c . T ,

Ž 6.

is an automorphism of ᑛ of period 3. Also it is easy to see that, although an algebra ᑛ of type C does not have a multiplicative identity, the

400

R. D. SCHAFER

element e s 12 Ž 1, 1, 1 . is a principal idempotent in ᑛ w5, p. 39x. We determine the Peirce decomposition of ᑛ relative to e as follows. Let R e be the right multiplication Ž a, b, c . ª Ž a, b, c .( e of ᑛ determined by e. Then a straightforward computation yields 4 R 4e y 5R 2e q I s 0, or

Ž R e y I . Ž R e q I . Ž 2 R e y I . Ž 2 R e q I . s 0. Hence ᑛ is the direct sum ᑛ s ᑛ 1 [ ᑛy1 [ ᑛ 1r2 [ ᑛy1r2

Ž 7.

of subspaces ᑛ i s Ž a, b, c . in ᑛ ¬ Ž a, b, c .( e s iŽ a, b, c .4 , and Ž7. is the Peirce decomposition of the algebra ᑛ of type C. It is easy to see Žusing characteristic / 3. that ᑛ 1 s Fe;

Ž 8.

ᑛy1 s  a ˜ s Ž a, a, a. ¬ a in ᑫ , t Ž a. s 0 4 ;

Ž 9.

ᑛ 1r2 s  Ž a, b, c . ¬ a q b q c s 0, t Ž a . s t Ž b . s 0 4 ;

Ž 10 .

ᑛy1 r2 s  Ž ␣ 1, ␤ 1, ␥ 1 . ¬ ␣ , ␤ , ␥ in F ; ␣ q ␤ q ␥ s 0 4 .

Ž 11 .

and

ŽNote that ᑛy1 s  04 if and only if ᑫ s F1.. The Žcommutative. products of the Peirce spaces Ž8. ᎐ Ž11. are included in the spaces which are indicated as

ᑛ1 ᑛy1 ᑛ 1r2 ᑛy1 r2

ᑛ1

ᑛy1

ᑛ 1r2

ᑛy1r2

ᑛ1

ᑛy1

ᑛ 1r2

ᑛy1r2

ᑛ1

ᑛ 1r2 [ ᑛy1r2

ᑛ 1r2

ᑛ 1 [ ᑛy1 [ ᑛy1 l 2

ᑛy1 [ ᑛ 1r2

Ž 12 .

ᑛ 1 [ ᑛy1r2

Since ᑛ 1 s Fe, it follows from the definition of ᑛ i Ž i s 1, y1, 12 , y 12 . that the first row of inclusions in Ž12. actually consists of equalities. Now 2 2 ᑛy1 : ᑛ 1 s Fe since ᑛy1 s  ⌺Ž a, a, a.(Ž b, b, b . ¬ t Ž a. s t Ž b . s 04 where

AN ALGEBRA OF CHEVALLEY

401

Ž a, a, a.(Ž b, b, b . s ˜ c s Ž c, c, c ., c s ab q ba s ab q abs t Ž ab.1. Also ᑛy1 ( ᑛ 1r2 s  ⌺Ž a, a, a.(Ž b, c, yb y c . ¬ t Ž a. s t Ž b . s t Ž c . s 04 where Ž a, a, a.(Ž b, c, yb y c . s Ž r, s, yr y s . q Ž ␳ 1, ␴ 1, yŽ ␳ q ␴ .1. with r s w c, ax y 12 w a, b x, s s w a, b x y 12 w c, ax, ␳ s y 12 t Ž ab., ␴ s y 12 t Ž ac .. Hence ᑛy1 ( ᑛ 1r2 : ᑛ 1r2 [ ᑛy1r2 . Next, ᑛy1 ( ᑛy1r2 s  ⌺Ž a, a, a.(Ž ␣ 1, ␤ 1, yŽ ␣ q ␤ .1. ¬ t Ž a. s 04 where Ž a, a, a.(Ž ␣ 1, ␤ 1, yŽ ␣ q ␤ .1. s Ž ␣ a, ␤ a, y␣ a y ␤ a. is in ᑛ 1r2 . ŽActually, ᑛy1( ᑛy1r2 s ᑛ 1r2 since t Ž a. s t Ž b . s 0 implies Ž a, b, ya y b . s Ž a, 0, ya. q Ž0, b, yb . is in ᑛy1 ( ᑛy1r2 .. Also 2 ᑛ 1r2 s  ⌺Ž a, b, ya y b .(Ž x, y, yx y y . ¬ t Ž a. s t Ž b . s t Ž x . s t Ž y . s 04 where Ž a, b, ya y b .(Ž x, y, yx y y . s ˜ z q Ž ␳ 1, ␴ 1, ␶ 1. with ˜ z s Ž z, z, z ., z s ybx y ya, ␳ s yt Ž by ., ␴ s yt Ž ax ., ␶ s t Ž bx . q t Ž ay .. Then ˜ z 2 is in ᑛ 1 [ ᑛy1 and Ž ␳ 1, ␴ 1, ␶ 1. is in ᑛ 1 [ ᑛy1r2 , implying ᑛ 1r2 : ᑛ1 [ ᑛy1 [ ᑛy1r2 . Next, ᑛ 1r2( ᑛy1r2 s  ⌺Ž a, b, ya y b .(Ž ␣ 1, ␤ 1, yŽ ␣ q ␤ .1. ¬ t Ž a. s t Ž b . s 04 where Ž a, b, ya y b .(Ž ␣ 1, ␤ 1, yŽ ␣ q ␤ .1. s ˜ c qŽ y, z, yy y z . with c s 13 Ž2 ␣ q ␤ . a q 13 Ž ␣ q 2 ␤ . b, y s 23 Žy␣ q ␤ . a q 23 Ž ␣ q 2 ␤ . b, z s 23 Ž2 ␣ q ␤ . a q 23 Ž ␣ y ␤ . b, so that ᑛ 1r2( ᑛy1r2 : 2 ᑛy1 [ ᑛ 1r2 . Finally, ᑛy1r2 s  ⌺Ž ␣ 1, ␤ 1, yŽ ␣ q ␤ .1.(Ž␥ 1, ␦ 1, yŽ␥ q . . 4 Ž ␦ 1 ¬ ␣ , ␤ , ␥ , ␦ in F : ␳ 1, ␴ 1, ␶ 1. ¬ ␳ , ␴ , ␶ in F 4 s ᑛ 1 [ ᑛy1r2 . This establishes Ž12.. Now Ž12. implies that ᑜ s ᑛ 1 [ ᑛy1 s Fe [ ᑛy1

Ž 13 .

ᑧ s ᑛ 1 [ ᑛy1r2 s Fe [ ᑛy1r2

Ž 14 .

and

are subalgebras of ᑛ. The elements of ᑜ are a ˜ s Ž a, a, a.

for a in ᑫ ,

Ž 15 .

implying that dim ᑜ s dim ᑫ Žpossibly infinite .. The elements of ᑧ are Ž ␣ 1, ␤ 1, ␥ 1. for ␣ , ␤ , ␥ in F, implying dim ᑧ s 3. Note that in the extreme case where ᑫ s F1, then ᑛy1 s ᑛ 1r2 s  04 and ᑛ s ᑛ 1 [ ᑛy1 r2 s ᑧ. Since ᑜ is commutative, products in ᑜ are completely determined by squares Ž2 a( ˜ ˜b s Ž a˜ q ˜b . 2 y a˜2 y ˜b 2 .. For a˜ in ᑜ Žas in Ž15.. we have 2 2 a ˜ s 2Ž a .˜ where a2 s Ž t Ž a. 2 y nŽ a..1 y t Ž a. a, so that 2

a ˜2 s y2 t Ž a. a˜ q 4 Ž t Ž a. y n Ž a. . e.

Ž 16 .

402

R. D. SCHAFER

Also a( ˜ e s a˜

for all a in ᑫ ,

Ž 17 .

˜ We note that all elements since a( ˜ e s 12 Ž a, a, a.(Ž1, 1, 1. s Ž a, a, a. s a. Ž . a ˜ s a, a, a in ᑜ are fixed under the automorphism T of ᑛ in Ž6.. In particular, the element e s 12 Ž1, 1, 1. is fixed under T. Hence the Peirce spaces ᑛ i in Ž7. are stable under T. Next we consider the 3-dimensional subalgebra ᑧ s Fe [ ᑛy1 r2 in Ž14.. T induces an automorphism of ᑧ of period 3, which we also denote by T. There are exactly four distinct idempotents in ᑧ. If Ž ␣ 1, ␤ 1, ␥ 1. in ᑧ is idempotent, then 2 ␤␥ s ␣ , 2␥␣ s ␤ , 2 ␣␤ s ␥ , implying 4␣␤ 2 s ␣ ,

8 ␣ 2␤ 2␥ 2 s ␣␤␥ .

Ž 18 .

Since ␣ / 0 Žotherwise ␤ s ␥ s 0., Ž18. implies ␤ s " 12 . Applying T, we have ␥ s " 21 , ␣ s " 21 . But then Ž18. implies that an even number Žtwo or zero. of ␣ , ␤ , ␥ is y 12 , and the possible idempotents in ᑧ are e, f s 21 Ž 1, y1, y1 . ,

g s 21 Ž y1, 1, y1 . ,

h s 21 Ž y1, y1, 1 . .

Ž 19 . Actually, all four of the elements in Ž19. are idempotent. Note that eqfqgqhs0

Ž 20 .

and eT s e,

fT s g ,

gT s h,

hT s f .

Ž 21 .

Also,  f, g, h4 is a basis for the 3-dimensional algebra ᑧ. Finally, f ( g s y 12 f y 12 g , since f ( g s 14 Ž1, y1, y1.(Žy1, 1, y1. s 12 Ž0, 0, 1. s y 12 f y 12 g, implying that f and g span a subalgebra of ᑧ. THEOREM 1. There is a unique 3-dimensional commutati¨ e algebra ᑧ o¨ er any field F of characteristic / 2, 3 satisfying the following conditions: Ža. ᑧ has an automorphism T of period 3; Žb. there are exactly four idempotents e, f, g, h in ᑧ, and these idempotents span ᑧ Ž since these idempotents are permuted by T, the notation may be chosen so that Ž21. holds .; Žc. Equation Ž20. holds Ž so  f, g, h4 is a basis for the 3-dimensional algebra ᑧ .;

403

AN ALGEBRA OF CHEVALLEY

Žd. ᑧ has Peirce decomposition ᑧ s ᑧ 1 [ ᑧy1r2 relati¨ e to e, where ᑧ i s  m in ᑧ ¬ me s im4 and ᑧ 1 s Fe; Že. some pair of the idempotents f, g, h spans a subalgebra of ᑧ Ž implying, since T is an automorphism, that any pair does .. Proof. We have seen above that such an ᑧ exists. To see uniqueness, it is sufficient to prove that the basis  f, g, h4 given by Žc. determines a unique multiplication table. Now eT s e by Ž21. implies that ᑧy1 r2 is stable under T. Hence, writing f s ␣e q fX;

␣ in F , f X in ᑧy1 r2

Ž 22 .

by Žd., we have g s fT s ␣ e q g X ,

h s gT s ␣ e q hX ;

g X , hX in ᑧy1 r2 Ž 23 .

by Ž21.. Then Žc. implies ye s f q g q h s 3 ␣ e q Ž f X q g X q hX ., so that ␣ s y 13 in Ž22. and Ž23.. Then ef s y 13 e y 12 f X s y 12 Ž e qf . s 12 Ž g qh. by Ž20.. Applying the automorphism T, we have ef s 12 g q 12 h,

eg s 12 h q 12 f ,

eh s 12 f q 12 g

Ž 24 .

by Ž21.. Now Že. implies that f, g span a subalgebra of ᑧ, so that fg s ␳ f q ␴ g for some ␳ , ␴ in F. Applying T, we have fg s ␳ f q ␴ g ,

gh s ␳ g q ␴ h,

hf s ␳ h q ␴ f .

Ž 25 .

Then Ž25. and Ž20. imply fg q gh q hf s Ž ␳ q ␴ .Ž f q g q h. s yŽ ␳ q ␴ . e, so that e s e 2 s Ž f q g q h. 2 s f q g q h q 2Ž fg q gh q hf . s ye y 2Ž ␳ q ␴ . e, implying

␴ s yŽ 1 q ␳ .

Ž 26 .

in Ž25.. Thus f q g s yŽ e q h. in Ž20. implies f q 2 fg q g s Ž f q g . 2 s Ž e q h. 2 s e q 2 eh q h s e q f q g q h s 0 by Ž24.. Then Ž25. and Ž26. imply 0 s f q 2 fg q g s Ž1 q 2 ␳ .Ž f y g ., so that ␳ s y 12 in Ž25. and Ž26.; that is, fg s y 12 f y 12 g ,

gh s y 12 g y 12 h,

hf s y 12 h y 12 f .

Ž 27 .

Together with f 2 s f, g 2 s g, h2 s h, this completes the multiplication table for ᑧ with respect to the basis  f, g, h4 . A simpler multiplication table for this ᑧ is given by taking the basis  i, j, k 4 where i s e q f,

j s e q g,

k s e q h.

404

R. D. SCHAFER

Then i 2 s 0 by Ž24. and Ž20.; applying T, we have j 2 s k 2 s 0. Also ij s k by Ž24. and Ž27.. Applying T, we obtain the multiplication table i 2 s j 2 s k 2 s 0,

ij s k,

jk s i ,

ki s j.

We now return to algebras ᑛ of type C obtained from any quadratic algebra ᑫ with 1 and involution Ž1. by taking ᑛ to be the algebra of all triples Ž a, b, c . of elements of ᑫ with multiplication Ž2.. We have seen that the Peirce decomposition Ž7. of ᑛ yields two subalgebras ᑜ in Ž13. and ᑧ in Theorem 1. We have also seen that all elements a ˜ s Ž a, a, a. s ␣ e q a˜X in ᑜ are fixed under the automorphism T of ᑛ in Ž6., and that multiplication in the commutative algebra ᑜ is determined by the squares a ˜2 in Ž16.. Now ᑜ(ᑧ s ᑛ,

Ž 28 .

since ᑜ ( ᑧ s Ž ᑛ 1 [ ᑛy1 .(Ž ᑛ 1 [ ᑛy1r2 . s ᑛ 1 q ᑛy1 q ᑛ 1r2 q ᑛy1r2 s ᑛ by Ž7. and Ž12. and the remarks about equalities in the proof of Ž12.. We may sharpen Ž28. to ᑛ s ᑜ( f [ ᑜ( g [ ᑜ(h

Ž 29 .

for f, g, h in Ž19.. ᑜ ( f is the set of all a( ˜ f s 12 Ž a, a, a.(Ž1, y1, y1. s yŽ a, 0, 0. for a in ᑫ by Ž19. and Ž2.. Applying T, we have a( ˜ f s y Ž a, 0, 0 . ,

˜b( g s y Ž 0, b, 0 . ,

˜c( h s y Ž 0, 0, c . , Ž 30.

from which Ž29. follows. That is, Ž29. says that every element of ᑛ may be written uniquely in the form a( ˜ f q ˜b( g q ˜c( h

Ž a, b, c in ᑫ . .

Also Ž30. implies that Ž ᑜ ( f . 2 is the set of all sums of Ž a( ˜ f .(Ž ˜b( f . s Ž a, 0, 0.(Ž b, 0, 0. s 0 by Ž30. and Ž2.. Applying T, we have 2 2 2 Ž ᑜ ( f . s Ž ᑜ ( g . s Ž ᑜ ( h . s  04 .

Similarly, Ž30. implies that Ž a( ˜ f .(Ž ˜b( g . s Ž a, 0, 0.(Ž0, b, 0. s Ž0, 0, ab. ˜ Ž . s y ab ( h, or

Ž a( ˜ f . ( Ž ˜b( g . s y Ž Ž ab . ˜( e . ( h by Ž17.. Hence Ž ᑜ ( f .(Ž ᑜ ( g . : ᑜ ( h and, applying T, we have

Ž ᑜ ( f . ( Ž ᑜ ( g . : ᑜ ( h,

Ž ᑜ ( g . (Ž ᑜ ( h. : ᑜ ( f ,

Ž ᑜ ( h. (Ž ᑜ ( f . : ᑜ ( g .

Ž 31 .

405

AN ALGEBRA OF CHEVALLEY

We note that

Ž a( ˜ f . ( Ž a( ˜ g . s y 12 a˜2 ( h

for all a in ᑜ .

Equations Ž31., Ž5., and Ž17. imply that

Ž a( ˜ f . ( Ž a( ˜ g . s y Ž Ž a2 . ˜( e . ( h s Ž Ž yt Ž a. a˜ q n Ž a.˜1 . ( e . ( h

˜ q 2 n Ž a. e . ( h s Ž yt Ž a . a 2

s y 12 y2 t Ž a . a ˜ q 4 Ž t Ž a. y n Ž a. . e ( h s y 12 a˜2 ( h

ž

/

by Ž16.. Also

Ž a( ˜ f . ( Ž e( g . s Ž e( f . ( Ž a( ˜ g.

for all a in ᑜ .

Equation Ž31. implies that Ž a( ˜ f .(Ž e( g . s 12 Ž a( ˜ f .(Ž˜1( g . s y 12 Ž a( ˜ e. 1 ˜ . Ž . Ž . ( h s 2 Ž1( f .(Ž a( g s e( f ( a( g . Finally, we note that, for all a, b ˜ ˜ in ᑜ,

Ž a( ˜ f . ( Ž ˜b( g . y Ž ˜b( f . ( Ž a( ˜ g.

is in ᑛy1 ( h,

Ž 32 .

since t Žw a, b x. s 0, while Ž31. and Ž9. imply that the element in Ž32. equals yŽw a, b x˜( e .( h s w a, b x˜( h in ᑛy1 ( h. THEOREM 2. Let ᑛ be a commutati¨ e algebra o¨ er a field F of characteristic / 2, 3. Then ᑛ is of type C if and only if ᑛ satisfies the following conditions: ŽA. ᑛ has an automorphism T of period 3; ŽB. ᑛ contains an idempotent e with respect to which ᑛ has Peirce 2 decomposition Ž7. where ᑛ 1 s Fe and ᑛy1 : ᑛ 1 Ž implying that ᑜ s ᑛ 1 [ Ž . ᑛy1 s Fe [ ᑛy1 in 13 is a subalgebra of ᑛ .; ŽC. e¨ ery element a s ␣ e q aX of ᑜ Ž ␣ in F, aX in ᑛy1 . is fixed under T Ž implying that eT s e, so that the Peirce spaces ᑛ i in Ž7. are stable under T .; ŽD. ᑛ 1 [ ᑛy1r2 is the 3-dimensional commutati¨ e algebra ᑧ characterized in Theorem 1, where T in ŽA. induces by ŽC. an automorphism of ᑧ which we denote also by T ; ŽE. ᑛ s ᑜ ᑧ s ᑜ f [ ᑜ g [ ᑜ h where Ž ᑜ f . 2 s 0 and Ž ᑜ f .Ž ᑜ g . : ᑜ h Ž implying that e¨ ery element of ᑛ may be written uniquely in the form af q bg q ch for a, b, c in ᑜ and, applying T in ŽA. and ŽC., we ha¨ e 2 2 2 Ž ᑜ f . s Ž ᑜ g . s Ž ᑜ h . s  04

406

R. D. SCHAFER

and

Ž ᑜ f . Ž ᑜ g . : ᑜ h,

Ž ᑜ g . Ž ᑜ h. : ᑜ f ,

Ž ᑜ h. Ž ᑜ f . : ᑜ g

by Ž21..; ŽF. for e¨ ery element a in ᑜ, we ha¨ e

Ž af . Ž ag . s y 12 a2 h

Ž 33 .

Ž af . Ž eg . s Ž ef . Ž ag . ;

Ž 34 .

and

ŽG.

for all elements a, b in ᑜ, we ha¨ e

Ž af . Ž bg . y Ž bf . Ž ag .

in ᑛy1 h.

Ž 35 .

Proof. We have seen the ‘‘only if’’ part of this theorem. For the ‘‘if’’ part, let V and Vy1 be the vector spaces underlying ᑜ and ᑛy1 , so that V s Fe [ Vy1 . Then a in V has the form a s ␣ e q aX ,

␣ in F , aX in Vy1 , X

Ž 36 . X

and ŽB. implies that the map a s ␣ e q a ª ae s ␣ e y a is bijective on V. For any pair a, b in V, ŽE. implies there is a unique c in V satisfying Ž af .Ž bg . s ch. Let this c be denoted by aI b, and let a) b be the unique element of V satisfying aI b s yŽ a) b . e. Then

Ž af . Ž bg . s y Ž Ž a) b . e . h

for all a, b in V .

Ž 37 .

Hence a) b in Ž37. is a bilinear product on V defining a nonassociative algebra ᑫ s Ž V, ). over F. Linearize Ž33. to obtain Ž af .Ž bg . q Ž bf .Ž ag . s yŽ ab. h for all a, b in ᑜ. Putting b s e, this yields

Ž af . Ž eg . q Ž ef . Ž ag . s y Ž ae . h

for all a in ᑜ .

Ž 38 .

for all a in ᑜ .

Ž 39 .

Then Ž34. and Ž38. imply

Ž af . Ž eg . s Ž ef . Ž ag . s y 12 Ž ae . h Let u s 2 e. Then Ž37. and Ž39. imply a) u s u) a s a

for all a in V ,

so that u is a Žs the unique. multiplicative identity for ᑫ s Ž V, ).. Now ŽB. implies that, for all aX in Vy1 , 2 Ž aX . s q Ž aX . e

AN ALGEBRA OF CHEVALLEY

407

for some quadratic form q on Vy1 . Then Ž36. implies ae s ␣ e y aX s 2 ␣ e y a, so that a2 s ␣ 2 e q 2 ␣ aX e q Ž aX . 2 s Ž ␣ 2 q q Ž aX .. e y 2 ␣ aX s 2 ␣ ae y Ž ␣ 2 y q Ž aX .. e 2 . Hence Ž33. implies Ž af .Ž ag . s y 12 Ž2 ␣ ae y Ž ␣ 2 y q Ž aX .. e 2 . h s yŽŽ ␣ a y 14 Ž ␣ 2 y q Ž aX .. u. e . h, implying a) a s ␣ a y 14 Ž ␣ 2 y q Ž aX .. u by Ž37.; that is, a) a y t Ž a . a q n Ž a . u s 0,

Ž 40 .

where t Ž a. s ␣ ,

n Ž a. s

1 4

Ž ␣ 2 y q Ž aX . . ,

Ž 41 .

so that ᑫ s Ž V, ). is a quadratic algebra over F by Ž40. and Ž5.. If a is in ᑫ s Ž V, )., let a s t Ž a. u y a

Ž 42 .

for t Ž a. in Ž40. and Ž41.. We have seen that, treating e and a in Ž36. as elements of ᑜ, we have ae s a

for all a in V .

Ž 43 .

We claim that a ª a is an involution of ᑫ s Ž V, ).. ŽFor completeness we indicate a proof of the remark made earlier that Ž1. is an involution if and only if t Ž xy . s t Ž yx .. Equation Ž5. implies that t Ž x 2 . y t Ž x . 2 q 2 nŽ x . s 0, which by linearization yields t Ž xy . q t Ž yx . y 2 t Ž x . t Ž y . q 2 nŽ x q y . y 2 nŽ x . y 2 nŽ y . s 0; together with xy q yx s Ž x q y . 2 y x 2 y y 2 s t Ž x q y .Ž x q y . y nŽ x q y .1 y t Ž x . x q nŽ x .1 y t Ž y . y q nŽ y .1, this implies that xy y yx s 12 Ž t Ž xy . y t Ž yx ..1, as desired.. Thus we need only to prove that t Ž a) b . s t Ž b) a .

for a, b in ᑫ s Ž V , ) . .

Ž 44 .

Now Ž35. and Ž37. imply that ŽŽ a) b . e y Ž b) a. e . h is in ᑛy1 h, so that Ž a) b . e y Ž b) a. e is in ᑛy1 . Hence a) b y b) a s ŽŽ a) b y b) a. e . e is in ᑛy1 , implying Ž44. by Ž36. and Ž41.. It remains to show that ᑛ is isomorphic to the algebra of type C which is obtained from the quadratic algebra ᑫ s Ž V, ). with involution Ž42.. Map the triples Ž a, b, c . in this algebra of type C into ᑛ by

Ž a, b, c . ª y Ž ae . f y Ž be . g y Ž ce . h

Ž 45 .

for all a, b, c in ᑫ s Ž V, ).. The mapping Ž45. is bijective by ŽE., Ž42., and Ž43.. Applying T in ŽA. and ŽC. to Ž37., we have

Ž ag . Ž bh . s y Ž Ž a) b . e . f ,

Ž ah . Ž bf . s y Ž Ž a) b . e . g

Ž 46 .

408

R. D. SCHAFER

for all a, b in ᑜ by ŽD.. It follows from ŽE., Ž43., Ž37., and Ž46. that

Ž y Ž ae . f y Ž be . g y Ž ce . h . Ž y Ž xe . f y Ž ye . g y Ž ze . h . s Ž bg . Ž zh . q Ž ch . Ž yg . q Ž ch . Ž xf . q Ž af . Ž zh . q Ž af . Ž yg . q Ž bg . Ž xf . s y b) z q y ) c e f y Ž Ž c) x q z ) a . e . g

žž

/ /

y Ž Ž a) y q x ) b . e . h, which is the image under Ž45. of the product Ž a, b, c .(Ž x, y, z . s Ž b) z q y ) c, c) x q z ) a, a) y q x ) b . given by Ž2..

REFERENCES 1. C. C. Chevalley, ‘‘The Algebraic Theory of Spinors,’’ Columbia Univ. Press, New York, 1954. 2. A. J. Feingold, I. B. Frenkel, and J. F. X. Ries, ‘‘Spinor Construction of Vertex Operator Algebras, Triality, and E Ž1. 8 ,’’ Contemporary Mathematics, Vol. 121, Amer. Math. Soc., Providence, 1991. 3. M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, ‘‘The Book of Involutions,’’ Amer. Math. Soc. Colloq. Publ., Vol. 44, Amer. Math. Soc., Providence, 1998. 4. J. M. Osborn, Quadratic division algebras, Trans. Amer. Math. Soc. 105 Ž1962., 202᎐221. 5. R. D. Schafer, ‘‘An Introduction to Nonassociative Algebras,’’ Academic Press, New YorkrLondon, 1966; reprint, Dover, New York, 1995.