A Generalization of an Algebra of Chevalley

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1 Journal of Algebra 33, doi:0.006jabr , available online at on A Generalization of an Algebra of Chevalley R. D. Schafer Massachusetts Institute of Technology, Cambridge, Massachusetts 039 Communicated by Efim Zelmano Received March 5, 000 C. Chevalley, in The Algebraic Theory of Spinors, defined by means of a Clifford algebra a certain 4-dimensional commutative algebra which he used to prove the principle of triality to define the Ž Cayley. algebra of octonions. Reversing the construction, he used its involution x x tž x. x Ž. to obtain as the algebra of triples Ž a, b, c. of octonions with multiplication Ž a, b, c. Ž x, y, z. ž bz yc, cx za, ay xb/ Ž., p. 5, Ž. 3. Chevalley s algebra resonates in more recent research, 3. is related to the 7-dimensional exceptional simple Jordan algebra Ž Albert algebra. 5, p. 0 as follows: 3 3 consists of those 3 3 matrices 0 c b X c a b a 3 with elements in which are self-adjoint with respect to conjugate transposition, multiplication is given by X Y XY YX Žhere the usual are isomorphic under X. is deleted in order to give, but the Jordan algebras X. Consider the 4-dimensional subspace of $35.00 Copyright 000 by Academic Press All rights of reproduction in any form reserved. 398

2 consisting of matrices 3 AN ALGEBRA OF CHEVALLEY c b X0 c 0 a Ž 3. b a 0 define the product of X 0 in Ž z y Y0 z 0 x y x 0 to be the truncated product 0 ya bx cx za Ž X Y. ay xb 0 cy zb. Ž xc az bz yc 0 Then the resulting 4-dimensional algebra is clearly isomorphic by Ž. Ž. 4 to Chevalley s algebra under the mapping X Ž a, b, c. 0. The algebra of octonions is an example of a quadratic algebra with over a field F, where x tž x. x nž x. 0 for all x in Ž 5. with tž x. nž x. in F, Ž. is an involution of Žequivalently, tž xy. tž yx. 4, p ŽBy abuse of language we may include F as a quadratic algebra with t, n the identity map as involution.. Assume throughout this paper that F is a field of characteristic, 3. We generalize Chevalley s algebra by taking any Ž possibly infinite-dimensional. quadratic algebra with over F having Ž. as an involution, letting be the set of all triples Ž a, b, c. of elements in with multiplication Ž.. Equivalently, is the set of matrices Ž. 3 with elements in, together with the truncated product Ž. 4. We shall call any such commutative algebra 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. Ž c, a, b. Ž a, b, c. T, Ž 6. is an automorphism of of period 3. Also it is easy to see that, although an algebra of type C does not have a multiplicative identity, the

3 400 R. D. SCHAFER element e,, is a principal idempotent in 5, p. 39. We determine the Peirce decomposition of relative to e as follows. Let R e be the right multipli- cation Ž a, b, c. Ž a, b, c. e of determined by e. Then a straightforward computation yields 4R 4 e 5R e I 0, or Ž R e I.Ž Re I.Ž R e I.Ž R e I. 0. Hence is the direct sum Ž 7. of subspaces Ž a, b, c. in Ž a, b, c. e ia, Ž b, c.4, Ž. i 7 is the Peirce decomposition of the algebra of type C. It is easy to see Ž using characteristic 3. that Fe; Ž 8. a Ž a, a, a. a in, tž a. 04; Ž 9. Ž a, b, c. a b c 0, tž a. tž b. 04; Ž 0. Ž,,.,, in F ; 04. Ž. ŽNote that 04 if only if F.. The Ž commutative. products of the Peirce spaces Ž 8. Ž. are included in the spaces which are indicated as Ž. l i Since Fe, it follows from the definition of i,,, that the first row of inclusions in Ž. actually consists of equalities. Now Fe since Ž a, a, a. Ž b, b, b. ta tb 04 where

4 AN ALGEBRA OF CHEVALLEY 40 Ž a, a, a. Ž b, b, b. c Ž c, c, c., c ab ba ab ab tab. Also Ž a, a, a. Ž b, c, b c. ta tb tc 04 where Ž a, a, a. Ž b, c, b c. Ž r, s, r s. Ž,, Ž.. with r c, a a, b, s a, b c, a, tab, tac. Hence. Next, Ž a, a, a. Ž,, Ž.. ta 04 where Ž a, a, a. Ž,, Ž.. Ž a, a, a a. is in. ŽActually, since ta tb 0 implies Ž a, b, a b. Ž a,0,a. Ž 0, b, b. is in.. Also Ž a, b, a b. Ž x, y, x y. ta tž b. tž x. tž y. 04 where Ž a, b, a b. Ž x, y, x y. z Ž,,. with z Žz, z, z., z bx ya, tby, tax, tbx tay. Then z is in,, is in, implying. Next, Ž a, b, a b. Ž,, Ž.. ta Ž. tb 04 where Ž a, b, a b. Ž,, Ž.. c Ž y, z, y z. with c Ž. a Ž. b, y Ž a Ž. b, z Ž. a Ž b, so that. Finally, Ž,, Ž.. Ž,, Ž..,,, in F4 Ž,,.,, in F4. This establishes Ž.. Now Ž. implies that Fe Ž 3. Fe Ž 4. are subalgebras of. The elements of are a Ž a, a, a. for a in, Ž 5. implying that dim dim Ž possibly infinite.. The elements of are Ž,,. for,, in F, implying dim 3. Note that in the extreme case where F, then 04. Since is commutative, products in are completely determined by squares Ž a b Ž a b. a b.. For a in Žas in Ž 5.. we have Ž. Ž a a where a ta na. taa, so that a tž a. a 4 tž a. nž a. e. Ž 6.

5 40 R. D. SCHAFER Also a e a for all a in, Ž 7. since a e Ž a, a, a. Ž,,. Ž a, a, a. a. We note that all elements a Ž a, a, a. in are fixed under the automorphism T of in Ž 6.. In particular, the element e Ž,,. is fixed under T. Hence the Peirce spaces in Ž. i 7 are stable under T. Next we consider the 3-dimensional subalgebra Fe in Ž 4.. T induces an automorphism of of period 3, which we also denote by T. There are exactly four distinct idempotents in. IfŽ,,. in is idempotent, then,,, implying 4, 8. Ž 8. Since 0 Ž otherwise 0., Ž 8. implies. Applying T, we have,. But then Ž 8. implies that an even number Ž two or zero. of,, is, the possible idempotents in are Ž 9. e, f,,, g,,, h,,. Actually, all four of the elements in 9 are idempotent. Note that e f g h 0 Ž 0. et e, ft g, gt h, ht f. Ž. Also, f, g, h4 is a basis for the 3-dimensional algebra. Finally, f g f g, since f g Ž,,. Ž,,. Ž 0, 0,. 4 f g, implying that f g span a subalgebra of. THEOREM. There is a unique 3-dimensional commutatie algebra oer any field F of characteristic, 3 satisfying the following conditions: Ž. a has an automorphism T of period 3; Ž b. there are exactly four idempotents e, f, g, hin, these idempotents span Žsince these idempotents are permuted by T, the notation may be chosen so that Ž. holds.; Ž. c Equation Ž 0. holds Žso f, g, h4 is a basis for the 3-dimensional algebra.;

6 AN ALGEBRA OF CHEVALLEY 403 Ž d. has Peirce decomposition relatie to e, where minme im4 i Fe; Ž. e some pair of the idempotents f, g, h spans a subalgebra of Ž implying, since T is an automorphism, that any pair does.. Proof. We have seen above that such an exists. To see uniqueness, it is sufficient to prove that the basis f, g, h4 given by Ž. c determines a unique multiplication table. Now et e by Ž. implies that is stable under T. Hence, writing by d, we have f e f ; in F, f in Ž. g ft e g, h gt e h ; g, h in Ž 3. by Ž.. Then Ž c. implies e f g h 3 e Žf g h., so that in Ž. Ž 3.. Then ef e f Ž e f. Ž g h. 3 3 by Ž 0.. Applying the automorphism T, we have ef g h, eg h f, eh f g 4 by. Now e implies that f, g span a subalgebra of, so that fg f g for some, in F. Applying T, we have fg f g, gh g h, hf h f. Ž 5. Then Ž 5. Ž 0. imply fg gh hf Ž.Ž f g h. Ž. e, so that e e Ž f g h. f g h Ž fg gh hf. e Ž. e, implying Ž. Ž 6. in Ž 5.. Thus f g Ž e h. in Ž 0. implies f fg g Ž f g. Ž e h. e eh h e f g h 0 by Ž 4.. Then Ž 5. Ž 6. imply 0 f fg g Ž.Ž f g., so that in Ž 5. Ž 6.; that is, fg f g, gh g h, hf h f. 7 Together with f f, g g, h h, this completes the multiplication table for with respect to the basis f, g, h 4. A simpler multiplication table for this is given by taking the basis i, j, k4 where i e f, j e g, k e h.

7 404 R. D. SCHAFER Then i 0 by 4 0 ; applying T, we have j k 0. Also ij k by Ž 4. Ž 7.. Applying T, we obtain the multiplication table i j k 0, ij k, jk i, ki j. We now return to algebras of type C obtained from any quadratic algebra with involution Ž. by taking to be the algebra of all triples Ž a, b, c. of elements of with multiplication Ž.. We have seen that the Peirce decomposition Ž. 7 of yields two subalgebras in Ž 3. in Theorem. We have also seen that all elements a Ž a, a, a. e a in are fixed under the automorphism T of in Ž. 6, that multiplication in the commutative algebra is determined by the squares a in Ž 6.. Now, Ž 8. since Ž. Ž. by Ž 7. Ž. the remarks about equalities in the proof of Ž.. We may sharpen Ž 8. to f g h Ž 9. for f, g, h in 9. f is the set of all a f a, a, a,, Ž a, 0, 0. for a in by Ž 9. Ž.. Applying T, we have a f Ž a,0,0., b g Ž 0, b,0., c h Ž 0,0, c., Ž 30. from which 9 follows. That is, 9 says that every element of may be written uniquely in the form a f b g c h Ž a, b, c in.. Also Ž 30. implies that Ž f. is the set of all sums of Ž a f. Ž b f. Ž a,0,0. Ž b,0,0. 0 by Ž 30. Ž.. Applying T, we have Ž f. Ž g. Ž h Similarly, Ž 30. implies that Ž a f. Ž b g. Ž a,0,0. Ž 0, b,0. Ž 0, 0, ab. Ž ab. h, or Ž a f. b g Ž ab. e h Ž 3. by Ž 7.. Hence Ž f. Ž g. h, applying T, we have Ž f. Ž g. h, Ž g. Ž h. f, Ž h. Ž f. g.

8 AN ALGEBRA OF CHEVALLEY 405 We note that Ž. Ž. a f a g a h for all a in. Equations 3, 5, 7 imply that Ž. Ž a f. Ž a g. Ž a. e h tž a. a nž a. e h by 6. Also ž / tž a. a nž a. e h tž a. a 4 tž a. nž a. e h a h Ž a f. Ž e g. Ž e f. Ž a g. for all a in. Equation 3 implies that a f e g a f g a e h in, Ž f. Ž a g. Ž e f. Ž a g.. Finally, we note that, for all a, b Ž a f. b g b f Ž a g. is in h, Ž 3. since tža, b. 0, while Ž 3. Ž 9. imply that the element in Ž 3. equals Ža, b e. h a, b h in h. THEOREM. Let be a commutatie algebra oer a field F of characteristic, 3. Then is of type C if only if satisfies the following conditions: Ž A. has an automorphism T of period 3; Ž B. contains an idempotent e with respect to which has Peirce Ž. decomposition 7 where Fe Ž implying that Fe in Ž 3. is a subalgebra of. ; Ž C. eery element a e a of Ž in F, a in. is fixed under T Žimplying that et e, so that the Peirce spaces in Ž 7. i are stable under T.; Ž D. is the 3-dimensional commutatie algebra characterized in Theorem, where T in Ž A. induces by Ž C. an automorphism of which we denote also by T; Ž E. f g h where Ž f. 0 Ž f.ž g. h Žimplying that eery element of may be written uniquely in the form af bg ch for a, b, cin, applying T in Ž A. Ž C,we. hae Ž f. Ž g. Ž h. 04

9 406 R. D. SCHAFER Ž f.ž g. h, Ž g.ž h. f, Ž h.ž f. g. by ; F G for eery element a in, we hae Ž af.ž ag. ah Ž 33. Ž af.ž eg. Ž ef.ž ag.; Ž 34. for all elements a, bin, we hae Ž af.ž bg. Ž bf.ž ag. in h. Ž 35. Proof. We have seen the only if part of this theorem. For the if part, let V V be the vector spaces underlying, so that V Fe V. Then a in V has the form a e a, in F, a in V, Ž 36. Ž B. implies that the map a e a ae e a is bijective on V. For any pair a, b in V, Ž E. implies there is a unique c in V satisfying Ž af.ž bg. ch. Let this c be denoted by a b, let a b be the unique element of V satisfying a b Ž a be.. Then Ž af.ž bg. Ž Ž a b. e. h for all a, b in V. Ž 37. Hence a b in Ž 37. is a bilinear product on V defining a nonassociative algebra Ž V,. over F. Linearize Ž 33. to obtain Ž af.ž bg. Ž bf.ž ag. Ž ab. h for all a, b in. Putting b e, this yields Ž af.ž eg. Ž ef.ž ag. Ž ae. h for all a in. Ž 38. Then imply Ž af.ž eg. Ž ef.ž ag. Ž ae. h for all a in. Ž 39. Let u e. Then imply au u a a for all a in V, so that u is a Ž the unique. multiplicative identity for Ž V,.. Now Ž B. implies that, for all a in V, a q a e

10 AN ALGEBRA OF CHEVALLEY 407 for some quadratic form q on V. Then Ž 36. implies ae e a e Ž. Ž Ž a, so that a e ae a qa.. e a ae Ž. Ž Ž qa e. Hence 33 implies af ag ae.. ŽŽ Ž... Ž qa e h a 4 qa ueh, implying a a a 4 qa Ž.. uby Ž 37.; that is, where a a tž a. a nž a. u 0, Ž t a, n a q a, 4 so that Ž V,. is a quadratic algebra over F by Ž 40. Ž 5.. If a is in Ž V,., let a tž a. u a Ž 4. for t a in We have seen that, treating e a in 36 as elements of, we have ae a for all a in V. Ž 43. We claim that a a is an involution of Ž V,.. ŽFor completeness we indicate a proof of the remark made earlier that Ž. is an involution if only if tž xy. tž yx.. Equation Ž 5. implies that tžx. tž x. nž x. 0, which by linearization yields tž xy. tž yx. tž x. tž y. nž x y. nž x. nž y. 0; together with xy yx Ž x y. x y tž x y.ž x y. nž x y. tž x. x nž x. tž y. y nž y., this implies that xy yx ŽtŽ xy. tž yx.., as desired.. Thus we need only to prove that tž a b. tž b a. for a, b in Ž V,.. Ž 44. Now Ž 35. Ž 37. imply that ŽŽ a be. Ž b aehis.. in h, so that Ž a be. Ž b aeis. in. Hence a b b a ŽŽ a b b aeeis.. in, implying Ž 44. by Ž 36. Ž 4.. It remains to show that is isomorphic to the algebra of type C which is obtained from the quadratic algebra Ž V,. with involution Ž 4.. Map the triples Ž a, b, c. in this algebra of type C into by Ž a, b, c. Ž ae. f Ž be. g Ž ce. h Ž 45. for all a, b, c in Ž V,.. The mapping Ž 45. is bijective by Ž E., Ž 4., Ž 43.. Applying T in Ž A. Ž C. to Ž 37., we have Ž ag.ž bh. Ž Ž a b. e. f, Ž ah.ž bf. Ž Ž a b. e. g Ž 46.

11 408 R. D. SCHAFER for all a, b in by D. It follows from E, 43, 37, 46 that Ž Ž ae. f Ž be. g Ž ce. h.ž Ž xe. f Ž ye. g Ž ze. h. Ž bg.ž zh. Ž ch.ž yg. Ž ch.ž xf. Ž af.ž zh. Ž af.ž yg. Ž bg. Ž xf. ž ž / / b z yc e f Ž c x z a. e g Ž Ž a y x b. e. h, which is the image under Ž 45. of the product Ž a, b, c. Ž x, y, z. Žb z yc, c x z a, a y x b. given by Ž.. REFERENCES. C. C. Chevalley, The Algebraic Theory of Spinors, Columbia Univ. Press, New York, A. J. Feingold, I. B. Frenkel, J. F. X. Ries, Spinor Construction of Vertex Operator Algebras, Triality, E Ž. 8, Contemporary Mathematics, Vol., Amer. Math. Soc., Providence, M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, Amer. Math. Soc. Colloq. Publ., Vol. 44, Amer. Math. Soc., Providence, J. M. Osborn, Quadratic division algebras, Trans. Amer. Math. Soc. 05 Ž 96., R. D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New YorkLondon, 966; reprint, Dover, New York, 995.