- Email: [email protected]

On the Tensor Product of Composition Algebras Patrick J. Morandi Department of Mathematical Sciences, New Mexico State Uni¨ ersity, Las Cruces, New Mexico 88003 E-mail: [email protected] 1 Jose ´ M. Perez-Izquierdo ´

Departamento de Matematicas y Computacion, ´ ´ Uni¨ ersidad de La Rioja, 26004 Logrono, ˜ Spain E-mail: [email protected]

and S. Pumplun ¨ Facultat fur ¨ Mathematik, Uni¨ ersitat ¨ Regensburg, Uni¨ ersitatstrasse 31, D-93040 Regensburg, Germany E-mail: [email protected] Communicated by Georgia Benkart Received December 10, 1999 DEDICATED TO H. P. PETERSSON ON THE OCCASION OF HIS

60TH BIRTHDAY

1. INTRODUCTION Let C1 mF C2 be the tensor product of two composition algebras over a field F with charŽ F . / 2. Brauer w8x and Albert w1᎐3x seemed to be the first mathematicians who investigated the tensor product of two quaternion algebras. Later their results were generalized to this more general 1 The second author thanks DGES for support from grants PB-1291-C03-02 and API00rB04.

41 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

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MORANDI, PEREZ ´ -IZQUIERDO, AND PUMPLUN ¨

situation by Allison w4᎐6x and to biquaternion algebras over rings by Knus w20x. In Section 2 we give some new results on the Albert form of these algebras. We also investigate the F-quadric defined by this Albert form, generalizing a result of Knus w21x. Since Allison regarded the involution s ␥ 1 m ␥ 2 as an essential part of the algebra C s C1 mF C2 , he only studied automorphisms of C which are compatible with . In Section 3 we determine, if charŽ F . / 2, the automorphism group of a tensor product of octonion algebras. We also show that any automorphism of such a tensor product is compatible with the canonical tensor product involution. As a consequence, we determine the forms of a tensor product of octonion algebras. Furthermore, we show that any such algebra does not satisfy the Skolem᎐Noether Theorem. Our results of Section 3 arise from a study of the generalized alternative nucleus of an algebra, since a tensor product of octonion algebras is generated by its generalized alternative nucleus. In Section 4, using Lie algebra-theoretic techniques, we classify finite dimensional simple unital algebras over an algebraically closed field of characteristic 0 which are generated by their generalized alternative nucleus, proving that such an algebra is the tensor product of a simple associative algebra and a symmetric tensor product of octonion algebras. This result is used in Section 5 to sketch a variation of the Allison᎐Smirnov proof of the classification of finite dimensional central simple structurable algebras over a field of characteristic 0. Finally, in Section 6, we prove that if A is generated by its generalized alternative nucleus, then the associated bilinear form Ž x, y . s traceŽ L x L y . is associative. Let F be a field and C a unital, nonassociative F-algebra. Then S is a composition algebra if there exists a nondegenerate quadratic form n: C ª F such that nŽ x ⭈ y . s nŽ x . nŽ y . for all x, y g C. The form n is uniquely determined by these conditions and is called the norm of C. We will write n s n C . Composition algebras only exist in rank 1, 2, 4, or 8 Žsee w17x.. Those of rank 4 are called quaternion algebras and those of rank 8 octonion algebras. A composition algebra C has a canonical in¨ olution ␥ given by ␥ Ž x . s t Ž x .1 C y x, where the trace map t: C ª F is given by t Ž x . s nŽ1, x .. An example of an eight-dimensional composition algebra is Zorn’s algebra of vector matrices ZorŽ F . Žsee w22, p. 507x for the definition.. The norm form of ZorŽ F . is given by the determinant and is a hyperbolic form. Composition algebras are quadratic. That is, they satisfy the identities x 2yt Ž x . xqn Ž x . 1 C s 0 n Ž 1C . s 1

for all x g C,

ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS

43

and are alternati¨ e algebras; i.e., xy 2 s Ž xy . y and x 2 y s x Ž xy . for all x, y g C. In particular, nŽ x . s ␥ Ž x . x s x␥ Ž x . and t Ž x .1 C s ␥ Ž x . q x. For any composition algebra D over F with dim F Ž D . F 4 and any g F=, the F-vector space D [ D becomes a composition algebra via the multiplication

Ž u, ¨ . Ž u⬘, ¨ ⬘ . s Ž uu⬘ q ␥ Ž ¨ ⬘ . ¨ , ¨ ⬘u q ¨ ␥ Ž u⬘ . . for all u, ¨ , u⬘, ¨ ⬘ g D, with norm n Ž Ž u, ¨ . . s n D Ž u . y n D Ž ¨ . . This algebra is denoted by CayŽ D, .. Note that the embedding of D into the first summand of CayŽ D, . is an algebra monomorphism. The norm form of CayŽ D, . is obviously isometric to ²1, y : m n D . Since two composition algebras are isomorphic if and only if their norm forms are isometric, we see that if C is a composition algebra whose norm form satisfies n C ( ²1, y : m n D for some D then C ( CayŽ D, .. In particular, ZorŽ F . ( CayŽ D, 1. for any quaternion algebra D since ²1, y1: m n D is hyperbolic. A composition algebra is split if it contains an isomorphic copy of F [ F as a composition subalgebra, which is the case if and only if it contains zero divisors. Over algebraically closed fields any composition algebra of dimension G 2 is split.

2. ALBERT FORMS From now on we consider only the fields F with charŽ F . / 2 unless stated otherwise. It is well known that any norm of a composition algebra is a 3-fold Pfister form, and conversely any 3-fold Pfister form is the norm of some composition algebra. Let C be a composition algebra. Define C⬘ s ² F1:H s x g C : t Ž x . s nŽ x, 1. s 04 . Then n⬘ s n < C ⬘ is the pure norm of C. Note that C⬘ s x g C : x s 0 or x f F1 C and x 2 g F1 C 4 s x g C : ␥ Ž x . s yx 4 . Moreover, C is split if and only if its norm n is hyperbolic, two composition algebras are isomorphic if and only if their norms are isometric, and C is a division algebra if and only if n is anisotropic. We first investigate tensor products of two composition algebras. Following Albert, we associate to the tensor product C s C1 mF C2 of two composition algebras with dimŽ Ci . s ri and n C i s n i the Ž r 1 q r 2 y 2.dimensional form nX1 H ² y1: nX2 of determinant y1. This definition, for

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MORANDI, PEREZ ´ -IZQUIERDO, AND PUMPLUN ¨

C1 or C2 an octonion algebra, was first given by Allison in w5x. In the Witt ring W Ž F ., obviously this form is equivalent to n1 y n 2 . Like the norm form of a composition algebra, this Albert form contains crucial information about the tensor product algebra C. For biquaternion algebras, this is well-known w1, Theorem 3; 19, Theorem 3.12x. We introduce some notation and terminology. If q is a quadratic form and if ⺘ s ²1, y1: is the hyperbolic plane, then q s q0 H i⺘ for some anisotropic form q0 and integer i. The integer i is called the Witt index of q and is denoted by i W Ž q .. In the proof of the following proposition, we use the notion of linkage of Pfister forms Žsee w12, Section 4x.. Recall that two n-fold Pfister forms q1 and q2 are r-linked if there is an r-fold Pfister form h with q1 s h m qXi for some Pfister forms qXi . Finally, we call a two-dimensional commutative F-algebra that is separable over F a quadratic ´ etale algebra. Note that any quadratic ´ etale algebra either is a quadratic field extension of F or is isomorphic to F [ F. Part of the following result has been proved in w15, Theorem 5.1x. PROPOSITION 2.1. Let C1 and C2 be octonion algebras o¨ er F with norms n1 and n 2 , and let i s i W Ž N . be the Witt index of the Albert form N s nX1 H ² y1: nX2 . Ži. i s 0 m C1 and C2 do not contain isomorphic quadratic ´ etale subalgebras. Žii. i s 1 m C1 and C2 contain isomorphic quadratic ´ etale subalgebras, but no isomorphic quaternion subalgebras. Žiii. i s 3 m C1 and C2 contain isomorphic quaternion subalgebras, but C1 and C2 are not isomorphic. Živ. i s 7 m C1 ( C2 . Proof. By w12, Propositions 4.4 and 4.5x, the Witt index of n1 H ² y1: n 2 is 2 r , where r is the linkage number of n1 H ² y1: n 2 . Note that the Witt index of N is one less than the Witt index of n1 H ² y1: n 2 since n1 H ² y 1: n 2 s ⺘ H N. If C1 ( C2 , then n1 ( n 2 , so i s 7. Conversely, if i s 7, then n1 H ² y 1: n 2 is hyperbolic, so n1 ( n 2 , which forces C1 ( C2 . If C1 and C2 are not isomorphic but contain a common quaternion algebra Q, then Ci s CayŽ Q, i . for some i. Therefore, n1 s n Q m ²1, y 1 : and n 2 s n Q m ²1, 2 :. These descriptions show that n1 and n 2 are 2-linked, so i s 3. Conversely, if i s 3, then n1 and n 2 are 2-linked but not isometric. If ²² a, b :: is a factor of both n1 and n 2 , then n1 s ²² a, b, c :: and n 2 s ²² a, b, d :: for some c, d g F=. If Q s Žya, yb ., we get C1 s CayŽ Q, yc . and CayŽ Q, yd ., so C1 and C2 contain a common quaternion algebra. If C1 and C2 contain a common quadratic ´ etale algebra F w t xrŽ t 2 y a. but no common quaternion algebra,

ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS

45

then ²1, ya: is a factor of n1 and n 2 , which means they are 1-linked. If n1 and n 2 are 2-linked, then the previous step shows that C1 and C2 have a common quaternion subalgebra, which is false. Conversely, if n1 and n 2 are 1-linked but not 2-linked, then C1 and C2 do not have a common quaternion subalgebra, and if ²² a:: is a common factor to n1 and n 2 , then C1 and C2 both contain the ´ etale algebra F w t xrŽ t 2 y a.. PROPOSITION 2.2. Let C1 be an octonion algebra o¨ er F and C2 be a quaternion algebra o¨ er F, with norms n1 and n 2 . Again consider the Witt index i of the Albert form N s nX1 H ² y1: nX2 . Ži. i s 0 m C1 and C2 do not contain isomorphic quadratic ´ etale subalgebras. Žii. i s 1 m C1 and C2 contain isomorphic quadratic ´ etale subalgebras, but C2 is not a quaternion subalgebra of C1. Žiii. i s 3 m C1 ( CayŽ C2 , . for a suitable g F= and C2 is a di¨ ision algebra. Živ. i s 5 m C1 ( ZorŽ F . and C2 ( M2 Ž F .. Proof. In the case that both algebras C1 and C2 are division algebras, this is an immediate consequence of w15, Lemma 3.2x. If both C1 and C2 are split, then clearly N has Witt index 5. If C2 is a division algebra and C1 s CayŽ C2 , . for some , then n 2 is anisotropic and N H ⺘ s n 2 m ²1, y : H ² y1: n 2 s 4⺘ H ² y : n 2 , so N has Witt index 3. Note that the converse is easy, since if i s 3 then n 2 is isomorphic to a subform of n1 , which forces n 2 to be a factor of n1. If n1 s ²1, a: m n 2 , then C1 ( CayŽ C2 , ya., so C2 is a subalgebra of C1. If C1 and C2 contain a common quadratic ´ etale algebra F w t xrŽ t 2 y a. but C2 is not a quaternion subalgebra of C1 , then n1 and n 2 have ²² a:: as a common factor, so i s 1. Finally, if N is isotropic, there are x i g Ci , both skew, with n1Ž x 1 . s n 2 Ž x 2 .. Then, as t 1Ž x 1 . s 0 s t 2 Ž x 2 ., the algebras F w x 1 x and F w x 2 x are isomorphic, so C1 and C2 share a common quadratic ´ etale subalgebra. This finishes the proof. If C is a biquaternion algebra Ži.e., C ( C1 mF C2 for two quaternion algebras C1 and C2 ., then the Albert form nX1 H ² y1: nX2 is determined up to similarity by the isomorphism class of the algebra C w19, Theorem 3.12x. Allison generalizes this result w5, Theorem 5.4x to tensor products of arbitrary composition algebras. However, he always considers the involution s ␥ 1 m ␥ 2 as a crucial part of the algebra C s C1 mF C2 . Allison proves that Ž C1 mF C2 , ␥ 1 m ␥ 2 . and Ž C3 mF C4 , ␥ 3 m ␥4 . are isotropic algebras if and only if they have similar Albert forms, for the cases that C1 ,C3 are octonion and C2 , C4 are quaternion or octonion algebras.

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MORANDI, PEREZ ´ -IZQUIERDO, AND PUMPLUN ¨

The fact that any F-algebra isomorphism : Ž C1 mF C2 , ␥ 1 m ␥ 2 . ª Ž C3 mF C4 , ␥ 3 m ␥4 . between arbitrary products of composition algebras yields an isometry nX1 H ² y1: nX2 ( Ž nX3 H ² y1: nX4 . for a suitable g F= is easy to see. Also, since for C s C1 mF C2 the map ² , :: C = C ª F given by ² x 1 m x 2 , y 1 m y 2 : s n1Ž x 1 , y 1 . m n 2 Ž x 2 , y 2 . is a nondegenerate symmetric bilinear form on C such that ² Ž x ., Ž y .: s ² x, y :, the equation ² zx, y : s ² x, Ž z . y : holds Žthat is, ² , : is an in¨ ariant form; cf. w6, p. 144x or Section 6 below. and : C = C ª k, Ž x, y . s ² x, Ž y .: is an associative nondegenerate symmetric bilinear form which is proper, it follows easily that n1 m n 2 ( n 3 m n 4 . Suppose that we have two algebras that each are a tensor product of an octonion algebra and a quaternion algebra. We obtain a necessary and sufficient condition for when their Albert forms are similar. We use the notation DŽ q . to denote the elements of F= represented by a quadratic form q. THEOREM 2.3. Let C1 , C2 be octonion algebras and Q1 , Q 2 quaternion algebras o¨ er F. Let N1 and N2 be the Albert forms of C1 mF Q1 and C2 mF Q 2 , respecti¨ ely. If N1 ( N2 for some g F=, then Q1 ( Q 2 . Moreo¨ er, there is a quaternion algebra Q and elements c, d g F= such that C1 ( CayŽ Q, c ., C2 ( CayŽ Q, d ., CayŽ Q1 , . ( CayŽ Q, cd ., and y c g DŽ n C 2 .. Con¨ ersely, if there is a quaternion algebra Q and elements c, d, g F= such that C1 , Q1 s Q 2 , and C2 satisfy the conditions of the pre¨ ious sentence, then N1 ( N2 . Proof. Suppose that N1 ( N2 for some g F=. If c: W Ž F . ª BrŽ F . is the Clifford invariant, then cŽ N1 . s cŽ N2 . s cŽ N2 .. Since c is trivial on I 3 Ž F ., we have cŽ N1 . s cŽyn Q 1 . and cŽ N2 . s cŽyn Q 2 . Žsee w23, Chap. 5.3x.. Therefore, cŽ n Q 1 . s cŽ n Q 2 .. However, the Clifford invariant of the norm form of a quaternion algebra is the class of the quaternion algebra, by w23, Corollary V.3.3x. This implies that cŽ N1 . s w Q1 x and cŽ N2 . s w Q 2 x. Since w Q1 x s w Q 2 x, we get Q1 ( Q 2 . As a consequence of this, n Q 1 ( n Q 2 . Thus, n C 1 ( y²1, y : m n Q 1 ( n C 1 H Ž yn Q 1 H n Q 1 . ( Ž n C 2 H yn Q 2 . H n Q 1 ( n C 2 H Ž n Q 1 H y n Q 2 . ( n C 2 H 4⺘. The forms n C 1 and ²1, y : m n Q 1 are Pfister forms. The line above shows that these Pfister forms are 2-linked, in the terminology of w12x. Therefore, there is a 2-fold Pfister form ²² y a, yb :: with n C 1 ( ²² y a, yb, yc :: and ²1, y : m n Q ( ²² y a, yb, ye :: for some c, e g F=. An elemen1

ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS

47

tary calculation shows that ²² y a, yb, yc :: H y²² y a, yb, ye :: ( 4⺘ H ² y c, e : m ²² y a, yb :: . Therefore, n C 2 ( ² y c, e : m ²² y a, yb ::. Thus, n C 2 ( y c²² y a, yb, yce ::. Since n C 2 and ²² ya, yb, yce :: are Pfister forms, we get n C 2 ( ²² ya, yb, yce ::. If we set d s ce and let Q be the quaternion algebra Ž a, b .F , then the isomorphisms n C 1 ( ²² ya, yb, yc :: and n C 2 ( ²² ya, yb, yd :: give C1 ( CayŽ Q, c . and C2 ( CayŽ Q, d .. Moreover, n C 2 ( y cn C 2 , so y c g DŽ n C 2 .. Finally, the isomorphism ²1, y : m n Q 2 ( ²² ya, yb, ye :: gives CayŽ Q1 , . ( CayŽ Q, e . ( CayŽ Q, cd .. It is a short calculation to show that if C1 s CayŽ Q, c ., C2 s CayŽ Q, d ., and Q1 s Q 2 is a quaternion algebra with CayŽ Q1 , . ( CayŽ Q, ydc ., then nXC 1 H ² y 1: nXQ 1 ( Ž nXC 2 H ² y 1: nXQ 2 .. The argument of the previous theorem does not work for a tensor product of two octonion algebras since the Albert form is an element of I 3 Ž F ., whose Clifford invariant is trivial. COROLLARY 2.4. With the notation in the pre¨ ious theorem, suppose that N1 ( N2 for some g F=. If one of C1 and C2 is split, then the other algebra is isomorphic to CayŽ Q1 , .. Proof. We saw in the proof of the previous proposition that n C 1 H y²1, y : m n Q 1 ( n C 2 H 4⺘. Suppose that C2 is split. Then n C 1 H y²1, y : m n Q 1 is hyperbolic, so n C 1 ( ²1, y : m n Q 2 . Therefore, C1 ( CayŽ Q1 , .. On the other hand, if C1 is split, then n C 1 ( 4⺘, so by Witt cancellation y n C 2 ( ²1, y : m n Q 1. Since n C 2 and ²1, y : m n Q 1 are both Pfister forms, this implies that n C 2 ( ²1, y : m n Q 1, and so C2 ( CayŽ Q1 , .. In Theorem 2.3 above, it is possible for N1 ( N2 without C1 ( C2 . Moreover, the quaternion algebra Q of the proposition need not be isomorphic to Q1. We verify both of these claims in the following example. EXAMPLE 2.5. In this example we produce nonisomorphic octonion algebras C1 and C2 and a quaternion algebra Q1 that is not isomorphic to a subalgebra of either C1 or C2 and is such that the Albert forms of C1 mF Q1 and C2 mF Q1 are similar. To do this we produce nonisometric Pfister forms ²² x, y, z :: and ²² x, y, w :: and elements u, ¨ , with ²² x, y, zw :: ( ²² u, ¨ , :: such that the Witt indexes of ²² x, y, z :: H y²² u, ¨ , :: and ²² x, y, w :: H y²² u, ¨ , :: are both 2 and z g DŽ²² x, y, w ::.. We then set Q s Žyx, yy ., C1 s CayŽ Q, yz ., C2 s

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CayŽ Q, yw ., and Q1 s Žyu, y¨ .. From Theorem 2.3, we have N1 ( N2 . However, Proposition 2.2 shows that Q1 is not isomorphic to a subalgebra of either C1 or C2 . Moreover, C1 and C2 are not isomorphic since their norm forms are not isometric. Note that Q and C2 are not isomorphic since Q1 is not a subalgebra of C1. Let k be a field of characteristic not 2, and let F s k Ž x, y, z, w . be the rational function field in four variables over k. Set s xyzw, n1 s ²² x, y, z ::, and n 2 s ²² x, y, w ::. By embedding F in the Laurent series field k Ž x, y, z .ŽŽ w .., we see that n1 and n 2 s ²² x, y :: H w ²² x, y :: are not isomorphic over this field by Springer’s theorem w23, Proposition VI.1.9x, so n1 and n 2 are not isomorphic over F. Also, z s z 2 Ž xyw ., which is clearly represented by n 2 . Set Q1 s Žyzw, yxzw .. A short calculation shows that ²² x, y, zw :: s ²² zw, xzw, ::. Finally, for the Witt indices, we have n1 H yn Q 1 s ²1, x, y, xy, z, xz, yz , xyz : H y²1, zw, xzw, x : s 2⺘ H ² y, xy, z, xz, yz , xyz, yzw, yxzw : s 2⺘ H ² y, xy, z, xz, yz , xyz : H w ² y z, yxz : . The Springer theorem shows that this form has Witt index 2. Similarly, n 2 H yn Q 1 s ²1, x, y, xy, w, xw, yw, xyw : H y²1, zw, xzw, x : s 2⺘ H ² y, xy, w, xw, yw, xyw, yzw, yxzw : s 2⺘ H ² y, xy : H w ²1, x, y, xy, yz, yxz : has Witt index 2. For the remainder of this section we will also consider the case that charŽ F . s 2. Let C1 and C2 be composition algebras over F of dim F Ž Ci . s ri G 2, and let n i be the norm form of Ci . Using the notation of w21x, the subspace Q Ž C1 , C2 . s u s x 1 m 1 y 1 m x 2 : t 1Ž x 1 . s t 2 Ž x 2 .4 has dimension r 1 q r 2 y 2, and Q Ž C1 , C2 . s z y Ž␥ 1 m ␥ 2 .Ž z . : z g C1 mF C2 4 is the set of alternating elements of C1 mF C2 with respect to ␥ 1 m ␥ 2 . The nondegenerate quadratic form N: Q Ž C1 , C2 . ª F given by N Ž x 1 m 1 y 1 m x 2 . s n1Ž x 1 . y n 2 Ž x 2 . is isometric to the Albert form nX1 H ² y 1: nX2 of C1 mF C2 . Let VN ; ⺠ r 1qr 2y3 be the F-quadric defined via N. In the case that charŽ F . / 2, VN coincides with the open subvariety UN of closed points x 1 m 1 y 1 m x 2 with x 1 f F1 and x 2 f F1. We now generalize w21, Propositionx in the following two propositions. We will make use of the following fact that comes from Galois theory: Let F w z i x be the commutative F-subalgebra of dimension two of Ci generated by z i g Ci for i s 1, 2.

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ON THE TENSOR PRODUCT OF COMPOSITION ALGEBRAS

; Then there exists an isomorphism ␣ : F w z1 x ª F w z 2 4 such that ␣ Ž z1 . s z 2 if and only if n1Ž z1 . s n 2 Ž z 2 . and t 1Ž z1 . s t 2 Ž z 2 ..

PROPOSITION 2.6. There exists a bijection ⌽ between the set of F-rational points of UN and the set of triples Ž K 1 , K 2 , ␣ ., where K i is a two-dimensional ; commutati¨ e subalgebra of Ci and where ␣ : K 1 ª K 2 is an F-algebra isomorphism, ;

⌽ : P g UN : P an F-rational point 4 ª Ž K 1 , K 2 , ␣ . : K 1 , K 2 , ␣ as abo¨ e 4 F w z1 x , F w z 2 x , ␣ : F w z1 x ª F w z 2 x . z1 ¬ z 2 ;

P s z1 m 1 y 1 m z 2 ¬

ž

/

Proof. Any F-rational point P g UN corresponds with an element x 1 m 1 y 1 m x 2 g Q Ž C1 , C2 . with t 1Ž x 1 . s t 2 Ž x 2 . and n1Ž x 1 . s n 2 Ž x 2 .. ; Then there exists an F-algebra isomorphism ␣ : F w x 1 x ª F w x 2 x with x 1 ¬ x 2 . For x 1 m 1 y 1 m x 2 s z1 m 1 y 1 m z 2 it can easily be verified that F w x1 x , F w x 2 x , ␣ : F w x1 x ª F w x 2 x x1 ¬ x 2 ;

ž

/

F w z1 x , F w z 2 x ,  : F w z1 x ª F w z 2 x . z1 ¬ z 2 ;

s

ž

/

Therefore, the mapping ⌽ is well defined. Given a triple Ž K 1 , K 2 , ␣ ., there are elements z i g CiX such that K i s ; F w z i x and ␣ : F w z1 x ª F w z 2 x with z1 ¬ z 2 . By the remark before the proposition, we have n1Ž z1 . s n 2 Ž z 2 . and t 1Ž z1 . s t 2 Ž z 2 .; thus N Ž z1 m 1 y 1 m z 2 . s 0 and the triple defines the F-rational point P g UN corresponding to z1 m 1 y 1 m z 2 . So ⌽ is surjective. To prove injectivity, suppose that ⌽ Ž x 1 m 1 y 1 m x 2 . s ⌽ Ž z1 m 1 y 1 ; m z 2 .. Then F w x 1 x s F w z1 x, F w x 2 x s F w z 2 x, and the maps ␣ : F w x 1 x ª ; F w x 2 x, x 1 ¬ x 2 , and  : F w z1 x ª F w z 2 x, z ¬ z 2 , are equal. Since F w x i x s F w z i x, we write x 1 s a q bz1 and x 2 s c q dz 2 with a, b, c, d g F. We have a s c since t 1Ž x 1 . s t 2 Ž x 2 .. Therefore, we may replace x 1 with bz1 and x 2 with dz 2 without changing x 1 m 1 y 1 m x 2 . Thus, x 1 m 1 y 1 m x 2 s bz1 m 1 y 1 m dz 2 , and n1Ž x 1 . s n 2 Ž x 2 ., n1Ž z1 . s n 2 Ž z 2 . imply that n1Ž x 1 . s b 2 n1Ž z1 . and n 2 Ž x 2 . s d 2 n 2 Ž z 2 .. Therefore, b 2 s d 2 , so b s "d. Now x 2 s ␣ Ž x 1 . s ␣ Ž bz1 . s bz 2 yields b s d, and we get x 1 m 1 y 1 m x 2 s bŽ z1 m 1 y 1 m z 2 . which shows that ⌽ is injective. In the case that charŽ F . s 2, the set UN s x 1 m 1 y 1 m x 2 : x 1 f F1, x 2 f F14 is a proper open subvariety of VN . The proof of the previous proposition shows that ⌽ again is a bijection between the F-rational points

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of UN and the triples Ž K 1 , K 2 , ␣ ., where K i is a two-dimensional commu; tative F-subalgebra of Ci and ␣ : K ª L is an F-algebra isomorphism. We can say more in this situation. PROPOSITION 2.7. Let charŽ F . s 2. There exists an F-rational point in VN if and only if there exists a triple Ž K 1 , K 2 , ␣ . such that K i is a quadratic ´ etale ; subalgebra of Ci and ␣ : K 1 ª K 2 is an F-algebra isomorphism. In addition, there exists an F-rational point in VN l t 1Ž x 1 . s 04 if and only if there exists a triple Ž K 1 , K 2 , ␣ . such that K 1 and K 2 are purely inseparable quadratic ; extensions and ␣ : K 1 ª K 2 is an F-algebra isomorphism. Proof. As pointed out before the proposition, there is a bijection between F-rational points in UN and triples Ž K 1 , K 2 , ␣ . with K i : Ci commutative subalgebras of dimension 2 over F. To prove the first statement, only one half needs further argument. Suppose VN has an F-rational point. Since VN is a quadric hypersurface, VN is then birationally equivalent to ⺠ r 1qr 2y3 . The F-rational points of projective space are dense, so there is an F-rational point in UN . Therefore, we get a triple Ž K 1 , K 2 , ␣ . with K i a quadratic ´ etale subalgebra of Ci . For the second statement, an F-rational point in VN l t 1Ž x 1 . s 04 corresponds with an element x 1 m 1 y 1 m x 2 such that n1Ž x 1 . s n 2 Ž x 2 . and t 1Ž x 1 . s t 2 Ž x 2 . s 0, so F w x i x is a purely inseparable extension. There ; exists an isomorphism ␣ : F w x 1 x ª F w x 2 x with ␣ Ž x 1 . s x 2 and thus a triple Ž F w x 1 x, F w x 2 x, ␣ .. Conversely, if there is a triple Ž K 1 , K 2 , ␣ . with K i purely inseparable, there are x i g Ci with t i Ž x i . s 0 such that K s F w x 1 x, ; L s F w x 2 x, and ␣ : F w x 1 x ª F w x 2 x, x 1 ¬ x 2 , so n1Ž x 1 . s n 2 Ž x 2 . and x 1 m 1 y 1 m x 2 defines an F-rational point in VN l t 1Ž x 1 . s 04 .

3. THE AUTOMORPHISM GROUP OF A TENSOR PRODUCT OF OCTONION ALGEBRAS In this section we compute the automorphism group, the derivation algebra, and the forms of a tensor product of a finite number of octonion algebras over a field F with charŽ F . / 2. Let C s C1 mF ⭈⭈⭈ mF Cn be the tensor product of octonion algebras. As we will see, the subspace C1 mF F mF ⭈⭈⭈ mF F q ⭈⭈⭈ qF mF ⭈⭈⭈ mF F mF Cn is responsible for many properties of C, so our first goal is to characterize it. Recall that the associati¨ e nucleus of an algebra A is NŽ A. s a g A : Ž a, A, A. s Ž A, a, A. s Ž A, A, a. s 04 , where Ž x, y, z . s Ž xy . z y x Ž yz . denotes the usual associator. The commutati¨ e nucleus of A is the set KŽ A. s a g A : wŽ a, A x s 04 .

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DEFINITION 3.1. The subspace Nalt Ž A . : s a g A : Ž a, x, y . s y Ž x, a, y . s Ž x, y, a . ᭙ x, y g A4 will be called the generalized alternative nucleus of A. Remark 3.2. The alternati¨ e nucleus was introduced by Thedy w35x as

a g A : Ž x, a, x . s 0 and Ž a, x, y . s Ž x, y, a. s Ž y, a, x . for all x, y g A4 and is a subalgebra of A. The generalized alternative nucleus differs from this nucleus and, in general, it may not be closed under products, although it possesses an interesting algebraic structure Žsee Proposition 4.6.. PROPOSITION 3.3. Let A1 , A 2 be unital algebras with NŽ A1 . s F s KŽ A 2 . or NŽ A 2 . s F s KŽ A1 .. Then Nalt Ž A1 mF A 2 . s Nalt Ž A1 . mF F q F mF Nalt Ž A 2 .. Proof. By symmetry we can assume that NŽ A1 . s F s KŽ A 2 .. Let a s Ýa i m aXi g Nalt Ž A1 mF A 2 . with aXi linearly independent. The identities defining the generalized alternative nucleus with x replaced with x m 1 and y with y s y m 1 show that a i g Nalt Ž A1 .. A similar argument with a i linearly independent Žin Nalt Ž A1 .. shows that Nalt Ž A1 mF A 2 . : Nalt Ž A1 . mF Nalt Ž A 2 .. Now the identity Ž a, x, y . s yŽ x, a, y . with x replaced with x m x⬘ and y with y m 1 leads to ÝŽ a i , x, y . m aXi x⬘ s yÝŽ x, a i , y . m x⬘aXi . Since a i g Nalt Ž A1 ., this implies that wÝŽ a i , x, y . m aXi , 1 m x⬘x s 0. But, by hypothesis, the centralizer of A 2 in A1 mF A 2 is A1 mF F, hence ÝŽ a i , x, y . m aXi g A1 mF F. By choosing aX1 s 1 and aXi linearly independent, we get that Ž a i , x, y . s 0 if i G 2, and since a i g Nalt Ž A1 ., it follows that Ž x, a i , y . s 0 s Ž x, y, a i ., too. Therefore, a i g NŽ A1 . s F if i G 2 and Nalt Ž A1 mF A 2 . ; Nalt Ž A1 . mF F q F mF Nalt Ž A 2 .. The other inclusion is obvious. In general, in a tensor product of algebras A1 mF ⭈⭈⭈ mF A n we will identify the factors A i with the subalgebra F mF ⭈⭈⭈ mF A i mF ⭈⭈⭈ mF F without mention; thus, for instance, we will write A1 mF ⭈⭈⭈ mF A n s Ł i Ai. COROLLARY 3.4. Nalt Ž C1 mF ⭈⭈⭈ mF Cn . s C1 q ⭈⭈⭈ qCn . Proof. It is well known w37, p. 41x that NŽ Ci . s F s KŽ Ci . and that NŽ A1 mF A 2 . s NŽ A1 . mF NŽ A 2 . and KŽ A1 mF A 2 . s KŽ A1 . mF KŽ A 2 ..

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Recall that in any algebra with product denoted by w , x the element J Ž x, y, z . s ww x, y x, z x q ww y, z x, x x q ww z, x x, y x is called the Jacobian of x, y, z. The algebra is called a Malce¨ algebra if it is anticommutative and J Ž x, y, w x, z x. s w J Ž x, y, z ., x x. One important example of a simple Malcev algebra is the algebra of elements of zero trace in an octonion algebra with the product given by the commutator w24, 26, 29, 30x. Therefore Nalt Ž C . is a Malcev algebra, and Nalt Ž C . s F1 [ C1X [ ⭈⭈⭈ [ CnX with CiX minimal ideals that are simple Malcev algebras and F1 the center. The derived algebra of Nalt Ž C . is X Nalt Ž C . s Nalt Ž C . , Nalt Ž C . s C1X [ ⭈⭈⭈ [ CnX .

Remark 3.5. Let 0 : C1X ª C2X be an isomorphism of Malcev algebras. Since w a, w a, b xx s y4n1Ž a. b q 2 n1Ž a, b . a, we have n 2 Ž 0 Ž a., 0 Ž b .. s n1Ž a, b ., so we can define : C1 ª C2 by ␣ 1 q a ¬ ␣ 1 q 0 Ž a., which is an isomorphism because of the identity 2 ab s w a, b x y n1Ž a, b .. That is, any isomorphism from C1X onto C2X is the restriction of an isomorphism between C1 and C2 . Moreover, given an automorphism g AutŽ F . then any -semilinear isomorphism 0 between C1X and C2X is induced by a -semilinear isomorphism : ␣ 1 q a ¬ Ž ␣ .1 q 0 Ž a. between C1 and C2 . In the same way, any derivation of C1X is the restriction of a derivation of C1. Something similar holds for C1 mF ⭈⭈⭈ mF Cn . Let 0 be an automorphism of C1X [ ⭈⭈⭈ [ CnX . Since CiX are the minimal ideals there exists a permutation g Ý n such that 0 Ž CiX . s CX Ž i. . Therefore, by the previous, 0 < C Xi is the restriction of an isomorphism i : Ci ª C Ž i. and we can define an automorphism of C1 mF ⭈⭈⭈ mF Cn such that the restriction to Ci is i . Hence, any automorphism of C1X [ ⭈⭈⭈ [ CnX is the restriction of an automorphism of C1 mF ⭈⭈⭈ mF Cn . The same is true for -semilinear automorphisms. PROPOSITION 3.6. The restriction map gi¨ es the isomorphisms X Aut Ž C . ( Aut Ž Nalt ŽC.., X Der Ž C . ( Der Ž Nalt Ž C . . ( Der Ž C1 . [ ⭈⭈⭈ [ Der Ž Cn . .

X Ž C . invariProof. Any automorphism Žresp. derivation. of C leaves Nalt X Ž C .. Since ant, so it induces an automorphism Žresp. derivation. of Nalt X Ž C . generates C as an algebra, the restriction map induces monomorNalt X X Ž C .. and DerŽ C . ª DerŽNalt Ž C ... In the case of phisms AutŽ C . ª AutŽNalt

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AutŽ C . this monomorphism is also an epimorphism by Remark 3.5. In the X Ž C .. we have case of DerŽ C ., given d g DerŽNalt d Ž CiX . s d Ž w CiX , CiX x . : d Ž CiX . , CiX q CiX , d Ž CiX . : CiX X Ž C .. s DerŽ C1X . [ ⭈⭈⭈ [ DerŽ CnX .. Since any derivation d i of CiX so DerŽNalt is induced by a derivation d i of Ci , and d i m id m ⭈⭈⭈ m id q ⭈⭈⭈ qid m ⭈⭈⭈ m id m d n is a derivation of C, it follows that DerŽ C1X . [ ⭈⭈⭈ [ DerŽ CnX . s DerŽ C ..

Let be the tensor product of the canonical involutions of the Ci and AutŽ C, . the automorphisms of C that commute with . COROLLARY 3.7. With the pre¨ ious notation, AutŽ c . s AutŽ C, .. mn m Remark 3.8. We can write C ( C1mn1 mF ⭈⭈⭈ mF Cm with Cimn i the tensor product of n i copies of Ci and Ci \ C j if i / j and n1 q ⭈⭈⭈ qn m s X X Ž C . ( n1C1X [ ⭈⭈⭈ [ n m Cm n. Thus, Nalt with n i CiX isomorphic to the direct X X X sum of n i copies of Ci , and Ci \ C j if i / j. Since AutŽ n i CiX . is the wreath product of AutŽ CiX . and the symmetric group Ý n i , we obtain that AutŽ C mn i . ( AutŽ Ci . n i , the wreath product of AutŽ Ci . and Ý n i , and thus AutŽ C . ( AutŽ C1 . n1 = ⭈⭈⭈ = AutŽ Cm . n m . X Ž C . ( C1X [ ⭈⭈⭈ [ CnX gives the The uniqueness of the decomposition Nalt following uniqueness of the factorization of C.

PROPOSITION 3.9. Let A1 , A 2 be unital algebras such that C1 mF ⭈⭈⭈ mF Cn ( A1 mF A 2 . Then there exists a partition 1, . . . , n4 s ⌳ 1 j ⌳ 2 such that A1 ( mi g ⌳ 1 Ci and A 2 ( mj g ⌳ 2 C j . In particular, the factors C1 , . . . , Cn4 in C1 mF ⭈⭈⭈ mF Cn are uniquely determined up to order and isomorphism. Proof. Since the associative and commutative nuclei of C s C1 mF ⭈⭈⭈ mF Cn are each the base field, NŽ A i . s F s KŽ A i . for i s 1, 2. By Proposition 3.3, Nalt Ž A1 . mF q F mF Nalt Ž A 2 . ( C1 q ⭈⭈⭈ qCn s F1 [ C1X [ ⭈⭈⭈ [ CnX . The CiX are minimal ideals; therefore, there exists a partition 1, . . . , n4 s ⌳ 1 j ⌳ 2 such that Nalt Ž A i . ( F1 [ [j g ⌳ i CXj . Thus, the image in A1 mF A 2 of the subalgebra mj g ⌳ i C j generated by [j g ⌳ i CXj is contained in A i . Since the two algebras have the same dimension, they must be equal. COROLLARY 3.10. Two algebras C1 mF ⭈⭈⭈ mF Cn and C˜1 mF ⭈⭈⭈ mF C˜m are isomorphic if and only if n s m and there exists a permutation such that Ci is isomorphic to C˜ Ž i. . EXAMPLE 3.11. It is well known that if the Albert forms of two biquaternion algebras are similar, then the algebras are isomorphic. We

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give an example to show that the analogue of this result is false for octonion algebras. Let C1 and C2 be nonisomorphic octonion F-algebras and consider the tensor products C1 mF C1 and C2 mF C2 . Then their Albert forms are n C 1X H ynC 1X and n C X2 H ynC X2 , respectively. Therefore, these forms are isomorphic as they are both hyperbolic. However, C1 mF C1 is not isomorphic to C2 mF C2 by the previous corollary since C1 and C2 are not isomorphic. The following result points out the special role played by the involution . COROLLARY 3.12. The only in¨ olution of C1 mF ⭈⭈⭈ mF Cn which commutes with all automorphisms is . Proof. Let ⬘ be another involution of C s C1 mF ⭈⭈⭈ mF Cn commuting with AutŽ C .. The elements fixed by AutŽ Ci . : AutŽ C . are exactly Cˆi s mj/ i C j . Therefore, ⬘Ž Cˆi . s Cˆi . Looking at the centralizer of Cˆi yields ⬘Ž Ci . s Ci . The automorphism ⬘ induces an automorphism of Ci which commutes with AutŽ Ci .. That is, ⬘ s id w17x and ⬘ s . We will call the canonical in¨ olution of C1 mF ⭈⭈⭈ mF Cn . COROLLARY 3.13. Let : C s C1 mF ⭈⭈⭈ mF Cn ª C˜ s C˜1 mF ⭈⭈⭈ mF C˜n be an isomorphism. If and ⬘ are the canonical in¨ olutions of C and C˜ respecti¨ ely, then s ⬘ . Proof. Since AutŽ C˜. s AutŽ C . y1 , then y1 commutes with AutŽ C˜.. Therefore, ⬘ s y1 . We now show that the Skolem-Noether theorem does not hold for C. COROLLARY 3.14. There exist simple F-subalgebras B and B⬘ of C and an F-algebra isomorphism f : B ª B⬘ such that there is no F-algebra automorphism of C with < B s f. Proof. Let Q i be a quaternion subalgebra of Ci for i s 1,2 and let f be an F-algebra automorphism of A s Q1 mF Q 2 that is not compatible with < A ; such maps exist since we can take f to be the inner automorphism of an element t g A with Ž t . t f F. The condition Ž t . t g F is precisely the condition needed to ensure that f is compatible with < A . For example, we can take t s 1 q i1 i 2 g A s Q1 mF Q 2 Žwhere the standard generators of Q r are i r and jr .. If f extends to an automorphism of C, then Ž A. s A, so is compatible with . This forces < A s f to be compatible with < A , and f is chosen so that this does not happen. We devote the remainder of this section to computing the forms of tensor products of octonions, that is, F-algebras A such that A K s K mF

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A ( C1 mK ⭈⭈⭈ mK Cn for some extension KrF and octonion algebras Ci over K. We will denote C1 mK ⭈⭈⭈ mK Cn by T. X X Ž A. ( Nalt Ž K mF A. ( C1X [ ⭈⭈⭈ [ CnX , the algebra Since K mF Nalt X Nalt Ž A. is separable. It is worth noting that for a finite dimensional separable F-algebra R and a field extension KrF such that K mF R ( R1 [ ⭈⭈⭈ [ R n , with R i central simple K-algebras, there exists a subfield K 0 of K such that K 0rF is a finite Galois extension and K 0 mF R ( R˜1 [ ⭈⭈⭈ [ R˜n with R˜i central simple K 0-algebras and R i ( K mK 0 R˜i . LEMMA 3.15. Let A be a form of a tensor product of octonion algebras o¨ er F. There exists a finite Galois extension F : K 0 : K such that A K 0 is the tensor product of octonion algebras o¨ er K 0 . Proof. Let K 0 be a finite Galois extension of F contained in K such X Ž A. s C˜1X [ ⭈⭈⭈ [ C˜nX with C˜i octonion algebras over K 0 that K 0 mF Nalt X and K mK 0 C˜i ( CiX . By Remark 3.5, this isomorphism is induced by an isomorphism K mK 0 C˜i ( Ci . Thus we have an isomorphism K mK 0 Ž C˜1 mK 0 ⭈⭈⭈ mK 0 C˜n . ( T which restricts to an isomorphism C˜1 mK 0 ⭈⭈⭈ mK 0 C˜n ( K 0 mF A. This proposition allows us to assume in the following that KrF is a finite Galois extension. We denote the F-subalgebra generated by S by alg F ² S : and the subspace spanned by S by span F ² S :. Since X X K mF alg K² Nalt Ž A .: ( alg F² Nalt Ž K mF A .: s alg K ² C1X [ ⭈⭈⭈ [ CnX :

s T s K mF A, X X Ž A. since A s alg F ²Nalt Ž A.:. we obtain A from Nalt

PROPOSITION 3.16. The map

F-forms of C1 mK ⭈⭈⭈ mK Cn 4 ª F-forms of C1X [ ⭈⭈⭈ [ CnX 4 X A ¬ Nalt Ž A.

is a bijection with in¨ erse gi¨ en by N⬘ ¬ alg² N⬘:. Moreo¨ er, if A and B are X X Ž A. ( Nalt Ž B .. F-forms of C1 mK ⭈⭈⭈ mK Cn then A ( B if and only if Nalt X Ž A. is an F-form of C1X [ ⭈⭈⭈ [ CnX if A is an Proof. It is clear that Nalt F-form of C1 mK ⭈⭈⭈ mK Cn . Conversely, let N⬘ be an F-form of C1X [ ⭈⭈⭈ [ CnX and U : g GalŽ KrF .4 the semilinear automorphisms of C1X [ ⭈⭈⭈ [ CnX such that N⬘ is the set of fixed elements. By Remark 3.5, we can assume that U is the restriction of a -semilinear automorphism U˜ of C1 mK ⭈⭈⭈ mK Cn . The algebra A of fixed elements by U˜ : g GalŽ KrF .4 is an F-form of C1 mK ⭈⭈⭈ mK Cn containing N⬘. In fact, since

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X Ž A., and thus alg F ² N⬘: s A N⬘ extends to C1X [ ⭈⭈⭈ [ CnX , we get N⬘ s Nalt is an F-form of C1 mK ⭈⭈⭈ mK Cn . X X X X Ž A. ( Nalt Ž B .. Conversely, if 0 : Nalt Ž A. ª Nalt Ž B. If A ( B then Nalt X X is an isomorphism, it induces an automorphism ˜0 of C1 [ ⭈⭈⭈ [ Cn that, by Remark 3.5, is the restriction of an automorphism ˜ of C1 mK ⭈⭈⭈ mK Cn . Since X X X Ž A . : . s alg F² ˜ Ž Nalt Ž A . .: s alg F² Ž Nalt Ž B . .: s B, ˜ Ž A . s ˜ Ž alg F ²Nalt

it follows that A ( B. This proposition allows us to construct easily the forms of a tensor X Ž A. s N1 [ N2 then product of octonion algebras. First, observe that if Nalt X K mF Ni ( [j g ⌳ i C j , i s 1, 2 for some partition 1, . . . , n4 s ⌳ 1 j ⌳ 2 . By Proposition 3.16, A i s alg F ² Ni : is an F-form of mj g ⌳ i C j and hence A ( A1 mF A 2 . Therefore, it is enough to construct the forms of a tensor product of octonion algebras with simple generalized alternative nucleus. X Ž A. simple and K a finite Galois Let A be an F-algebra with Nalt extension such that K mF A ( C1 mK ⭈⭈⭈ mK Cn for some octonion algeX X Ž A. is simple, the centroid ⌫ s ⌫ ŽNalt Ž A.. is a bras Ci over K. Since Nalt finite separable extension of F. In fact, X K mF ⌫ ( ⌫ Ž K mF Nalt Ž A . . ( ⌫ Ž C1X [ ⭈⭈⭈ [ CnX . ( K [ ⭈⭈⭈ [ K

implies that ⌫ is an extension of degree n and that we have n different F-monomorphisms i : ⌫ ª K. Every i allows us to define a right ⌫-vector space structure on K by ␣ (␥ s ␣i Ž␥ ., ␣ g K, ␥ g ⌫. We denote this new vector space by K 1 . Now, X X X K mF Nalt Ž A . ( K mF Ž ⌫ m⌫ Nalt Ž A . . ( Ž K mF ⌫ . m⌫ Nalt Ž A. X ( Ž K 1 [ ⭈⭈⭈ [ K n . m⌫ Nalt Ž A. X X ( Ž K 1 m⌫ Nalt Ž A . . [ ⭈⭈⭈ [ Ž K n m⌫ Nalt Ž A. .

( C1X [ ⭈⭈⭈ [ CnX X Ž A. ( CiX . Therefore, we can think of implies that, up to order, K i m⌫ Nalt X the ⌫-algebra Nalt Ž A. as a form of CiX . By the arguments in w24, pp. X Ž A. ( C⬘ for some 240᎐241x, for instance, we can conclude that Nalt octonion algebra C over ⌫. Under the isomorphism K mF C⬘ ( Ž K 1 m⌫ C⬘. [ ⭈⭈⭈ [ Ž K n m⌫ C⬘. we identify x g C⬘ with Ž1 m x . q ⭈⭈⭈ qŽ1 m x .. Since we can view Ž K 1 m⌫ C⬘. [ ⭈⭈⭈ [ Ž K n m⌫ C⬘. as the generalized alternative nucleus of Ž K 1 m⌫ C⬘. mK ⭈⭈⭈ mK Ž K n m⌫ C⬘., the algebra A corresponds with the F-subalgebra generated by Ž1 m⌫ x . mK ⭈⭈⭈ mK Ž1 m⌫ 1. q ⭈⭈⭈ qŽ1 m⌫ 1. mK ⭈⭈⭈ mK Ž1 m⌫ x .. Conversely, given an octonion ⌫-al-

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gebra C with ⌫ a finite separable extension of F, it is easy to check that the algebra A constructed as above is a form of a tensor product of octonion algebras with a simple generalized alternative nucleus.

4. THE GENERALIZED ALTERNATIVE NUCLEUS The generalized alternative nucleus is responsible for many properties of the tensor product of octonion algebras. In this section we pay special attention to this nucleus. We classify the simple finite dimensional unital algebras which are generated by their generalized alternative nucleus. Our methods rely on the representation theory of some Lie algebras; therefore, in this section we will assume that charŽ F . s 0 and that F is algebraically closed. We make free use of Lie algebra terminology and refer the reader to the books of Humphreys w16x and Jacobson w18x for definitions and results. Let C be an octonion algebra over F and Sym n Ž C . the symmetric tensors of C mF ⭈⭈⭈ mF C, the tensor product of n copies of C. The elements a m 1 m ⭈⭈⭈ m 1 q ⭈⭈⭈ q1 m ⭈⭈⭈ m 1 m a with a g C lie in Nalt Ž C mF ⭈⭈⭈ mF C . and generate Sym n Ž C .. Therefore Sym n Ž C . is an algebra generated by its generalized alternative nucleus. However, Sym n Ž C . is no longer simple. The contraction Sym n Ž C . ª Sym ny 2 Ž C . induced by x m ⭈⭈⭈ m x ¬ nŽ x . x m ⭈⭈⭈ m x is an epimorphism whose nucleus we will denote by TnŽ C ., n G 2. We recover the Kantor᎐Smirnov structurable algebra when n s 2 w33, 7x. We will see that TnŽ C . is a unital simple algebra generated by Nalt ŽTnŽ C ... We set T1Ž C . s C and T0 Ž C . s F. We now give our classification result. THEOREM 4.1. Any simple finite dimensional unital algebra o¨ er an algebraically closed field of characteristic zero which is generated by its generalized alternati¨ e nucleus is isomorphic to the tensor product of a simple associati¨ e algebra and TnŽ C . for some n. Recall from w28x that a ternary deri¨ ation of an algebra A is a triple Ž d1 , d 2 , d 3 . g End F Ž A. = End F Ž A. = End F Ž A. such that d1 Ž xy . s d 2 Ž x . y q xd 3 Ž y .

Ž 1.

for any x, y g A. The Lie algebra of ternary derivations is denoted by TderŽ A.. If d1 s d 2 s d 3 then Ž1. says that d1 is a derivation, and in that case we will say that Ž d1 , d 2 , d 3 . represents a deri¨ ation. Let Ta s L a q R a . It is worth noting that a g Nalt Ž A . m Ž L a , Ta , yL a . and Ž R a , yR a , Ta . g Tder Ž A . . Ž 2 .

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The following identities will be useful. LEMMA 4.2.

Let a, b g Nalt Ž A. and x g A. Then

Ži. L a x s L a L x q w R a , L x x, L x a s L x L a q w L x , R a x. Žii. R a x s R x R a q w R x , L a x, R x a s R a R x q w L a , R x x. Žiii. w L a , R b x s w R a , L b x. Živ. w L a , L b x s Lw a, bx y 2w R a , L b x, w R a , R b x s yRw a, bx y 2w L a , R b x. Žv. The map Da, b s w L a , L b x q w L a , R b x q w R a , R b x is a deri¨ ation of A, Da, b s ad w a, bx y 3w L a , R b x and 2 Da, b s ad w a, bx q wad a , ad b x, where ad a : x ¬ w a, x x. Proof. Parts Ži. and Žii. follow from the identities Ž a, x, y . s Ž x, y, a., Ž x, a, y . s yŽ x, y, a., Ž y, a, x . s yŽ a, y, x ., and Ž y, x, a. s Ž a, y, x .. Part Žiii. follows from Ž b, x, a. s yŽ a, x, b ., while Živ. is an easy consequence of parts Ži., Žii., and Žiii.. Now, by Ž2. we have that Žw L a , L b x, w Ta , Tb x, w L a , L b x., Žw L a , R b x, yw Ta , R b x, yw L a , Tb x., and Žw R a , R b x, w R a , R b x, w Ta , Tb x. lie in TderŽ A.. Adding up these elements and using Žii., we obtain a ternary derivation that represents the derivation Da, b . From Živ. we get Da,b s ad w a, bx y 3w L a , R b x. Finally, ad w a, bx q w ad a , ad b x s ad w a, bx q w L a , L b x q w R a , R b x y 2 w L a , R b x s 2 Ž ad w a, bx y 3 w L a , R b x . s 2 Da, b . As we saw in the case of tensor products of octonions, Nalt Ž A. may not be a subalgebra of A. The natural product on Nalt Ž A. seems to be the commutator w a, b x s ab y ba. PROPOSITION 4.3. Gi¨ en a, b g Nalt Ž A. then w a, b x g Nalt Ž A.. Moreo¨ er, ŽNalt Ž A., w , x. is a Malce¨ algebra. Proof. By Lemma 4.2Živ., Lw a, bx s w L a , L b x q 2w R a , L b x and Rw a, bx s yw R a , R b x y 2w L a , R b x, thus by Ž2. we obtain

Ž Lw a, bx , w Ta , Tb x q 2 wyR a , Tb x , w L a , Lb x q 2 w Ta , yLb x . g TderŽ A . . Since Tw a, bx s w L a , L b x y w R a , R b x s w Ta , Tb x q 2 w yR a , Tb x and yLw a, bx s y w L a , L b x y 2 w R a , L b x s w L a , L b x q 2 w Ta , yL b x , it follows that Ž Lw a, bx , Tw a, bx , yLw a, bx . g TderŽ A.. Similarly, Ž Rw a, bx , yRw a, bx , Tw a, bx . g TderŽ A.. Therefore, w a, b x g Nalt Ž A.. The same arguments as those in w27, p. 9x show that ŽNalt Ž A., w , x. is a Malcev algebra.

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In the following we will always assume that A is simple, finite dimensional and generated by Nalt Ž A.. In our discussion, the Lie algebra T Ž A. generated by L a , R a : a g Nalt Ž A.4 will play a prominent role. Given a subset S of an algebra we say that ␦ Ž x . is the degree of x on S if x can be written as x s pŽ s1 , . . . , sm . with s1 , . . . , sn g S and pŽ x 1 , . . . , x n . some nonassociative polynomial Žconstants are allowed. of degree ␦ Ž x ., and if there is no other such expression for a polynomial of degree - ␦ Ž x .. By convention the degree of 0 is set to y⬁. LEMMA 4.4. If S : Nalt Ž A. then the degree of x on S is the same as the degree of L x , R x on span F ² L a , R a : a g S :. Proof. We proceed by induction to see that the degree of L x and R x is F ␦ Ž x .. The case ␦ Ž x . s y⬁ is trivial. If ␦ Ž x . s 0 then 0 / x g F and therefore ␦ Ž L x . s 0 s ␦ Ž R x .. Now let x be a monomial of degree n ) 1, so x s x 1 x 2 with ␦ Ž x i . - ␦ Ž x .. By induction ␦ Ž L x 1 . - ␦ Ž x ., and therefore x s ax 0 or x s x 0 a with a g S and ␦ Ž x 0 . - ␦ Ž x .. By Lemma 4.2 Ži and ii. and by the hypothesis of induction we get ␦ Ž L x ., ␦ Ž R x . F ␦ Ž x .. Finally, since x s L x Ž1. s R x Ž1., it follows that ␦ Ž x . F ␦ Ž L x ., ␦ Ž R x .. PROPOSITION 4.5. A is an irreducible TŽ A.-module and TŽ A. s T⬘Ž A. [ F id, with T⬘Ž A. s wTŽ A., TŽ A.x a semisimple Lie algebra. Proof. By Lemma 4.4, the multiplication algebra of A is generated by the left and right multiplication maps L a , R a for a g Nalt Ž A.. Therefore, any TŽ A.-submodule is an ideal, hence A irreducible. Since A is irreducible and faithful, T⬘Ž A. is semisimple and TŽ A. is the direct sum of T⬘Ž A. and the center w19, p. 47x. But any element in the center commutes with the multiplication algebra of A and therefore lives in the centroid of A. Since A is simple and F is algebraically closed, we conclude that the center is F id. Recall that a Malcev algebra is semisimple if 0 is the only Abelian ideal w24x. X Ž A. is a semisimple Malce¨ algebra. PROPOSITION 4.6. Nalt X Ž A . and consider TI s Proof. Let I be an ideal of Nalt X span F ² L a , R a , Da, c : a g I, c g Nalt Ž A.:. By Lemma 4.2 Živ., TI : T ⬘Ž A.. Moreover, Part Žv. of the same lemma shows that w L a , R b x s w R a , L b x g TI X Ž A.. Then, by Part Živ., it follows that w L a , L b x, if a g I and b g Nalt w R a , R b x g TI , too. Finally, Da, c Ž b . g I by Žv., so w Da, c , L b x s L D Ž b. , a, c w Da,c , R b x s R D Ž b. g TI . Therefore TI is an ideal of T ⬘Ž A.. By the a, c semisimplicity of T ⬘Ž A. we must have w TI , TI x s TI . In particular, I s X Ž A. TI Ž1. s w TI , TI xŽ1. s w I, I x and therefore the only abelian ideal of Nalt X is 0, so Nalt Ž A. is semisimple.

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In a Malcev algebra M the subspace generated by the Jacobians is an ideal of M denoted by J Ž M, M, M .. The subspace N Ž M . s x g M : J Ž x, M, M . s 04 is also an ideal and is called the J-nucleus of M. It is well-known that N Ž M . J Ž M, M, M . s 0. In fact, any finite dimensional semisimple Malcev algebra M over a perfect field of characteristic not two can be decomposed as M s N Ž M . [ J Ž M, M, M . with N Ž M . a semisimple Lie algebra and J Ž M, M, M . the direct sum of simple non-Lie Malcev algebras. If the field has characteristic 0 then N Ž M . is the direct sum of simple Lie algebras, by w16, Theorem 5.3x. X X Ž A. s [i NiX be the decomposition of Nalt Ž A. PROPOSITION 4.7. Let Nalt as the direct sum of ideals that are simple Malce¨ algebras, and let A i s alg² NiX , 1:. Then, A i is a simple unital algebra generated by Nalt Ž A i . s F1 q NiX , and A ( mi A i .

Proof. Given a g NiX and b g NjX with i / j, then by Lemma 4.2Žv. we have Da, b s 3w L a , R b x and Da, b ŽNalt Ž A.. s 0. Since A is generated by Nalt Ž A., it follows that w L a , R b x s 0. By Lemma 4.2Živ., we also get w L a , L b x s w R a , R b x s 0. By Lemma 4.4, the left and right multiplication operators by elements of A i commute with those by elements of A j . Therefore, we have an epimorphism : mi A i ª A given by the multiplication of the factors. Consider the ideals TN iX of T ⬘Ž A. as in the proof of Proposition 4.6. Since T ⬘Ž A. is semisimple so is TN iX . The subalgebra A i is a TN iX-module, and by Weyl’s theorem it is completely reducible. In fact, by Lemma 4.4 any submodule is an ideal and the converse. Thus A i is the direct sum of simple Žunital. ideals. Fix AXi to be one of these simple ideals. Clearly Ł i AXi : A is a T Ž A.-submodule. By irreducibility it follows that A s Ł j AXj . Any other simple ideal AYi in the decomposition of A i verifies AYi s AAYi s ŽŁ j/ i AXj . AXi AYi s 0. So, A i s AXi is a central simple algebra as well as mi A i , and consequently is an isomorphism. Finally, we observe that Ý i Nalt Ž A i . : Nalt Ž A. : F1 q Ý i NiX implies Nalt Ž A i . s F1 q NiX . X This proposition allows us to distinguish two cases, algebras in which Nalt X is a simple Lie algebra and algebras in which Nalt is a simple non-Lie Malcev algebra.

PROPOSITION 4.8. associati¨ e algebra.

X Ž A. is a simple Lie algebra, then A is a simple If Nalt

Proof. Since J Ž a, c, b . s 6Ž a, c, b . for any a, b, c g Nalt Ž A. w27, 37x, the hypothesis implies that Ž a, b, c . s 0 and thus Da, b Ž c . s ad w a, bxŽ c . s X ww a, b x, c x by Lemma 4.2. Since Nalt Ž A. is a simple Lie algebra, any X derivation of Nalt Ž A. has the form ad w a, bx; thus, we obtain an epimorphism X Ž A. that is in fact an from the derivations of A onto the derivations of Nalt X Ž A., we isomorphism because A is generated by Nalt Ž A.. Given a g Nalt

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X Ž A.. denote by Da the unique derivation of A that restricts to ad a over Nalt X It is not difficult to check that span F ² Da y ad a : a g Nalt Ž A.: is an ideal of T ⬘Ž A. that kills Nalt Ž A.. Since T ⬘Ž A. is semisimple then the subspace killed by an ideal is a submodule of A and, by irreducibility, it must be all of A. Therefore, ad a s Da is a derivation. But Ž L a y R a , Ta q R a , yL a y Ta ., Ž L a y R a , L a y R a , L a y R a . g TderŽ R . implies Ž0, 3 R a , y3L a . g TderŽ A. which can be written as Ž x, a, y . s 0. Thus Ž a, x, y . s Ž x, y, a. s X Ž A. and NŽ A. yŽ x, a, y . s 0 and a g NŽ A.. Since A is generated by Nalt is a subalgebra, this finishes the proof. X Ž A. is a simple non-Lie Malcev algebra, Now we will assume that Nalt X that is, Nalt Ž A. s C⬘ where C s ZorŽ F . denotes the split octonion algebra, which is the only octonion algebra up to isomorphism over an algebraically closed field.

PROPOSITION 4.9. We ha¨ e that X Ži. DerŽ A. s span F ² Da, b : a, b g Nalt Ž A.: is a simple Lie algebra of type G 2 . X Žii. span F ² Da, b , ad a : a, b g Nalt Ž A.: is a simple Lie algebra of type B3 . Žiii. T ⬘Ž A. is a simple Lie algebra of type D4 . Živ. The maps , : T ⬘Ž A. ª T ⬘Ž A. gi¨ en by : L a ¬ Ta , R a ¬ yR a , Da, b ¬ Da, b , and : L a ¬ yL a , R a ¬ Ta , Da, b ¬ Da, b can be identified with the automorphisms corresponding to the permutations Ž13. and Ž14. of the Dynkin diagram of D4 . X Ž A., and since Proof. Any derivation of A induces a derivation of Nalt X Ž A. A is generated by Nalt Ž A. then any two derivations that agree on Nalt X X must be equal. It is known that DerŽNalt Ž A.. s ² Da, b < N alt Ž A. : a, b g X Ž A.:; therefore, this yields the first part of Ži.. Consider a standard Nalt basis e1 , e2 , u1 , u 2 , u 3 , ¨ 1 , ¨ 2 , ¨ 34 of C w11x and f s e1 y e2 . Relative to the subalgebra H s span F ² Du1 , ¨ 1, Du 2 , ¨ 2 , L f , R f :, T ⬘Ž A. decomposes as the direct sum of root spaces in the way given in Table I Žnote that the elements in the right column are not 0 by evaluating them in appropriate elements of the standard basis.. Therefore, H is a Cartan subalgebra of T ⬘Ž A. and the root system corresponds to a simple Lie algebra of type D4 . The automorphism leaves H invariant and permutes the root spaces corresponding to ␣ 1 and ␣ 3 , but fixes those corresponding to ␣ 2 and ␣ 4 . Thus, can be thought of as the automorphism Ž13. of the Dynkin diagram of D4 . Similarly, corresponds to Ž14.. X Ž A.: is the algebra fixed by The subalgebra span F ² Da, b , ad a : a, b g Nalt the automorphism which corresponds to the automorphism Ž34. of the Dynkin diagram. This algebra is known to be a simple Lie algebra of type

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TABLE I Root

Element that spans the root space

␣1 ␣2 ␣3 ␣4 ␣1 q ␣2 ␣2 q ␣3 ␣ 2 q ␣4 ␣1 q ␣2 q ␣3 ␣1 q ␣ 2 q ␣4 ␣ 2 q ␣ 3 q ␣4 ␣1 q ␣ 2 q ␣ 3 q ␣4 ␣1 q 2 ␣ 2 q ␣ 3 q ␣4 Negative roots

L u 3 y R u 3 y De1, u 3 Du 2 , ¨ 3 L u 3 q 2 R u 3 y De1, u 3 2 L u 3 q R u 3 q De1, u 3 L u 2 y R u 2 y De1, u 2 L u 2 q 2 R u 2 y De1, u 2 2 L u 2 q R u 2 q De1, u 2 2 L¨ 1 q R ¨ 1 q De 2 , ¨ 1 L¨ 1 q 2 R ¨ 1 y De 2 , ¨ 1 L¨ 1 y R ¨ 1 y De 2 , ¨ 1 D¨ 1, u 3 D¨ 1, u 2 Change ␣ i by y␣ i , e1 by e2 and u i by ¨ i in the previous rows

X Ž A.: is the algebra fixed by B3 . Finally, DerŽ A. s span F ² Da, b : a, b g Nalt the automorphism , which corresponds to Ž134. as an automorphism of the Dynkin diagram, therefore it is a Lie algebra of type G 2 .

Let ␣ 1 , . . . , ␣ 4 4 be a basis of a root system of D4 as in the above table and 1 , . . . , 4 be the corresponding fundamental weights. If is a dominant weight, we will denote by V Ž . the irreducible module of highest weight . PROPOSITION 4.10. The Lie algebra A is isomorphic as a D4-module to V Ž n 1 . for some n. Proof. Since F1 is a trivial submodule for B3 , the branching rules for the inclusion B3 : D4 w14, Theorem 8.1.4x imply that there exists an n such that A ( V Ž n 1 .. Since C ( V Ž 1 ., this proposition allows us to identify A with the submodule of C mF ⭈⭈⭈ mF C generated by ¨ 0 m ⭈⭈⭈ m ¨ 0 with ¨ 0 the highest weight of C. This submodule obviously lies in Sym n Ž C . and it is killed by the contraction Sym n Ž C . ª Sym ny 2 Ž C ., x m ⭈⭈⭈ m x ¬ nŽ x . x m ⭈⭈⭈ m x. In fact, this is the kernel of this contraction w13, Example 19.21x. Finally, in order to prove Theorem 4.1 we have to determine the product on A. This product is not a D4-homomorphism of V Ž n 1 . mF V Ž n 1 . ª V Ž n 1 . since that would imply that D4 acts as derivations, which is not true. Given an automorphism of D4 and V a module, we denote by V the vector space V but with a new action given by d( x s Ž d . x for all

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d g D4 and x g V. Then Ž2. implies that V Ž n 1 . mF V Ž n 1 . ª V Ž n 1 . x m y ¬ xy is a D4-homomorphism. Since V Ž n 1 . ( V Ž n 3 . and V Ž n 1 . ( V Ž n 4 ., the product is a D4-homomorphism from V Ž n 3 . mF V Ž n 4 . onto V Ž n 1 .. However, since dim Ž Hom D 4Ž V Ž n 3 . mF V Ž n 4 . , V Ž n 1 . . . s 1 w25x, then we only have a possibility that is fulfilled by the induced product of Sym n Ž C .. This proves Theorem 4.1. Remark 4.11. The commutative nucleus K ŽTnŽ C .. is killed by ad a for any a, so it is killed by the action of B3 . Since the decomposition of V Ž n 1 . as a B3-module is multiplicity free this implies that K ŽTnŽ C .. s F. The generalized alternative nucleus Nalt ŽTnŽ C .. is the direct sum of simple Malcev ideals. One of these ideals is F and the other is C⬘. Since any other ideal would be killed by the action of B3 we have that Nalt ŽTnŽ C .. s C. In particular, N ŽTnŽ C .. s F. If A is as in Theorem 4.1 and A s A1 mF ⭈⭈⭈ mF A m is the decomposition as a tensor product given by the theorem, then Proposition 3.3 implies that Nalt Ž A . s Nalt Ž A1 . mF F mF ⭈⭈⭈ mF q ⭈⭈⭈ qF mF ⭈⭈⭈ mF F mF Nalt Ž A m . . As in Corollary 3.10, this implies that the decomposition is unique up to order and isomorphism of the factors.

5. CONNECTIONS WITH STRUCTURABLE ALGEBRAS In w6x Allison classified the finite dimensional central simple structurable algebras over fields of characteristic zero. Later, Smirnov w33, 34x showed that there was a gap in the list provided by Allison, and one has to include in the previous list the algebra of symmetric octonion tensors w7x, a 35-dimensional algebra which in our notation corresponds to T2 Ž C .. We want to analyze the connection between structurable algebras and algebras generated by its generalized alternative nucleus. Recall from w6x that any structurable algebra Ž A, y. is skew-alternative; that is, the skewsymmetric elements for the involution lie in the generalized alternative nucleus. In his work, Allison first reduces the classification of finite dimensional central simple Žas algebras with involution. structurable alge-

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bras to the case in which the algebra is central simple, so up to a scalar extension one may assume that the field is algebraically closed. After that, he splits the proof into two cases, depending on whether or not the algebra is generated by the skew-symmetric elements. If the algebra is not generated by the skew-symmetric elements ŽCase 1., then one obtains either a central simple Jordan algebra with the identity as involution, an algebra constructed from a nondegenerate Hermitian form on a module over a unital central simple associative algebra with involution, or an algebra with involution constructed from an admissible triple. The second case ŽCase 2., where the Kantor᎐Smirnov structurable algebra is missed, deals with algebras generated by the skew-symmetric elements. In this case Allison and Smirnov obtain that the only possibilities are either a central simple associative algebra with involution, the tensor product of an octonion algebra with a composition algebra, or the Kantor᎐Smirnov structurable algebra. Since the skew-symmetric elements lie in the generalized alternative nucleus, then Case 2 falls naturally into our context. So we can use Theorem 4.1 to give a new proof of this case. We will assume that A is a finite dimensional central simple structurable algebra, over an algebraically closed field of characteristic zero, which is generated by the skew-symmetric elements. The key point is Lemma 14 in Allison’s paper which establishes that A is spanned by s, s 2 : s is skew-symmetric4 . Let us write A s A1 mF ⭈⭈⭈ mF A m as given by Theorem 4.1. Since for any skewsymmetric element s, s, s 2 g

Ý F mF

⭈⭈⭈ mF F mF A i mF F mF ⭈⭈⭈

i, j

mF F mF A j mF F mF ⭈⭈⭈ mF F , then m F 2. If m s 1, then A is either associative or isomorphic to TnŽ C . with n G 2. Since Nalt ŽTnŽ C .. ( C then, in the latter case, Lemma 14 also implies that dim TnŽ C . s dim V Ž n 1 . F 35, so n F 2, and we obtain the octonions and the Kantor᎐Smirnov structurable algebra. Finally, if m s 2 then A s A1 mF A 2 and Lemma 14 implies that A s A1 mF F q F mF A 2 q Nalt Ž A1 . mF Nalt Ž A 2 .. In particular, A1 and A 2 are alternative, so A is either the tensor product of two octonion algebras or the tensor product of an octonion algebra and an associative algebra. In the second case, the involution of A preserves the associative nucleus and its centralizer so it preserves each factor in the tensor product. If Si denotes the skew-symmetric elements of A i , then Lemma 14 implies that A s A1 mF F q F mF A 2 q S1 mF S2 , so the set of symmetric elements of A i must be F and A i are quadratic algebras. Therefore the associative factor must be isomorphic to the two-by-two matrices, which is isomorphic to the quaternions.

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6. INVARIANT BILINEAR FORMS A symmetric bilinear form Ž , . : A = A ª F on an F-algebra A is said to be associati¨ e if Ž xy, z . s Ž x, yz . for any x, y, z g A. If the algebra has an involution x ¬ x with Ž x, y . s Ž x, y ., the new bilinear form ² x, y : s Ž x, y . is symmetric and verifies ² x, y : s ² x, y : and ² xy, z : s ² y, xz :; that is, ² , : is invariant. In w33x, Schafer proves that, up to scalar multiples, there is only one invariant symmetric bilinear form on a finite dimensional central simple structurable algebra over a field of characteristic zero. That invariant form was constructed by Allison in w6x. In this section we construct an associative symmetric bilinear form on any algebra generated by its generalized alternative nucleus. PROPOSITION 6.1. Let A be an algebra generated by its generalized alternati¨ e nucleus; then the symmetric bilinear form

Ž x, y . s trace Ž L x L y . is associati¨ e. If A is unital, then Ž x, y . s traceŽ L x y .. Proof. We prove that Ž xy, z . s Ž y, zx . by induction on the degree of x on Nalt Ž A.. If x s a g Nalt Ž A. then trace Ž L a y L z . s trace Ž L a L y L z q R a , L y L z . s trace Ž L y L z L a q R a , L y L z . s trace Ž L y L z a y L y w L z , R a x q R a , L y L z . s trace Ž L y L z a q R a , L y L z

.

s trace Ž L y L z a . , where we have used Lemma 4.2. Thus Ž ay, z . s Ž y, za. and we get the first step in the induction. Now suppose that x s ax 0 or x s x 0 a with a g Nalt Ž A. and that Ž x 0 y, z . s Ž y, zx 0 . for any y, z. In the first case it follows that

Ž xy, z . s Ž Ž ax 0 . y, z . s Ž aŽ x 0 y . , z . q Ž Ž a, x 0 , y . z . s Ž y, Ž za. x 0 . q Ž Ž a, x 0 , y . z . s Ž y, zx . q Ž y, Ž z, a, x 0 . . q Ž Ž a, x 0 , y . z . s Ž y, zx . y Ž y, Ž a, z, x 0 . . q Ž Ž x 0 , y, a . z . s Ž y, zx . y Ž y, R x 0 , L a Ž z . . q Ž R a , L x 0 Ž y . , z . s Ž y, zx . y Ž R a , L x 0 Ž y . , z . q Ž R a , L x 0 Ž y . , z . s Ž y, zx . .

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The second case is analogous. This completes the induction. If A is unital, then Ž x, y . s Ž xy, 1. s traceŽ L x y .. We denote the radical of this bilinear form by Rad. Since the form is associative, Rad is an ideal. We remark that if this form is nondegenerate then it is, up to scalar multiples, the only nondegenerate associative bilinear form on A w9x. COROLLARY 6.2. If Ž , . is nondegenerate then A is the direct sum of algebras as in Theorem 4.1. Proof. Let I be an ideal with I 2 s 0 and x g I. Given y g A, x Ž y Ž x Ž yA... : xI s 0; thus L x L y is nilpotent and Ž x, y . s 0. Since Rad s 0, we obtain I s 0. By w31, Theorem 2.6x, A is the direct sum of ideals A i that are simple unital algebras. Moreover, Nalt Ž A. s Nalt Ž[i A i . s [i Nalt Ž A i . implies that A i s alg²Nalt Ž A i .:. Remark 6.3. Let A be generated by Nalt Ž A. and suppose that the associative bilinear form Ž x, y . s traceŽ L x L y . is nondegenerate. If A is unital, the corollary yields the classification of A. In general, we consider the unital algebra A噛 s A [ F1, which contains A as an ideal. Since the bilinear forms on A and A噛 agree, it follows that A噛 s A [ Fe with Fe the orthogonal complement of A, which is an ideal. Now e s ␣ 1 q x with x g A, and ␣ / 0 implies that 0 / e 2 g Fe, and we can assume that e is an idempotent. Therefore traceŽ L e L e . s 1 and the bilinear form on A噛 is nondegenerate. By the corollary, A噛 is the direct sum of simple unital ideals, but A is an ideal and thus it is the sum of some of these ideals. This implies that A must be unital if the bilinear form is nondegenerate.

REFERENCES 1. A. A. Albert, On the Wedderburn norm condition for cyclic algebras, Bull. Amer. Math. Soc. 37 Ž1931., 301᎐312. 2. A. A. Albert, A construction of non-cyclic normal division algebras, Bull. Amer. Math. Soc. 38 Ž1932., 449᎐456. 3. A. A. Albert, Tensor products of quaternion algebras, Proc. Amer. Math. Soc. 35 Ž1972., 65᎐66. 4. B. N. Allison, Structurable division algebras and relative rank one simple Lie algebras, Can. Math. Soc. Conf. Proc. 5 Ž1986., 149᎐156. 5. B. N. Allison, Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras, Trans. Amer. Math. Soc. 306, No. 2 Ž1988., 667᎐695. 6. B. N. Allison, A class of nonassociative algebras with involution containing the class of Jordan algebras, Math. Ann. 237 Ž1978., 133᎐156. 7. B. N. Allison and J. R. Faulkner, The algebra of symmetric octonion tensors, No¨ a J. Algebra Geom. 2, No. 1 Ž1993., 47᎐57.

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