On The Tensor Product of Two Composition Algebras

Let C1 ⊗F C2 be the tensor product of two composition algebras over a field F with char(F ) 6= 2. R. Brauer [7] and A. A. Albert [1], [2], [3] seemed to be the first mathematicians who investigated the tensor product of two quaternion algebras. Later their results were generalized to this more general situation by B. N. Allison [4], [5], [6] and to biquaternion algebras over rings by Knus [12]. In the second section 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 ([13]). Since Allison regarded the involution σ = γ1 ⊗ γ2 as an essential part of the algebra C = C1 ⊗ C2, he only studied automorphisms of C which are compatible with σ. In the last section we show that any automorphism of C that preserves a certain biquaternion subalgebra also is compatible with σ. As a consequence, if C is the tensor product of two octonion algebras, we show that C does not satisfy the Skolem-Noether Theorem. Let F be a field and C a unital, nonassociative F -algebra. Then C is a composition algebra if there exists a nondegenerate quadratic form n : C → F such that n(x · y) = n(x)n(y) for all x, y ∈ C. The form n is uniquely determined by these conditions and is called the norm of C. We will write n = nC . Composition algebras only exist in ranks 1, 2, 4 or 8 (see [10]). Those of rank 4 are called quaternion algebras, and those of rank 8 octonion algebras. A composition algebra C has a canonical involution γ given by γ(x) = t(x)1C − x, where the trace map t : C → F is given by t(x) = n(1, x). An example of an 8-dimensional composition algebra is Zorn’s algebra of vector matrices Zor(F ) (see [14, p. 507] for the definition). The norm form of Zor(F ) is given by the determinant and is a hyperbolic form. Composition algebras are quadratic; i.e., they satisfy the identities