Identities for algebras of matrices over the octonions
نویسندگان
چکیده
This paper describes all the identities of degree 7 satisfied by algebras of 2 × 2 matrices over the octonions. There are three cases: (1) the full matrix algebra under the usual matrix product, (2) the algebra of Hermitian matrices under the symmetric product, and (3) the algebra of skewHermitian matrices under the antisymmetric product. In case (1) we present seven new identities in degree 7 which were discovered by a computer search but which are proved to hold for matrices with entries in any alternative ring. In case (2) we recover the identities of Vasilovsky in degrees 5 and 6 for the special Jordan algebra of a nondegenerate symmetric bilinear form. In case (3) we describe a computational proof that there are no identities in degree 7 which are not implied by anticommutativity. 2004 Elsevier Inc. All rights reserved.
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