Identities of Cayley-Dickson algebras
When one studies the polynomial identities satisfied by a given algebra, it is of great importance to know as much as possible about its identities of ``small'' degrees. In general they are able to determine the structure of the T-ideal of the algebra. The paper under review considers algebras over a field of characteristic 0 (or prime \(p\) that is greater than the degrees of the identities to be considered). It describes all multilinear identities in Cayley-Dickson algebras whose degrees do not exceed 6. In addition, all central polynomials of degrees \({}\leq 6\) for these algebras are obtained. It is shown which ones are consequences of identities of lower degrees, and which are the ``new'' identities. It should be mentioned that when one studies the identities in a nonassociative algebra the location of the brackets in the monomials is important; this increases the number of polynomials to be considered. In order to reduce the computations to some ``reasonable'' amount (i.e., reasonable for quite powerful computers) the authors have established some properties of the group rings of the symmetric groups, and have used some interesting combinatorics in order to reduce significantly the number of polynomials to be checked on a computer. The paper is of interest for the specialists in the theory of algebras with polynomial identities as well as in algebraic combinatorics.
scientific article; zbMATH DE number 1013925