Identities of Cayley-Dickson algebras
type: Item
instance of: scholarly article
title: Identities of Cayley-Dickson algebras
zbMATH Open document ID: 0890.17001
published in: Journal of Algebra
publication date: 1997-07-17T00:00:00Z
review text: 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.
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 17-04
Mathematics Subject Classification ID: 17C05
Mathematics Subject Classification ID: 17D05
Mathematics Subject Classification ID: 16R99
Mathematics Subject Classification ID: 17A30
zbMATH DE Number: 1013925
zbMATH Keywords: Cayley-Dickson algebras
zbMATH Keywords: polynomial identities
zbMATH Keywords: computational methods
zbMATH Keywords: nonassociative algebras
zbMATH Keywords: free algebras
zbMATH Keywords: multilinear identities
zbMATH Keywords: central polynomials
author: Irvin Roy Hentzel
author: Luiz Antonio Peresi
MaRDI profile type: Publication
full work available at URL: jabr.1996.6814
OpenAlex ID: W1993070736
cites work: A Simplification of the Computation of the Natural Representation of the Symmetric Group S n
cites work: Q4294170
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cites work: Q3947798
DOI: 10.1006/JABR.1996.6814
description: scientific article; zbMATH DE number 1013925