Hentzel, Irvin R.; Peresi, Luiz A. Identities of Cayley-Dickson algebras. (English) Zbl 0890.17001 J. Algebra 188, No. 1, 292-309 (1997). 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. Reviewer: Plamen Koshlukov (Campinas) Cited in 1 ReviewCited in 14 Documents MSC: 17-04 Software, source code, etc. for problems pertaining to nonassociative rings and algebras 17C05 Identities and free Jordan structures 17D05 Alternative rings 16R99 Rings with polynomial identity 17A30 Nonassociative algebras satisfying other identities Keywords:Cayley-Dickson algebras; polynomial identities; computational methods; nonassociative algebras; free algebras; multilinear identities; central polynomials × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Clifton, J., A simplification of the computation of the natural representation of the symmetric group\(S_n\), Proc. Amer. Math. Soc., 83, 248-250 (1981) · Zbl 0443.20013 [2] Correa, I.; Hentzel, I. R.; Peresi, L. A., Minimal identities of Bernstein algebras, Algebras Groups Geom., 11, 181-199 (1994) · Zbl 0794.17017 [3] Hentzel, I. R., Processing identities by group representation, Computers in Nonassociative Rings and Algebras (1977), Academic Press: Academic Press New York, p. 13-40 [4] Hentzel, I. R., Alternators of a right alternative algebra, Trans. Amer. Math. Soc., 242, 141-156 (1978) · Zbl 0405.17001 [5] Isaev, I. M., Identities of a finite Cayley-Dickson algebra, Algebra and Logic, 23, 282-289 (1984) · Zbl 0598.17013 [6] Olsson, J. B.; Regev, A., An application of representation theory toPI, Proc. Amer. Math. Soc., 55, 253-257 (1976) · Zbl 0328.16017 [7] Racine, M. L., Minimal identities for Jordan algebras of degree 2, Comm. Algebra, 13, 2493-2506 (1985) · Zbl 0579.17014 [8] Racine, M. L., Minimal identities of Octonion algebras, J. Algebra, 115, 251-260 (1988) · Zbl 0651.17012 [9] Zhevlakov, K. A.; Slin’ko, A. M.; Shestakov, I. P.; Shirshov, A. I., Rings That Are Nearly Associative (1982), Academic Press: Academic Press New York · Zbl 0487.17001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.