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Classifying varieties of anti-commutative algebras. (English) Zbl 0876.17004

The author classifies the varieties of anti-commutative algebras that are defined by identities of degree \(n\leq 7\). His method is to compute the decomposition of the \(S_n\)-module of multilinear anti-commutative polynomials of degree \(n\) into simple submodules. Most of the calculations were done by computer, and so the results are only valid over any field of characteristic 0. In the case \(n=4\) these varieties were also classified by S. Kass and W. G. Witthoft [Proc. Am. Math. Soc. 26, 1-9 (1970; Zbl 0217.34501)]. Their work used J. M. Osborn’s method of irreducible identities [Adv. Math. 8, 163-369 (1972; Zbl 0232.17001)], and so did not explicitly involve any representation theory.

MSC:

17A30 Nonassociative algebras satisfying other identities
17-04 Software, source code, etc. for problems pertaining to nonassociative rings and algebras
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