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Algebras with polynomial identities. (English) Zbl 0976.16019

This survey is based on the notes for a minicourse delivered by the author at the 15th Escola de Álgebra held at Canela. It gives an introduction to selected combinatorial methods and results of PI theory. The topics touched include the Amitsur-Levitzki theorem, the Nagata-Higman-Dubnov-Ivanov theorem, Regev’s tensor product theorem, the Shirshov theorem, the Kurosh problem, applications of the representation theory of symmetric and general linear groups to the study of identities, the Specht problem, the T-ideal of identities of \(2\times 2\) matrices, identities and weak identities of Lie and Jordan algebras, identities in positive characteristic. Proofs are often substituted by hints or references.

MSC:

16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
17B01 Identities, free Lie (super)algebras
16R40 Identities other than those of matrices over commutative rings
17C05 Identities and free Jordan structures
20C30 Representations of finite symmetric groups