During the past forty years, a new trend in the theory of associative algebras, Lie algebras, and their representations has formed under the influence of mathematical logic and universal algebra, namely, the theory of varieties and identities of associative algebras, Lie algebras, and their representations. The last twenty years have seen the creation of the method of 2-words and $\alpha$-functions, which allowed a number of problems in the theory of groups, rings, Lie algebras, and their representations to be solved in a unified way. The possibilities of this method are far from exhausted. This book sums up the applications of the method of 2-words and $\alpha$-functions in the theory of varieties and gives a systematic exposition of contemporary achievements in the theory of identities of algebras and their representations closely related to this method. The aim is to make these topics accessible to a wider group of mathematicians.
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978-0-8218-4608-7 (9780821846087)
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Schweitzer Klassifikation
Preliminary results Characters and $\alpha$-functions on 2 -words and varieties of representations of Lie algebras distinguished by them $\alpha$-functions related to the killing form and to irreducible representations of semisimple Lie algebras. Central polynomials of irreducible representations of reductive Lie algebras $\alpha$-functions related to full matrix algebras. Trace identities and central polynomials of full matrix algebras $M_n$ and matrix superalgebras $M_{n,k}$ The $alpha$-function related to representations of the simple three-dimensional Lie algebra $\mathfrak g$ and its applications to varieties of groups and associative algebras Varieties generated by Lie algebras of Cartan type Algebraic supplements.
