# Identities of Algebras and their Representations by Yu. P. Razmyslov

By Yu. P. Razmyslov

In past times 40 years, a brand new development within the idea of associative algebras, Lie algebras, and their representations has shaped less than the effect of mathematical good judgment and common algebra, particularly, the speculation of sorts and identities of associative algebras, Lie algebras, and their representations. The final 20 years have visible the construction of the tactic of 2-words and α-functions, which allowed a couple of difficulties within the conception of teams, jewelry, Lie algebras, and their representations to be solved in a unified means. the probabilities of this system are faraway from exhausted. This publication sums up the purposes of the tactic of 2-words and α-functions within the conception of sorts and offers a scientific exposition of latest achievements within the thought of identities of algebras and their representations heavily relating to this system. the purpose is to make those issues obtainable to a much wider team of mathematicians.

Readership: study mathematicians.

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**Extra resources for Identities of Algebras and their Representations**

**Example text**

For pth Engel Lie algebras the converse is also true. (Higgins). 2) holds in it for 1 (see [Higgl, Bak]). 1 This result implies that the negative solution of the nilpotency problem forpth Engel Lie algebras is equivalent to the negative solution of the solvability problem for these algebras. 2. The variety of 3rd Engel Lie algebras over afield of characteristic 5 is nonsolvable (nonnilponent). 2) is valid in M. 3) [x, y, y, y] = 0. Let us substitute in it an element y + fix (/3 E K) instead of the variable y.

Choose in the space F; a basis consisting of monomials of the form x1, x1,, . Since any Lie commutator in ,C;, is representable in the form i+1 [y1,, [... ]] _ Egjvj ® [x1,, [... )), j=1 x1, x1 , in view of the fact that the F-module V is free, such where g j E F and Lie commutators are linearly independent in ,C', and thus we have dimK ,C;,, > dimK F), . 1) imply that all coefficients of the power series for the function (d/dz)c93, (z) are not less than the corresponding coefficients of the series for the function ez (z), and this is equivalent to the desired statement.

Thus, p is a well defined K-linear mapping of Q (A 1) into Q (A2) Obviously, p is a K-homomorphism that fixes the elements of the subalgebra A. Let V be an arbitrary finite-dimensional K-subspace in the algebra A1. Since 'FAI = A2, we see that rank (A2, V) = rank (A 1, V). This implies that p is a monomorphism and that there exists an isomorphism Q (A2) -- C (A2) ®6c (A,) p Q (A1). The proposition is proved. 2. , C (A I) = K and Q (A 1) = A 1). Then, for any extension K of the field K, the K-algebra A2 = K ®K A I 1 is prime, and C(A2) = K (Q(A2) = A2).