Alexei Kanel-Belov

REPRESENTABILITY AND SPECHT PROBLEM FOR G-GRADED ALGEBRAS

Abstract Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let id G ( W ) denote the T-ideal of G-graded identities of W. We prove: 1. [G-graded PI-equivalence] There exists a field extension K of F and a finite-dimensional Z / 2 Z × G -graded algebra A over K such that id G ( W ) = id G ( A ∗ ) where A ∗ is the Grassmann envelope of A. 2. [G-graded Specht problem] The T-ideal id G ( W ) is finitely generated as a T-ideal. 3. [G-graded PI-equivalence for affine algebras] Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite-dimensional algebra A over K such that id G ( W ) = id G ( A ) .

Kemer's theorem for affine PI algebras over a field of characteristic zero

Abstract We present a proof of Kemers representability theorem for affine PI algebras over a field of characteristic zero.

Hilbert series of PI relatively free G-graded algebras are rational functions

Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}s variables that correspond to the g_is, i=1,...,r. Let F be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F /I is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F /I is a rational function.

The images of Lie polynomials evaluated on matrices

ABSTRACT Kaplansky asked about the possible images of a polynomial f in several noncommuting variables. In this paper, we consider the case of f a Lie polynomial. We describe all the possible images of f in M2(K) and provide an example of f whose image is the set of non-nilpotent trace zero matrices, together with 0. We provide an arithmetic criterion for this case. We also show that the standard polynomial sk is not a Lie polynomial, for k>2.

Power-central polynomials on matrices

Any multilinear non-central polynomial p (in several noncommuting variables) takes on values of degree n in the matrix algebra Mn(F) over an infinite field F. The polynomial p is called ν-central for Mn(F) if pν takes on only scalar values, with ν minimal such. Multilinear ν-central polynomials do not exist for any ν, with n>3, answering a question of Drensky and Spenko. n nSaltman proved a result implying that a non-central polynomial p cannot be ν-central for Mn(F), for n odd, unless ν is a product of distinct odd primes and n=mν with m prime to ν; we extend this by showing for n even, that ν cannot be divisible by 4.