Search results for "Polynomial identity"

MR2966998 Aljadeff, Eli; Kanel-Belov, Alexei Hilbert series of PI relatively free G-graded algebras are rational functions. Bull. Lond. Math. Soc. 44…

2013

MR3038546, Brešar, Matej; Klep, Igor A local-global principle for linear dependence of noncommutative polynomials. Israel J. Math. 193 (2013), no. 1,…

2014

Let F be a eld of characteristic zero and FhXi the free associative algebra on X = fX1;X2; : : : g over F; i.e., the algebra of polynomials in the non-commuting variables Xi 2 X. A set of polynomials in FhXi is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. In [Integral Equations Operator Theory 46 (2003), no. 4, 399{454; MR1997979 (2004f:90102)], J. F. Camino et al., in the setting of free analysis, motivated by systems engineering, proved that a nite locally linearly dependent set of polynomials is linearly dependent. In this paper the authors give an alternative algebraic proof of this result based on the theory of polynomial i…

Classifying the Minimal Varieties of Polynomial Growth

2014

Let $\mathcal{V}$ be a variety of associative algebras generated by an algebra with $1$ over a field of characteristic zero. This paper is devoted to the classification of the varieties $\mathcal{V}$ which are minimal of polynomial growth (i.e., their sequence of codimensions growth like $n^k$ but any proper subvariety grows like $n^t$ with $t 4$, the number of minimal varieties is at least $|F|$, the cardinality of the base field and we give a recipe of how to construct them.

Varieties of algebras of polynomial growth

2008

Let V be a proper variety of associative algebras over a field F of characteristic zero. It is well-known that V can have polynomial or exponential growth and here we present some classification results of varieties of polynomial growth. In particular we classify all subvarieties of the varieties of almost polynomial growth, i.e., the subvarieties of var(G) and var(UT 2), where G is the Grassmann algebra and UT2 is the algebra of 2 x 2 upper triangular matrices.

On the growth of the identities of algebras

2006

Codimensions of algebras with additional structures

2022

Let A be an associative algebra endowed with an automorphism or an antiautomorphism phi of order <= 2. One associates to A, in a natural way, a numerical sequence c(n)(phi)(A), n = 1, 2, ... , called the sequence of phi-codimensions of A which is the main tool for the quantitative investigation of the polynomial identities satisfied by A. In [13] it was proved that such a sequence is eventually nondecreasing in case phi is an antiautomorphism. Here we prove that it still holds in case phi is an automorphism and present some recent results about the asymptotics of c(n)(phi)(A).

Growth of central polynomials of algebras with involution

2021

Let A be an associative algebra with involution ∗ over a field of characteristic zero. A central ∗-polynomial of A is a polynomial in non- commutative variables that takes central values in A. Here we prove the existence of two limits called the central ∗-exponent and the proper central ∗-exponent that give a measure of the growth of the central ∗-polynomials and proper central ∗-polynomials, respectively. Moreover, we compare them with the PI-∗-exponent of the algebra.