Search results for "central polynomials"

On central polynomials and codimension growth

2022

Let A be an associative algebra over a field of characteristic zero. A central polynomial is a polynomial of the free associative algebra that takes central values of A. In this survey, we present some recent results about the exponential growth of the central codimension sequence and the proper central codimension sequence in the setting of algebras with involution and algebras graded by a finite group.

MR2852326 Bekh-Ochir, C.; Rankin, S. A. Examples of associative algebras for which the T-space of central polynomials is not finitely based. Israel J…

2012

The central polynomials of the infinite-dimensional unitary and nonunitary Grassmann algebras

2010

For a fixed field $k$, let $k_0\langle X\rangle$ and $k_1\langle X \rangle$ denote respectively the free nonunitary associative $k$-algebra and the free unitary associative $k$-algebra on the countable set $X=\{x_1, x_2, \ldots\}.$ A polynomial $f\in k_i\langle X\rangle$ is called a central polynomial for an associative algebra $A$ if every evaluation of $f$ on $A$ lies in the center of $A.$ The set of all central polynomials of $A$ is a $T$-space of $k_i\langle X\rangle,$ i.e, a subspace closed under all endomorphisms of $k_i\langle X\rangle.$ In this paper the authors describe the T-space of central polynomials for both the unitary and the nonunitary infinite-dimensional Grassmann algebra…

Graded central polynomials for the matrix algebra of order two

2009

Let K be an infinite integral domain and $A=M_2(K)$ the algebra of $2\times 2$ matrices over $K$. The authors consider the natural $\mathbb{Z}_2$-grading of $A$ obtained by requiring that the diagonal matrices and the off-diagonal matrices are of homogeneous degree $0$ and $1$, respectively. When $K$ is a field, a basis of the graded identities of $A$ was described in [O. M. Di Vincenzo, On the graded identities of $M_{1,1}(E).$ Israel J. Math. 80 (1992), no. 3, 323-–335] in case $\mbox{char}\, K = 0$ and in [P. E. Koshlukov and S. S. de Azevedo, Graded identities for T-prime algebras over fields of positive characteristic. Israel J. Math. 128 (2002), 157-–176] when $K$ is infinite and $\mb…

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.