Varieties of algebras with polynomial identities: some invariants and numerical sequences

Speaker: Fabrizio Martino, University of Palermo.

Date: 23 sep 2022, 12h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: Let F be a field of characteristic zero, A be an associative F-algebra and F(X) be the free associative algebra, freely generated over F by the countable set X of variables. A non-zero polynomial f(x1, . . . , xn) ∈ F(X) is a polynomial identity of A if for all a1, . . . , an ∈ A, f(a1, . . . , an) = 0. The set of all polynomial identities of a given algebra is called T-ideal of identities and it is denoted by Id(A). Motivated by an idea of Regev, in characteristic zero one can consider the space of multilinear polynomials

P_n = span_F {x_σ(1)x_σ(2) · · · x_σ(n) | σ ∈ S_n}

and define the n-th codimension of A as the dimension of the quotient vector space P_n(A) of multilinear polynomials reduced modulo the identities of A, i.e.
c_n(A) = dim_F P_n / (P_n ∩ Id(A)).

The asymptotic behavior of such codimension sequence measures in some sense the number of polynomial identities satisfied by A. In this talk we will present the main results and problems about varieties of algebras with polynomial identities and their codimension sequences. Furthermore, we will generalize the idea of polynomial identity by studying the so-called central polynomials, i.e., polynomials whose evaluations belong to Z(A), the center of A. We will define, in fact, the central codimension sequence c^z_n(A) and we will connect it with c_n(A).