On the multiplicities of the central cocharacter of algebras with polynomial identities

TL;DR

This paper studies the structure of polynomial identities and central polynomials in associative algebras, providing explicit descriptions and classifications of algebras based on their cocharacter sequences.

Contribution

It offers explicit descriptions of cocharacter, central cocharacter, and proper central cocharacter sequences for specific PI-algebras, leading to a classification of algebras with bounded colengths.

Findings

Explicit descriptions of cocharacter sequences for several PI-algebras

Classification of algebras with bounded colengths and central colengths

Complete characterization up to PI-equivalence

Abstract

For an associative algebra $A$ over a field of characteristic zero, let $P_n(A)$ and $P_n^z(A)$ denote the spaces of multilinear polynomials of degree $n$ modulo the polynomial identities and the central polynomials of $A$, respectively. We also write $\Delta_n(A)$ for the space of multilinear central polynomials of degree $n$ modulo the polynomial identities of $A$. The corresponding sequences of colengths, central colengths and proper central colengths measure the number of irreducible components in the $S_n$-module decompositions of $P_n(A)$, $P_n^z(A)$ and $\Delta_n(A)$, respectively. In this paper, we investigate several examples of PI-algebras and explicitly describe their cocharacter, central cocharacter and proper central cocharacter sequences. As a consequence, we obtain a complete classification, up to PI-equivalence, of all algebras whose sequences of colengths and central colengths are bounded by a constant.