Central cocharacters of the subvarieties of varieties of superalgebras with almost polynomial growth

TL;DR

This paper investigates the structure of central polynomials and cocharacters in superalgebra varieties with near-polynomial growth, providing explicit decompositions and generators for these algebraic objects.

Contribution

It introduces detailed descriptions of central graded codimensions and cocharacters for specific supervarieties with almost polynomial growth, including explicit generators and decompositions.

Findings

Explicit formulas for central graded codimensions.

Decomposition of central graded cocharacters.

Generators of the space of central polynomials.

Abstract

In recent years, the study of the $T$-space of central polynomials of an algebra $A$ has become an object of great interest in the PI-theory. Such interest has been extended to the context of algebras with additional structures. The main goal of this paper is to present information about the central graded codimensions and the central graded cocharacters of the varieties of superalgebras $\mathrm{var}^{gr}(G)$, $\mathrm{var}^{gr}(UT_2)$, $\mathrm{var}^{gr}(G^{gr})$, $\mathrm{var}^{gr}(UT^{gr}_2)$ and $\mathrm{var}^{gr}(D^{gr})$, which are the only supervarieties with almost polynomial growth of the graded codimensions. Also we establish the generators of the space of central polynomials, determine the central codimensions and explicitly give the decomposition of the central graded cocharacters of each minimal subvariety of such supervarieties.