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

Ana Vieira, Thais Nascimento, Juan Cruz, Willer Costa

TL;DR

This work extends PI-theory to central graded polynomials in the setting of superalgebras with almost polynomial growth. It identifies the five base superalgebras $\mathcal{G},\mathcal{G}^{gr},UT_2,UT^{gr}_2,D^{gr}$ as the building blocks for minimal subvarieties and provides explicit generators for the central graded polynomials $C^{gr}(A)$, central graded codimensions $c^{gr,z}_n(A)$, and central graded cocharacters $\chi^{gr,z}_n(A)$ for these varieties and their minimal subvarieties. The authors give detailed, closed-form descriptions for $D^{gr}$, $UT_2$, $UT_2^{gr}$, $\mathcal{G}$, and $\mathcal{G}^{gr}$ (including finite Grassmann variants) and classify the central graded structure in terms of partition-indexed representations of the hyperoctahedral group, thereby enriching the understanding of graded identities and centrality in almost-polynomial-growth contexts. These results yield precise combinatorial and representation-theoretic data that illuminate the interaction between grading, centrality, and growth in the PI-theory of superalgebras.

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.

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

TL;DR

This work extends PI-theory to central graded polynomials in the setting of superalgebras with almost polynomial growth. It identifies the five base superalgebras as the building blocks for minimal subvarieties and provides explicit generators for the central graded polynomials , central graded codimensions , and central graded cocharacters for these varieties and their minimal subvarieties. The authors give detailed, closed-form descriptions for , , , , and (including finite Grassmann variants) and classify the central graded structure in terms of partition-indexed representations of the hyperoctahedral group, thereby enriching the understanding of graded identities and centrality in almost-polynomial-growth contexts. These results yield precise combinatorial and representation-theoretic data that illuminate the interaction between grading, centrality, and growth in the PI-theory of superalgebras.

Abstract

In recent years, the study of the -space of central polynomials of an algebra 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 , , , and , 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.

Paper Structure

This paper contains 5 sections, 17 theorems, 46 equations.

Table of Contents

  1. Introdution
  2. Generalities
  3. The varieties $\textnormal{var}^{gr}(D^{gr}), \ \textnormal{var}^{gr}(UT_{2})\ \ \hbox{and} \ \ \textnormal{var}^{gr}(UT_{2}^{gr})$
  4. The variety $\textnormal{var}^{gr}(\mathcal{G})$
  5. The variety $\textnormal{var}^{gr}(\mathcal{G}^{gr})$

Key Result

Proposition 2.2

Let $A$ be a unitary superalgebra such that $A \cong F \dotplus J(A)$ and $A_{0} \not\subset Z(A)$. Then $Z(A) \cong F \dot+ J'$, where $J' = J(A) \cap Z(A)$. Moreover, if $J'$ is a graded ideal of $A$, then $C^{gr}(A)$ is a $T_{2}$-ideal of $F \langle Y,Z \rangle$.

Theorems & Definitions (27)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • Theorem 3.5
  • ...and 17 more