$\mathbb{Z}_2$-graded $*$-polynomial identities and cocharacteres for $M_{1,1}(E)$, $UT_{1,1}(E)$ and $UT_{(0,1,0)}(E)$

TL;DR

This paper characterizes the polynomial identities and cocharacter sequences of certain $ ext{Z}_2$-graded matrix algebras with superinvolutions over an infinite-dimensional Grassmann algebra, advancing understanding of graded algebra structures.

Contribution

It provides a detailed description of polynomial identities and cocharacter sequences for specific $ ext{Z}_2$-graded matrix algebras with superinvolutions tensorized with the Grassmann algebra.

Findings

Explicit descriptions of polynomial identities for the algebras.

Determination of cocharacter sequences for the graded algebras.

Analysis of the impact of superinvolutions on identities.

Abstract

Let $K$ be a field of characteristic 0, and let $E$ be the infinite-dimensional Grassmann algebra over $K$. We consider $E$ as a $\mathbb{Z}_2$-graded algebra, where the grading is given by the vector subspaces $E_0$ and $E_1$, consisting of monomials of even and odd lengths, respectively. Thus, if $A = A_0 \oplus A_1$ is an associative $\mathbb{Z}_2$-graded algebra, we can consider the $\mathbb{Z}_2$-graded algebra $A \hat{\otimes} E = (A_0 \otimes E_0) \oplus (A_1 \otimes E_1)$. In case both $E$ and $A$ are endowed with superinvolutions, we can define a $\mathbb{Z}_2$-graded involution on $A \hat{\otimes} E$ induced by the respective superinvolutions. In this paper, we consider the $\mathbb{Z}_2$-graded matrix algebras $M_{1,1}(K)$, $UT_{1,1}(K)$, and $UT_{(0,1,0)}(K)$ endowed with superinvolutions. We shall provide a description of the polynomial identities and the cocharacter sequences of $M_{1,1}(K)\hat{\otimes} E$, $UT_{1,1}(K)\hat{\otimes} E$, and $UT_{(0,1,0)}(K)\hat{\otimes} E$, considering these resulting algebras as $\mathbb{Z}_2$-graded algebras with graded involution.