Superpolynomial identities of finite-dimensional simple algebras

TL;DR

This paper characterizes polynomial identities of Grassmann envelopes of finite-dimensional $bZ_2$-graded algebras, classifies when two such algebras share identities, and extends the Grassmann envelope concept to $bOmega$-algebras.

Contribution

It provides a detailed description of superpolynomial identities for Grassmann envelopes and classifies $bZ_2$-graded-simple algebras sharing the same identities, extending the theory to $bOmega$-algebras.

Findings

Describes polynomial identities of Grassmann envelopes of $bZ_2$-graded algebras.

Classifies conditions for two graded algebras to share identities.

Extends Grassmann envelope construction to $bOmega$-algebras.

Abstract

We investigate the Grassmann envelope (of finite rank) of a finite-dimensional $\mathbb{Z}_2$-graded algebra. As a result, we describe the polynomial identities of $G_1(\mathcal{A})$, where $G_1$ stands for the Grassmann algebra with $1$ generator, and $\mathcal{A}$ is a $\mathbb{Z}_2$-graded-simple associative algebra. We also classify the conditions under which two associative $\mathbb{Z}_2$-graded-simple algebras share the same set of superpolynomial identities, i.e., the polynomial identities of its Grassmann envelope (in particular, of finite rank). Moreover, we extend the construction of the Grassmann envelope for the context of $\Omega$-algebras and prove some of its properties. Lastly, we give a description of $\mathbb{Z}_2$-graded-simple $\Omega$-algebras.