Relatively free algebras of finite rank

TL;DR

This paper studies polynomial identities of relatively free algebras of finite rank within a specific algebraic variety defined by Grassmann envelopes of finite-dimensional superalgebras, including special cases like upper triangular matrices.

Contribution

It characterizes polynomial identities of these algebras and introduces the analysis of their Grassmann and $k$-th Grassmann envelopes, extending understanding of superalgebra identities.

Findings

Identifies polynomial identities for the variety defined by Grassmann envelopes.

Analyzes identities of $UT_2(G)$ and $UT_2(G^{(k)})$ algebras.

Provides explicit descriptions for identities of these specific superalgebras.

Abstract

Let $\mathbb{K}$ be a field of characteristic zero and $B=B_0+B_1$ a finite dimensional associative superalgebra. In this paper we investigate the polynomial identities of the relatively free algebras of finite rank of the variety $\mathfrak V$ defined by the Grassmann envelope of $B$. We also consider the $k$-th Grassmann Envelope of $B$, $G^{(k)}(B)$, constructed with the $k$-generated Grassmann algebra, instead of the infinite dimensional Grassmann algebra. We specialize our studies for the algebra $UT_2(G)$ and $UT_2(G^{(k)})$, which can be seen as the Grassmann envelope and $k$-th Grassmann envelope, respectively, of the superalgebra $UT_2(\mathbb{K}[u])$, where $u^2=1$.