Relatively free algebras of finite rank

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 $U{T}_2(G)$ and $U{T}_2(G^{(k)})$, which can be seen as the Grassmann envelope and $k$-th Grassmann envelope, respectively, of the superalgebra $U{T}_2(\mathbb{K}[u])$, where $u^2=1$.

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