Minimal Varieties and Identities of Relatively Free Algebras

Abstract

Let $K$ be a field of characteristic zero and let ${\mathfrak{M}}_5$ be the variety of associative algebras over $K$, defined by the identity $[x_1,x_2][x_3,x_4,x_5]$. It is well-known that such variety is a minimal variety and that is generated by the algebra $$A=\left(\begin{array}{cc}{E}_0& E\\ 0& E\end{array}\right),$$ where $E=E_0\oplus E_1$ is the Grassmann algebra. In this paper, for any positive integer $k$, we describe the polynomial identities of the relatively free algebras of rank $k$ of ${\mathfrak{M}}_5$, \[F_k(\mathfrak{M}_5)=\dfrac{K\langle x_1,\dots, x_k \rangle}{K\langle x_1,\dots, x_k \rangle\cap T(\mathfrak{M}_5)}.\] It turns out that such algebras satisfy the same polynomial identities of some algebras used in the description of the subvarieties of ${\mathfrak{M}}_5$, given by Di Vincenzo, Drensky and Nardozza.

19 pages, minor corrections

References in corpus (1)

• On the polynomial identities of the algebra $M_{11}(E)$

Cited by in corpus (1)

• Relatively free algebras of finite rank