Minimal Varieties and Identities of Relatively Free Algebras

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 $$A=\begin{pmatrix} E_0 & E\\ 0 & E\\ \end{pmatrix},$$ 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.