Minimal Varieties and Identities of Relatively Free Algebras

Abstract

Let K be a field of characteristic zero and let 𝔐5 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 it is generated by the algebra where E = E 0 ⊕ E 1 is the Grassmann algebra. In this article, for any positive integer k, we describe the polynomial identities of the relatively free algebras of rank k of 𝔐5, It turns out that such algebras satisfy the same polynomial identities of some algebras used in the description of the subvarieties of 𝔐5, given by Di Vincenzo, Drensky, and Nardozza.