ℤ2-gradings and minimal varieties of ℤ2-graded PI algebras

Abstract

Let [Formula: see text] be an algebraically closed field of characteristic zero and let [Formula: see text] be a minimal superalgebra with semisimple part [Formula: see text], where each [Formula: see text] is a simple superalgebra. We say that [Formula: see text] is [Formula: see text]-special if [Formula: see text] and there exists [Formula: see text] such that [Formula: see text], with [Formula: see text], whereas, for the remaining indices [Formula: see text], [Formula: see text]. In [1], Avelar et al. classified the [Formula: see text]-special minimal superalgebras and described some necessary and sufficient conditions for the varieties generated by such superalgebras to be minimal of [Formula: see text]-exponent [Formula: see text]. In this paper, we give continuity to a such study and we made explicit that the [Formula: see text]-gradings play an important role in the minimality of some varieties. We provide sufficient conditions, depending on the [Formula: see text]-grading of [Formula: see text], for the variety generated by [Formula: see text] to be minimal. Moreover, we characterize two families of [Formula: see text]-special minimal superalgebras and we give necessary and sufficient conditions (depending on their [Formula: see text]-gradings) for the varieties generated by them to be minimal.