On almost nilpotent varieties of subexponential growth

Abstract

Let ${}_2\mathcal{N}$ be the variety of left-nilpotent algebras of index two, that is the variety of algebras satisfying the identity $x(yz)\equiv 0$. We introduce two new varieties, denoted by ${\mathcal{V}}_{\mathit{sym}}$ and ${\mathcal{V}}_{\mathit{alt}}$, contained in the variety ${}_2\mathcal{N}$ and we prove that ${\mathcal{V}}_{\mathit{sym}}$ and ${\mathcal{V}}_{\mathit{alt}}$ are the only two varieties almost nilpotent of subexponential growth.