Non-finitely based varieties of right alternative metabelian algebras

TL;DR

This paper constructs a specific non-finitely based variety of right alternative metabelian algebras over certain fields, expanding understanding of their algebraic structure and Spechtian properties.

Contribution

It introduces a new non-finitely based variety generated by Grassmann algebra of rank 2, which can also be generated by a five-dimensional superalgebra, advancing the theory of these algebras.

Findings

Constructed a non-finitely based variety of rank 2 Grassmann algebra

Demonstrated the variety can be generated by a five-dimensional superalgebra

Extended the understanding of Spechtian properties in right alternative metabelian algebras

Abstract

Since 1976, it is known from the paper by V. P. Belkin that the variety $\mathrm{RA_2}$ of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains non-finitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann $\mathrm{RA_2}$-algebra of finite rank $r$ over a field $\mathcal{F}$, for $\mathrm{char}(\mathcal{F})\neq2$, is Spechtian iff $r=1$. We construct a non-finitely based variety $\mathfrak{M}$ generated by the Grassmann $\mathcal{V}$-algebra of rank $2$ of certain finitely based subvariety $\mathcal V\subset\mathrm{RA}_2$ over a field $\mathcal{F}$, for $\mathrm{char(\mathcal{F})}\neq2,3$, such that $\mathfrak{M}$ can also be generated by the Grassmann envelope of a five-dimensional superalgebra with one-dimensional even part.