On the topological rank of the variety of right alternative metabelian Lie-nilpotent algebras

TL;DR

This paper establishes that the topological rank of certain right alternative metabelian Lie-nilpotent algebras equals their Lie-nilpotency step, providing a new understanding of their structural complexity.

Contribution

It constructs varieties of right alternative metabelian Lie-nilpotent algebras with arbitrary finite topological rank, linking rank to Lie-nilpotency step.

Findings

Topological rank of these algebras equals their Lie-nilpotency step.

Provides explicit examples of varieties with arbitrary finite topological rank.

Enhances understanding of the structure of non-nilpotent subvarieties.

Abstract

In 1981, S. V. Pchelintsev introduced the notion of topological rank for Spechtian varieties of algebras as a certain tool for studying the structure of non-nilpotent subvarieties in a given variety. We provide a variety of right alternative algebras of arbitrary given finite topological rank. Namely, we prove that the topological rank of the variety of right alternative metabelian (solvable of index two) algebras that are Lie-nilpotent of step not more than $s$ over a field of characteristic distinct from two and three is equal to $s$.