On varieties of Lie algebras with infinite basis property

TL;DR

This paper constructs examples of locally finite Lie algebra varieties over fields of positive characteristic that lack finite bases of polynomial identities, demonstrating new possibilities in the structure of Lie algebra varieties.

Contribution

It introduces the first known example of a locally finite Lie algebra variety without a finite basis of identities over positive characteristic fields.

Findings

Existence of locally finite Lie algebra varieties without finite identity bases

Construction of varieties with prescribed properties

Implications for the theory of polynomial identities in Lie algebras

Abstract

Over an arbitrary field of positive characteristic we construct an example of a locally finite variety of Lie algebras which does not have a finite basis of its polynomial identities. As a consequence we construct varieties of Lie algebras with prescribed properties.