On Universal Equivalence of Partially Commutative Metabelian Lie Algebras

TL;DR

This paper investigates the universal equivalence of partially commutative metabelian Lie algebras based on their defining graphs, establishing conditions for equivalence and highlighting limitations in classifying these algebras.

Contribution

It characterizes when such Lie algebras are universally equivalent based on cycle length and shows the non-separability of tree-based algebras by universal theory.

Findings

Universal equivalence holds iff cycles have the same length.

Class of tree-based algebras is not universally separable.

Provides examples illustrating these properties.

Abstract

In this paper, we consider partially commutative metabelian Lie algebras whose defining graphs are cycles. We show that such algebras are universally equivalent iff the corresponding cycles have the same length. Moreover, we give an example showing that the class of partially commutative metabelian Lie algebras such that their defining graphs are trees is not separable by universal theory in the class of all partially commutative metabelian Lie algebras.