On Universal Equivalence of Partially Commutative Metabelian Lie Algebras

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.