The weak commutativity construction for Lie algebras

Abstract

We study the analogue of Sidki's weak commutativity construction, defined originally for groups, in the category of Lie algebras. This is the quotient χ(g) of the Lie algebra freely generated by two isomorphic copies g and gψ of a fixed Lie algebra by the ideal generated by the brackets [x,xψ], for all x. We exhibit an abelian ideal of χ(g) whose associated quotient is a subdirect sum in g⊕g⊕g and we give conditions for this ideal to be finite dimensional. We show that χ(g) has a subquotient that is isomorphic to the Schur multiplier of g. We prove that χ(g) is finitely presentable or of homological type FP2 if and only if g has the same property, but χ(f) is not of type FP3 if f is a non-abelian free Lie algebra.