The weak commutativity construction for Lie algebras

We study the analogue of Sidki’s weak commutativity construction, defined originally for groups, in the category of Lie algebras. This is the quotient $\chi (\mathfrak{g})$ of the Lie algebra freely generated by two isomorphic copies $\mathfrak{g}$ and ${\mathfrak{g}}^{\psi }$ of a fixed Lie algebra by the ideal generated by the brackets $[x,x^{\psi }]$, for all $x$. We exhibit an abelian ideal of $\chi (\mathfrak{g})$ whose associated quotient is a subdirect sum in $\mathfrak{g}\oplus \mathfrak{g}\oplus \mathfrak{g}$ and we give conditions for this ideal to be finite dimensional. We show that $\chi (\mathfrak{g})$ has a subquotient that is isomorphic to the Schur multiplier of $\mathfrak{g}$. We prove that $\chi (\mathfrak{g})$ is finitely presentable or of homological type $F{P}_2$ if and only if $\mathfrak{g}$ has the same property, but $\chi (\mathfrak{f})$ is not of type $F{P}_3$ if $\mathfrak{f}$ is a non-abelian free Lie algebra.