Weak commutativity and nilpotency

Abstract: We continue the analysis of the weak commutativity construction for Lie algebras. This is the Lie algebra $\chi(\mathfrak{g})$ generated by two isomorphic copies $\mathfrak{g}$ and $\mathfrak{g}^{\psi}$ of a fixed Lie algebra, subject to the relations $[x,x^{\psi}]=0$ for all $x \in \mathfrak{g}$. In this article we study the ideal $L =L(\mathfrak{g})$ generated by $x-x^{\psi}$ for all $x \in \mathfrak{g}$. We obtain an (infinite) presentation for $L$ as a Lie algebra, and we show that in general it cannot be reduced to a finite one. With this in hand, we study the question of nilpotency. We show that if $\mathfrak{g}$ is nilpotent of class $c$, then $\chi(\mathfrak{g})$ is nilpotent of class at most $c+2$, and this bound can improved to $c+1$ if $\mathfrak{g}$ is $2$-generated or if $c$ is odd. We also obtain concrete descriptions of $L(\mathfrak{g})$ (and thus of $\chi(\mathfrak{g})$) if $\mathfrak{g}$ is free nilpotent of class $2$ or $3$. Finally, using methods of Gr\"obner-Shirshov bases we show that the abelian ideal $R(\mathfrak{g}) = [\mathfrak{g}, [L, \mathfrak{g}^{\psi}]]$ is infinite-dimensional if $\mathfrak{g}$ is free of rank at least $3$.