On the finiteness homological properties of some modules over metabelian Lie algebras

We characterise the modules $B$ of homological type $FP_m$ over a finitely generated Lie algebra $L$ such that $L$ is a split extension of an abelian ideal $A$ and an abelian subalgebra $Q$ and $A$ acts trivially on $B$. The characterisation is in terms of the invariant $\Delta$ introduced by R. Bryant and J. Groves and is a Lie algebra version of the still open generalised $FP_m$-Conjecture for metabelian groups. The case $m=1$ is treated separately as there the characterisation is proved without restrictions on the type of the extension.