Coabelian ideals in $mathbb{N}$-graded Lie algebras and applications to right angled Artin Lie algebras

We consider homological finiteness properties $FP_n$ of certain $mathbb{N}$-graded Lie algebras. After proving some general results, see Theorem A, Corollary B and Corollary C, we concentrate on a family that can be considered as the Lie algebra version of the generalized Bestvina-Brady groups associated to a graph $Gamma$. We prove that the homological finiteness properties of these Lie algebras can be determined in terms of the graph in the same way as in the group case.