  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. In the last version we have corrected some
missprints, in particular the statement of Theorem D (from $n-1$-acyclicity to
$n-1 - |w|$-acyclicity).
