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

TL;DR

This paper investigates homological finiteness properties of $ $-graded Lie algebras, especially those related to right angled Artin Lie algebras, establishing graph-based criteria similar to the group case.

Contribution

It extends the understanding of homological finiteness properties to a class of $ $-graded Lie algebras, linking them to graph combinatorics and correcting previous results.

Findings

Homological finiteness properties are determined by graph structure.

Established criteria for $ $-graded Lie algebras analogous to group cases.

Corrected key theorem regarding acyclicity conditions.

Abstract

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).