Non-finitely Presented Lie Algebras of Type $F{P}_2$

Abstract

Building on work of Kassel--Loday and Zelmanov--Zhang, we show that a Steinberg Lie algebra of the complex group algebra of a (suitable) Bestvina--Brady group is of type $F{P}_2$ and is not finitely presented. This also allows us to answer the (homological) $0$-$1$-$2$ question for Lie algebras.