Homological finiteness properties of Lie algebras

We apply the main ideas behind the group theoretic methods developed in [P.H. Kropholler, Bull. London Math. Soc. 25 (1993) 558–566; J. Pure Appl. Math. 90 (1993) 55–67] to study Lie algebras of type FP∞. We show that every soluble Lie algebra of type FP∞ is finite dimensional. Some refinements of this result, when the algebra is abelian-by-finite dimensional and only type FPm is assumed, are obtained. It is also shown, using the complete cohomology of Vogel and Mislin, that for a wide class of Lie algebras, including all countable soluble ones, FP∞ implies finite cohomological dimension.