Cohomology of Lie semidirect products and poset algebras

Abstract

When $\mathfrak h$ is a toral subalgebra of a Lie algebra $\mathfrak g$ over a field $\mathbf k$, and $M$ a $\mathfrak g$-module on which $\mathfrak h$ also acts torally, the Hochschild-Serre filtration of the Chevalley-Eilenberg cochain complex admits a stronger form than for an arbitrary subalgebra. For a semidirect product $\mathfrak g = \mathfrak h \ltimes \mathfrak k$ with $\mathfrak h$ toral one has $H^*(\mathfrak g, M) \cong \bigwedge\mathfrak h^{\vee} \bigotimes H^*(\mathfrak k,M)^{\mathfrak h} = H^*(\mathfrak h, \mathbf k)\bigotimes H^*(\mathfrak k,M)^{\mathfrak h}$, and for a Lie poset algebra $\mathfrak g$, that $H^*(\mathfrak g, \mathfrak g)$, which controls the deformations of $\mathfrak g$, can be computed from the nerve of the underlying poset. The deformation theory of Lie poset algebras, analogous to that of complex analytic manifolds for which it is a small model, is illustrated by examples.