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 $\mathfrak{g}=\mathfrak{h}⋉\mathfrak{k}$ with $\mathfrak{h}$ toral one has $H^{\ast }(\mathfrak{g},M)\cong \bigwedge {\mathfrak{h}}^{\vee }⨂H^{\ast }(\mathfrak{k},M{)}^{\mathfrak{h}}=H^{\ast }(\mathfrak{h},\mathbf{k})⨂H^{\ast }(\mathfrak{k},M{)}^{\mathfrak{h}}$, and for a Lie poset algebra $\mathfrak{g}$, that $H^{\ast }(\mathfrak{g},\mathfrak{g})$, 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.

Revised, errors corrected, 16 pages

References in corpus (2)

• A note on the cohomology of Lie algebras
• Path algebras and de Broglie waves