Cohomology of Lie Semidirect Products and Poset Algebras

摘要

When h is a toral subalgebra of a Lie algebra g over a field k, and M a g-module on which 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 g = h (sic) t with h toral one has H* (g, M) (sic) Lambda h(V) circle times H* (t, M)(h) = H* (h,k) circle times H* (t, M)(h); if, moreover, g is a Lie poset algebra, then H* (g, g), which controls the deformations of 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.