
Journal of Lie Theory 26 (2016), No. 1, 079095 Copyright Heldermann Verlag 2016 Cohomology of Lie Semidirect Products and Poset Algebras Vincent E. Coll Jr. Dept. of Mathematics, Lehigh University, Bethlehem, PA 18015, U.S.A. vec208@lehigh.edu Murray Gerstenhaber Dept. of Mathematics, University of Pennsylvania, Philadelphia, PA 191046395, U.S.A. mgersten@math.upenn.edu [Abstractpdf] \def\g{{\frak g}} \def\h{{\frak h}} \def\k{{\frak k}} \def\dirs{\hbox{\hskip2pt$\mathrel{\vrule height 4.2 pt depth1pt} {\hskip 4pt \times}$}} When $\h$ is a toral subalgebra of a Lie algebra $\g$ over a field $\bf k$, and $M$ a $\g$module on which $\h$ also acts torally, the HochschildSerre filtration of the ChevalleyEilenberg cochain complex admits a stronger form than for an arbitrary subalgebra. For a semidirect product $\g = \h \dirs \bf k$ with $\h$ toral one has $H^*(\g, M)\cong \bigwedge\h^{\vee} \bigotimes H^*(\k,M)^{\h} = H^*(\h,{\bf k})\bigotimes H^*(\k,M)^{\h}$; 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. Keywords: Lie algebra, cohomology, semidirect products, poset algebras. MSC: 17B56 [ Fulltextpdf (547 KB)] for subscribers only. 