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Journal of Lie Theory 26 (2016), No. 1, 079--095
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.

Murray Gerstenhaber
Dept. of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, U.S.A.


\def\g{{\frak g}} \def\h{{\frak h}} \def\k{{\frak k}} \def\dirs{\hbox{\hskip2pt$\mathrel{\vrule height 4.2 pt depth-1pt} {\hskip -4pt \times}$}} 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 Hochschild-Serre filtration of the Chevalley-Eilenberg 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

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