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.
vec208@lehigh.edu
Murray Gerstenhaber
Dept. of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, U.S.A.
mgersten@math.upenn.edu
[Abstract-pdf]
\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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