Cohomology of Lie Semidirect Products and Poset Algebras

V. E. Coll Jr. ; M. Gerstenhaber

Journal of Lie theory, Tome 26 (2016) no. 1, pp. 79-95

Résumé

\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.

Détail

Classification : 17B56
Mots-clés : Lie algebra, cohomology, semidirect products, poset algebras