Total cohomology of solvable Lie algebras and linear deformations

Abstract

Given a finite dimensional Lie algebra $\mathfrak{g}$, let ${\Gamma }_{\circ }(\mathfrak{g})$ be the set of irreducible $\mathfrak{g}$-modules with non-vanishing cohomology. We prove that a $\mathfrak{g}$-module $V$ belongs to ${\Gamma }_{\circ }(\mathfrak{g})$ only if $V$ is contained in the exterior algebra of the solvable radical $\mathfrak{s}$ of $\mathfrak{g}$, showing in particular that ${\Gamma }_{\circ }(\mathfrak{g})$ is a finite set and we deduce that $H^{\ast }(\mathfrak{g},V)$ is an $L$-module, where $L$ is a fixed subgroup of the connected component of $\mathrm{Aut}(\mathfrak{g})$ which contains a Levi factor. We describe ${\Gamma }_{\circ }$ in some basic examples, including the Borel subalgebras, and we also determine ${\Gamma }_{\circ }({\mathfrak{s}}_n)$ for an extension ${\mathfrak{s}}_n$ of the 2-dimensional abelian Lie algebra by the standard filiform Lie algebra ${\mathfrak{f}}_n$. To this end, we described the cohomology of ${\mathfrak{f}}_n$. We introduce the \emph{total cohomology} of a Lie algebra $\mathfrak{g}$, as $T{H}^{\ast }(\mathfrak{g})={⨁}_{V\in {\Gamma }_{\circ }(\mathfrak{g})}{H}^{\ast }(\mathfrak{g},V)$ and we develop further the theory of linear deformations in order to prove that the total cohomology of a solvable Lie algebra is the cohomology of its nilpotent shadow. Actually we prove that $\mathfrak{s}$ lies, in the variety of Lie algebras, in a linear subspace of dimension at least $\dim (\mathfrak{s}/\mathfrak{n}{)}^2$, $\mathfrak{n}$ being the nilradical of $\mathfrak{s}$, that contains the nilshadow of $\mathfrak{s}$ and such that all its points have the same total cohomology.

arXiv commentAccepted for publication in Trans. Amer. Math. Soc