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 $H^*(\mathfrak{g},V)$ is an $L$-module, where $L$ is a fixed subgroup of the connected component of $\operatorname{Aut}(\mathfrak{g})$ 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 $TH^*(\mathfrak{g})=\bigoplus_{V\in \Gamma_\circ(\mathfrak{g})} H^*(\mathfrak{g},V)$ 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.