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 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.