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Total cohomology of solvable Lie algebras and linear deformations


Authors: Leandro Cagliero and Paulo Tirao
Journal: Trans. Amer. Math. Soc. 368 (2016), 3341-3358
MSC (2010): Primary 17B56; Secondary 17B30, 16S80
DOI: https://doi.org/10.1090/tran/6424
Published electronically: September 15, 2015
MathSciNet review: 3451879
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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.


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Additional Information

Leandro Cagliero
Affiliation: CIEM-FaMAF, Universidad Nacional de Córdoba, Argentina

Paulo Tirao
Affiliation: CIEM-FaMAF, Universidad Nacional de Córdoba, Argentina

DOI: https://doi.org/10.1090/tran/6424
Keywords: Lie algebra vanishing cohomology, total cohomology, linear deformations, nilshadow
Received by editor(s): December 2, 2012
Received by editor(s) in revised form: March 11, 2014
Published electronically: September 15, 2015
Article copyright: © Copyright 2015 American Mathematical Society