Total cohomology of solvable lie algebras and linear deformations

Leandro Cagliero, Paulo Tirao

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Given a finite-dimensional Lie algebra g, let Γo(g) be the set of irreducible g-modules with non-vanishing cohomology. We prove that a gmodule V belongs to Γo(g) only if V is contained in the exterior algebra of the solvable radical s of g, showing in particular that Γo(g) is a finite set and we deduce that H(g, V) is an L-module, where L is a fixed subgroup of the connected component of Aut(g) which contains a Levi factor. We describe Γo in some basic examples, including the Borel subalgebras, and we also determine Γo(sn) for an extension sn of the 2-dimensional abelian Lie algebra by the standard filiform Lie algebra fn. To this end, we described the cohomology of fn. We introduce the total cohomology of a Lie algebra g, as (formula presented) 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 s lies, in the variety of Lie algebras, in a linear subspace of dimension at least dim(s/n)2, n being the nilradical of s, that contains the nilshadow of s and such that all its points have the same total cohomology.

Original languageEnglish
Pages (from-to)3341-3358
Number of pages18
JournalTransactions of the American Mathematical Society
Issue number5
StatePublished - May 2016
Externally publishedYes


  • Lie algebra vanishing cohomology
  • Linear deformations
  • Nilshadow
  • Total cohomology


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