## Abstract

Given a finite-dimensional Lie algebra g, let Γ_{o}(g) be the set of irreducible g-modules with non-vanishing cohomology. We prove that a gmodule V belongs to Γ_{o}(g) only if V is contained in the exterior algebra of the solvable radical s of g, showing in particular that Γ_{o}(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 Γ_{o} in some basic examples, including the Borel subalgebras, and we also determine Γ_{o}(s_{n}) for an extension s_{n} of the 2-dimensional abelian Lie algebra by the standard filiform Lie algebra f_{n}. To this end, we described the cohomology of f_{n}. 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 language | English |
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Pages (from-to) | 3341-3358 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 368 |

Issue number | 5 |

DOIs | |

State | Published - May 2016 |

Externally published | Yes |

## Keywords

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