Total cohomology of solvable Lie algebras and linear deformations

LEANDRO CAGLIERO; PAULO TIRAO

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2016 vol. 368 p. 3341 - 3358

0002-9947

Given a finite dimensional Lie algebra g, let Γ◦ (g) be theset of irreducible g-modules with non-vanishing cohomology. We provethat a g-module V belongs to Γ◦ (g) only if V is contained in the exterioralgebra of the solvable radical s of g, showing in particular that Γ◦ (g) isa finite set and we deduce that H ∗ (g, V ) is an L-module, where L is afixed subgroup of the connected component of Aut(g) which contains aLevi factor.We describe Γ◦ in some basic examples, including the Borel sub-algebras, and we also determine Γ◦ (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 T H ∗ (g) =∗V ∈Γ◦ (g) H (g, V ) and we develop further the theory of linear deforma-tions in order to prove that the total cohomology of a solvable Lie algebrais 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 leastdim(s/n)2 , n being the nilradical of s, that contains the nilshadow of sand such that all its points have the same total cohomology.