Total cohomology of solvable lie algebras and linear deformations
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 ded...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2016 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/58326 |
| Acceso en línea: | http://hdl.handle.net/11336/58326 |
| Access Level: | acceso abierto |
| Palabra clave: | LIE ALGEBRA VANISHING COHOMOLOGY LINEAR DEFORMATIONS NILSHADOW TOTAL COHOMOLOGY https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | 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. |
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