Deformaciones de álgebras de Lie nilpotentes filiformes

Abstract

Michele Vergne inició el estudio de la geometría de la variedad algebraica de todas las álgebras o corchetes de Lie nilpotentes mostrando el rol distintivo de las álgebras de Lie nilpotentes filiformes, aquéllas de nilíndice máximo. Un concepto fundamental en este marco, es el de rigidez. Un corchete de Lie m se dice rígida si todas las álgebras de Lie en algún entorno de m, son isomorfas a m. Esta tesis está motivada por el problema conocido como Conjetura de Vergne, que afirma que ningún corchete de Lie nilpotente es rígido. Nosotros probamos que no existen álgebras de Lie filiformes complejas rígidas en la variedad de álgebras de Lie (filiformes) de dimensión menor o igual a 11. Más precisamente, mostramos que en cualquier entorno euclideo de un corchete de Lie filiforme hay un corchete de Lie filiforme no isomorfo. Este resultado se obtiene construyendo deformaciones lineales no triviales en un conjunto de abiertos densos de la variedad de álgebras de Lie filiformes de dimensión menor o igual a 11.

Michele Vergne started the study of the geometry of the algebraic variety of all nilpotente Lie algebras or brackets showing the distinctive role of the filiform nilpotent algebras, those of nilindice maximal. A fundamental concept in this frame is that of rigidity. A Lie algebra μ is said to be rigid if all other Lie algebra in a μ neighborhood are isomorphic to μ. This thesis is motivated by the problem known as Vergne’s Conjecture, open from 1970, about wich very little is know up to date, that states that no nilpotent Lie bracket is ever rigid. A linear deformation of a Lie bracket μ is a family of Lie brackets μ t , of the form μ t = μ + tφ where φ is Lie bracket which is a 2-cocycle of μ. If for all small t, μ t is not isomorphic to μ, then the deformations is non trivial and μ is not rigid. In this thesis we approached the problem of rigidity of complex filiform Lie algebras. First, we present a general method for constructing linear deformations of Lie algebras that adapt very well and result effective in the case of filiform algebras. Using this method we constructed linear deformations for any filiform. For dimension ≤ 11, in which it is possible to describe the variety of filiforms in an accessible manner, we showed that the deformations constructed are non trivial in a open dense, to deduce later the main result of the thesis: Theorem. There are not complex filiform Lie algebras of dimension ≤ 11. To prove this result we resorted to some tools of algebraic geometry and in particular to the decomposition in irreducible components of the considered varieties. This point is the main difficulty that we encountered to be able to advance in larger dimensions and in the general case.