Skip to Main content Skip to Navigation
Theses

Structures de Poisson sur les Algèbres de Polynômes, Cohomologie et Déformations

Abstract : Deformation quantization and McKay correspondence form the main themes of the study which deals with singular algebraic varieties, quotients of polynomial algebras, and polynomial algebras invariant under the action of a finite group. Our main tools are Poisson and Hochschild cohomologies and representation theory. Certain calculations are made with Maple and GAP. We calculate Hochschild homology and cohomology spaces of Klein surfaces by developing a generalization of HKR theorem in the case of non-smooth varieties and use the multivariate division and the Groebner bases. The closure of the minimal nilpotent orbit of a simple Lie algebra is a singular algebraic variety : on this one we construct invariant star-products, with the help of the BGS decomposition of Hochschild homology and cohomology, and of results on the invariants of the classical groups. We give the generators of the Joseph ideal associated to this orbit and calculate the infinitesimal characters. For simple Lie algebras of type B, C, D, we establish general results on the Poisson homology space in degree 0 of the invariant algebra, which support Alev's conjecture, then we are interested in the ranks 2 and 3. We compute Poincaré series of 2 variables for the finite subgroups of the special linear group in dimension 3, show that they are rational fractions, and associate to the subgroups a generalized Cartan matrix in order to obtain a McKay correspondence in dimension 3. All the study comes from 4 papers.
Document type :
Theses
Complete list of metadatas

Cited literature [80 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-00444232
Contributor : Frédéric Butin <>
Submitted on : Wednesday, January 6, 2010 - 11:14:36 AM
Last modification on : Wednesday, July 8, 2020 - 12:43:13 PM
Long-term archiving on: : Thursday, June 17, 2010 - 10:22:13 PM

Identifiers

  • HAL Id : tel-00444232, version 1

Citation

F. Butin. Structures de Poisson sur les Algèbres de Polynômes, Cohomologie et Déformations. Mathématiques [math]. Université Claude Bernard - Lyon I, 2009. Français. ⟨tel-00444232⟩

Share

Metrics

Record views

390

Files downloads

1847