Spelling suggestions: "subject:"poisson cohomology"" "subject:"poisson gohomology""
1 |
[en] A DEFORMATION OF POISSON STRUCTURE IN TORIC VARIETY AND COHOMOLOGICAL CONSIDERATIONS / [pt] UMA DEFORMAÇÃO DE ESTRUTURA POISSON EM VARIEDADE TÓRICA E CONSIDERAÇÕES COHOMOLÓGICASMARCELO SANTOS DA SILVA 13 July 2021 (has links)
[pt] O estudo de deformações e degenerações de estruturas de Poisson
ocupa posição especial dentro do marco clássico de análise de degenerações
de estruturas geométricas. Nesta tese como resultado principal construímos
uma deformação não trivial na qual a estrutura quadrática canônica do
espaço projetivo complexo n-dimensional é limite contínuo de estruturas
Kahlerianas. Além disso, como resultado segundário de estudos de
deformações mostramos que uma estrutura Poisson invariante numa
variedade tórica com número finito de folhas não pode ser exata na
cohomologia Poisson. Nosso estudo também inclui considerações sobre
cohomologia Poisson da estrutura quadrática canônica do espaço vetorial
complexo n-dimensional. / [en] The study of deformations and degenerations of Poisson structures
occupies a special position within the classical framework of analysis of
degenerations of geometric structures. In this thesis as the main result we
build a non-triavial deformation in which the canonical quadratic structure
in CP(n) is a continuous limit of Kahlerian structures. Furthermore, as a
secondary result of deformation studies we have shown that an invariant
Poisson structure in a toric variety with finite number of leaves cannot be
exact in Poisson cohomology. Our study also includes considerations about
Poisson cohomology of the canonical quadratic structure of C(n).
|
2 |
Structures de Poisson sur les Algèbres de Polynômes, Cohomologie et Déformations / Poisson Structures on Polynomial Algebras, Cohomology and DeformationsButin, Frédéric 13 November 2009 (has links)
La quantification par déformation et la correspondance de McKay forment les grands thèmes de l'étude qui porte sur des variétés algébriques singulières, des quotients d'algèbres de polynômes et des algèbres de polynômes invariants sous l'action d'un groupe fini. Nos principaux outils sont les cohomologies de Poisson et de Hochschild et la théorie des représentations. Certains calculs formels sont effectués avec Maple et GAP. Nous calculons les espaces d'homologie et de cohomologie de Hochschild des surfaces de Klein, en développant une généralisation du Théorème de HKR au cas de variétés non lisses et utilisons la division multivariée et les bases de Gröbner. La clôture de l'orbite nilpotente minimale d'une algèbre de Lie simple est une variété algébrique singulière sur laquelle nous construisons des star-produits invariants, grâce à la décomposition BGS de l'homologie et de la cohomologie de Hochschild, et à des résultats sur les invariants des groupes classiques. Nous explicitons les générateurs de l'idéal de Joseph associé à cette orbite et calculons les caractères infinitésimaux. Pour les algèbres de Lie simples B, C, D, nous établissons des résultats généraux sur l'espace d'homologie de Poisson en degré 0 de l'algèbre des invariants, qui vont dans le sens de la conjecture d'Alev et traitons les rangs 2 et 3. Nous calculons des séries de Poincaré à 2 variables pour des sous-groupes finis du groupe spécial linéaire en dimension 3, montrons que ce sont des fractions rationnelles, et associons aux sous-groupes une matrice de Cartan généralisée pour obtenir une correspondance de McKay algébrique en dimension 3. Toute l'étude a donné lieu à 4 articles / 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
|
Page generated in 0.0453 seconds