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Doctoral Thesis
DOI
Document
Author
Full name
Cristian Camilo Cárdenas Cárdenas
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2018
Supervisor
Committee
Struchiner, Ivan (President)
Bursztyn, Henrique
Fernandes, Rui Loja
Gonzalez, Cristian Andres Ortiz
Torres, David Francisco Martínez
Title in English
Deformation problems in Lie groupoids
Keywords in English
Deformations
Lie groupoid morphisms
Lie subgroupoids
Symplectic groupoids
Abstract in English
In this thesis we present the deformation theory of Lie groupoid morphisms, Lie subgroupoids and symplectic groupoids. The corresponding deformation complexes governing such deformations are defined and used to investigate a Moser argument in each of these contexts. We also apply this theory to the case of Lie group morphisms and Lie subgroups, obtaining rigidity results of these structures. Moreover, in the case of symplectic groupoids, we define a map between the differentiable and deformation cohomology of the underlying groupoid, which is regarded as the global counterpart of a map $i$ defined by Crainic and Moerdijk (2004) which relates the (Poisson) cohomology of the Poisson structure on the base $M$ of the groupoid to the deformation cohomology of the Lie algebroid $T^{*}M$ associated to it.
Title in Portuguese
Problemas de deformação em grupoides de Lie
Keywords in Portuguese
Deformações
Grupoides simpléticos
Morfismos de grupoides de Lie
Subgrupoides de Lie
Abstract in Portuguese
Nesta tese apresentamos a teoria de deformação de morfismos de grupoides de Lie, subgrupoides de Lie e grupoides simpléticos, definimos os correspondentes complexos de deformação que controlam as deformações destas estruturas, e usamos estes complexos para desenvolver o argumento de Moser em cada um destes contextos. Também aplicamos esta teoria ao caso de morfismos de grupos de Lie e subgrupos de Lie obtendo resultados de rigidez de tais estruturas. Ademais, no caso de grupoides simpléticos, definimos uma função entre a cohomologia diferenciável e a cohomologia de deformação do grupoide, que é interpretada como o análogo global da aplicação $i$ definida por Crainic e Moerdijk (2004) que relaciona a cohomologia de Poisson da estrutura de Poisson induzida na base $M$ do grupoide com a cohomologia de deformação do algebroide de Lie $T^{*}M$ associado à estrutura de Poisson.
 
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Publishing Date
2019-08-08
 
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