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Doctoral Thesis
DOI
https://doi.org/10.11606/T.45.2020.tde-09062020-192819
Document
Author
Full name
André Eduardo Zaidan
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2020
Supervisor
Committee
Futorny, Vyacheslav (President)
Billig, Yuly
Iusenko, Kostiantyn
Jardim, Marcos Benevenuto
Rodriguez, Daniel Nelson Panario
Title in Portuguese
Reduções na teoria de representações das álgebras de Lie de campos vetoriais
Keywords in Portuguese
Álgebra de Lie de campos vetoriais
Módulos de gauge
Módulos de Rudakov
Abstract in Portuguese
Nós estudamos representações para álgebras de Lie que não possuem uma subálgebra de Cartan. O estudo de tais representações requer novas técnicas, a que nós aplicamos consiste em restringir a ação de outras estruturas algébricas que contenham a álgebra de Lie. Nossas álgebras de Lie são obtidas a partir de campos vetoriais de variedades arbitrárias. Nós estudamos representações que admitem a ação da álgebra de Lie de campos vetoriais e a ação da álgebra de funções na variedade de uma maneira compatível. Mais especificamente, estudamos duas classes de tais módulos: módulos de gauge e módulos de Rudakov. Nós provamos que módulos de gauge e módulos de Rudakov com o correspondente gl_N-módulo simples continuam irredutíveis como módulos sobre a álgebra de Lie de campos vetoriais, a menos que o módulos apareça no complexo de de Rham. Nós também estudamos irreducibilidade do produto tensorial de módulos de Rudakov. Por fim, nós apresentamos uma descrição de módulos tensoriais que pertencem ao complexo de de Rham como gl_3-módulos. Nós também realizamos tais módulos via tabelas GT.
Title in English
Reductions in representation theory of Lie algebras of vector fields
Keywords in English
Gauge modules
Lie algebra of vector fields
Rudakov modules
Abstract in English
We study representations of Lie algebras that do not have a Cartan subalgebra. The study of such representations required new techniques, one that we applied was to restrict the action of other algebraic structures that contain the Lie algebra. Our Lie algebras came from the vector fields on arbitrary varieties. We studied representations that admit the actions of the Lie algebra of vector field and the algebra of functions on the variety in a compatible way. More specifically, we studied two such classes of modules: gauge modules and Rudakov modules. We proved that gauge modules and Rudakov modules corresponding to simple gl_N-modules remain irreducible as modules over the Lie algebra of vector fields unless they appear in the de Rham complex. We also studied the irreducibility of tensor products of Rudakov modules. Lastly, we present a complete description of tensor modules belonging to the de Rham complex as gl_3-modules. We also realize these modules using GT-tableaux.
 
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Publishing Date
2021-01-20
 
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