Representations of Lie algebras of vector fields on algebraic varieties and supervarieties

• Ano de defesa: 2024
• Autor(a) principal: Rocha, Henrique de Oliveira
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: (Super)álgebras de Lie de campos vetoriais; Feixes de operadores diferenciais; Lie (super)algebras of vector fields; Representações de álgebras de Lie; Representations of Lie algebras; Sheaves of differential operators
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-12072024-142540/

This thesis is devoted to a study of the structure and representation theory of some infinite-dimensional Lie algebras and Lie superalgebras. The first family studied is the Lie algebras of vector fields on smooth affine algebraic varieties. After an exposition of the structure of such Lie algebras, we consider representations that admit a compatible action of the coordinate ring of the algebraic variety and are finitely generated as modules over this commutative algebra. We prove that these representations can be associated with a vector bundle that admits a compatible action of the tangent sheaf. We also prove that the action of the tangent sheaf is given by a differential operator. These results allow us to solve a conjecture made in the first papers of this theory. The second family considered is a supergeometry version of the previous. After an investigation of the smoothness of algebraic supervarieties, we prove that the global sections of the tangent sheaf of a smooth integral affine supervariety form a simple Lie superalgebra. Subsequently, we consider representations of this Lie superalgebra that admit a compatible action of global sections of the structure sheaf of the affine supervariety. Analogously to the non-super case, we show that the associated sheaf of modules admits a compatible action of the tangent sheaf when it is coherent. We also prove that this action is defined by a differential operator. Lastly, we study the weight modules with finite multiplicities over the map superalgebra associated with a basic Lie superalgebra. We prove that these representations are either cuspidal or parabolically induced from a cuspidal bounded module over a subalgebra of the map superalgebra. We also show that cuspidal bounded modules are evaluation modules.