Problema do isomorfismo para álgebras envolventes universais de álgebra de lie

Detalhes bibliográficos
Ano de defesa: 2014
Autor(a) principal: Danilo Vilela Avelar
Orientador(a): Não Informado pela instituição
Banca de defesa: Não Informado pela instituição
Tipo de documento: Dissertação
Tipo de acesso: Acesso aberto
Idioma: por
Instituição de defesa: Universidade Federal de Minas Gerais
UFMG
Programa de Pós-Graduação: Não Informado pela instituição
Departamento: Não Informado pela instituição
País: Não Informado pela instituição
Palavras-chave em Português:
Link de acesso: http://hdl.handle.net/1843/EABA-9MQMBJ
Resumo: Let L and H be two Lie algebras and UL and UH be their respective universal enveloping algebras. The isomorphism problem for universal enveloping algebras of Lie algebras asks us to determine under what conditions the isomorphism between UL and UH implies the isomorphism between L and H. In this dissertation, we will investigate this problem in the context of threedimensional simple Lie algebras and finite-dimensional nilpotent Lie algebras, based on the articles of Malcolmson [Mal92] and of Riley and Usefi [RU07]. In order to present a detailed proof of the main result of Malcolmson [Mal92], we will describe the class of quaternion algebras, based on the article [Lew06] of David W. Lewis and on masters dissertation [Sha08] of Zi Yang Sham. We will characterize the quaternion algebras over specific fields, in order to obtain a characterization of three-dimensional simple Lie algebras. Moreover, we will show that isomorphism between universal enveloping algebras of two three-dimensional simple Lie algebras occurs if and only if the Lie algebras are isomorphic. We will also show, based on the article [RU07], that the isomorphism type of the universal enveloping algebra of a finite dimensional Lie algebra determines whether or not the Lie algebra is nilpotent and, in case it is, it also determines the nilpotency class and the minimal number of generators of the Lie algebra.