Álgebras de Lie, grupos de Lie e espaços girovetoriais de Lie

Abstract

The first objective in this work is to present in a organized and detailed way, the concepts of Lie theory. To reach this one, we did an extensive study about Lie groups and Lie algebras: we present the soluble, nilpotent, simple and semisimple Lie algebras; we give the Cartan criterions, which allow us to investigate the solubility and semisimplicity of these Lie algebras and, finally we introduce the definition of exponential map and homogenous manifolds. Our second objective is to establish criterions to define a structure of a Lie gyrovector space in a homogenous space. To reach this one, we did a study of gyrogroups through of loop theory, and using Lie theory, for an arbitrary section of the canonic map of the Lie group G on G=H, where H is a closed subgroup of G, we define an binary operation on the cosets. With this operation, we give conditions to obtain Lie left loops and from these, we obtain the Lie gyrovector spaces