Classificação das álgebras de Lie semi-simples complexas e reais

Abstract

In this work, we present the classification of real and complex semisimple Lie algebras. Actually, we only introduce the real case, while we show the complete classification in the complex setting. In order to achieve this aim, we present basic concepts, passing through category theory, a little of modules over rings theory, homological algebra and the basic theory of Lie algebras. Afterwards, we summarize the theory of semisimple Lie algebras, that constitute the main topic of the present work. After this, we introduce the cohomology of Lie algebras, in order to prove two important theorems (Weyl’s Theorem and Levi-Malcev Theorem), which help us to obtain the classification, as well motivate us to do so. Finally, we show the classification of complex semisimple Lie algebras and we apply it to realize the analogous classification in the real framework.