Representació de grups de Lie compactes i la seva aplicació en física de partícules

Abstract

[en] The main goal of this work is to introduce the notion of Lie groups and their representations. We start with a reminder of the basic concepts of differential geometry. Immediately after we introduce the concept of a Lie group and some of its most important related notions, namely its Lie algebra, the adjoint representation and the exponential map. Then the main results of representation theory are introduced, with a focus on the representations of compact Lie groups. Torus representations receive special consideration due to their later importance. We also present the notion of a representation of a Lie algebra, along with the weights and infinitesimal weights of a Lie group. Finally, we introduce maximal tori of compact connected Lie groups, together with the corresponding Weyl group. We show without proof that the weights of the adjoint representation of a compact connected Lie group form a root system. These concepts are exemplified for the Lie group SU (3) due to its importance in particle physics. This is the topic of the last section.