Lie Groups, Lie Algebras and Representation Theory

Abstract

In Chapter 1, the concepts of a Lie group and a matrix Lie group are introduced, and we construct and study the group homomorphism SU(2) → SO(3). In Chapter 2 we de ne the notion of a matrix power series, and we do so in a general setting, which allows to deduce properties of which the matrix case is a particular case. The idea of a matrix exponential and matrix logarithm is also de ned, and we give proofs of their most important properties. To close the chapter, we give the general statement and the proof of the di erentiability of a matrix power series. In Chapter 3, we start with the concepts of an abstract Lie algebra and the Lie algebra of a matrix Lie group, providing the reader with examples of the Lie algebras of the typical matrix Lie groups. Then, we present the idea of a category and a functor, with lots of examples which intend to hint at the value of category theory as a unifying language in mathematics. We do this because in the following section, we prove that there exists a functor from the category of matrix Lie groups into the category of real Lie algebras, which condenses the relation between these two families of mathematical objects. After de ning the concept of the complexi cation of a real Lie algebra, we prove the main general results for matrix Lie groups, and in particular, that matrix Lie groups are embedded submanifolds of GL(푛; C). In the beginning of Chapter 4, we show that representations and actions, of a group or a Lie algebra, are just two sides of the same coin. We later explain how to understand the class of representations of a Lie group or a Lie algebra as a category, and show that the representations of a matrix Lie group can be related with those of its Lie algebra through a functor. After that, we proceed to classify the nite-dimensional irreducible representations of the Lie algebra of SU(2), and from that, we determine which nite-dimensional irreducible representations of the Lie algebra of SO(3) come from representations of SO(3) itself. In the nal Chapter 5, we state the Lie group–Lie algebra correspondence, and we use it to show that the category of the nite-dimensional representations of any simply connected matrix Lie group is isomorphic to the category of the nite-dimensional representations of its Lie algebra.