Lie Groups and Lie Algebras

Abstract

The subject of Lie groups is one that slips by many a mathematician. Many claim that the topic is not accessible to undergraduate research. The book Lie Groups by Harriet Pollatsek came out a few years ago, and it was meant to be a new way to be introduced to the topic. However, the book does not quite get far enough to give a formal definition of a Lie group. The goal of this project is to bridge the gap. The objective of this thesis is to include all the introductory material required to get to where the definition of a Lie group is no longer something so complicated. We will illustrate the major concepts by examples. Many matrix groups are Lie groups. Matrix groups are well-known, and they are an ideal place to start learning about what a Lie group can do. We then look at tangent spaces of the matrix groups, or the Lie algebra that is associated with each Lie group. After some motivation behind Lie algebras, we finally get to the feature presentation: a group and a differentiable manifold, put together into one super structure known as a Lie group.