Lie Groups

Abstract

This dissertation serves as an introduction to Lie group theory, particularly matrix Lie groups. We begin with an overview of matrix Lie groups, as well as some key examples of matrix Lie groups and how they’re constructed. The exponential of a matrix is then discussed. The fundamental properties of Lie groups and Lie algebras, as well as the exponential map’s relationship between Lie groups and their Lie algebras. We continue our dissertation by introducing simple connected groups and semisimple Lie algebras, and then concluding that compact Lie group classification can be reduced to simple Lie group classification. Keywords: Lie groups, Matrix Lie groups, Lie Algebras,Simple Lie groups, semisimple Lie Algebras.