Some Aspects of Elementary Lie Theory

In this work, we present some of the basic concepts and constructions in the theory of matrix Lie groups. For each matrix Lie group, we use the matrix exponential to construct a Lie algebra, and we then use the matrix exponential to show how different properties of the Lie group affect the Lie algebra and vice versa. In particular, we use the Baker–Campbell–Hausdorff formula to prove a one-to-one correspondence between the representations of a path-connected, simply connected matrix Lie group and the representations of its Lie algebra. The physically motivated groups SO(3) and SU(2) are used as a case study.