Chapter 3 - Lie Groups

Abstract

This chapter discusses the definition and the properties of a Lie group. A Lie group embodies three different forms of mathematical structure. The elements of the group also form a topological space, so that it may be described as being a special case of a topological group. Every Lie group that is important in physical problems is of a type known as a linear Lie group, for which a relatively straightforward definition can be given. The basic feature of any Lie group is that it has a noncountable number of elements lying in a region near its identity, and that the structure of this region determines the structure of the entire group and is itself determined by its corresponding real Lie algebra. All the Lie groups of physical interest are linear, in the sense that they have at least one faithful finite-dimensional representation. This representation can be used to provide the necessary precise formulation of distance and to ensure that all the other topological requirements are automatically observed. In generalizing to a connected linear Lie group, it is natural to make the hypothesis that the sum can be replaced by an integral with respect to the parameters. The significance of the distinction between compact and noncompact Lie groups is elaborated.