ABSTRACT
In the last section we studied topological groups; in particular, we could answer certain questions about matrix groups by considering topological aspects. On the other hand, we have not yet ventilated the question to what extent the strong techniques of calculus can be used to explore the structure of matrix groups. In the same way in which we defined a topological group as a group with a topological structure such that the group operations are continuous, we could abstractly define a Lie group as a group with a differentiable structure such that the group operations are differentiable. However, such an approach would presume the reader’s knowledge of the concept of a manifold and the rudiments of differential geometry which we do not want to suppose in this algebra text. Therefore, we limit our investigation to the concrete setting of matrix groups (which are actually the most important types of Lie groups) and avoid the introduction of the abstract concept of a Lie group. Nevertheless, this section will help to acquaint the reader with the flavor of Lie theory and prepare him for a more thorough study of Lie groups in subsequent courses.
