Chapter 9 - The Relationships between Lie Groups and Lie Algebras Explored

Abstract

This chapter introduces a number of concepts for Lie algebras, and investigates the relationships between these concepts for a linear Lie group and its corresponding real Lie algebra. In the special case of a real Lie algebra of matrices, the theorems show that there is an intimate connection between these concepts and the corresponding concepts for linear Lie groups. The homomorphic and isomorphic mappings of Lie algebras are presented. It is found that the structure of a linear Lie group is not completely determined by its corresponding real Lie algebra; two or more linear Lie groups that are not isomorphic can have isomorphic real Lie algebras. It is observed that the duality between matrix representations and modules, the existence of similarity transformations and the concepts of reducible, completely reducible and irreducible representations re-appear. The adjoint representations of Lie algebras and linear Lie groups are elaborated. The relationship of the concept for real Lie algebras to that of direct products of linear Lie groups is also analyzed.