Chapter 8 - The Role of Lie Algebras

Abstract

This chapter describes determination of local properties by the corresponding real Lie algebra. The matrix exponential function provides the link between a linear Lie group and its corresponding real Lie algebra. The multiplication properties of matrix exponential functions are more complicated than those of the exponential functions of real or complex numbers. It is possible to introduce the corresponding real Lie algebra in a very direct way by a combination of algebraic and geometric arguments. For the other linear Lie groups, the essential results are similar, but the arguments are rather longer and less direct. Every real Lie algebra is isomorphic to the real Lie algebra of some linear Lie group. The real Lie algebras that correspond to general linear Lie groups are analyzed. It is found that the exponential mapping provides a direct way of determining the real Lie algebras corresponding to a number of important linear Lie groups that does not require an explicit parameterization.