chapter  10
68 Pages

Lie Groups and Lie Algebras

WithL. S. Pontryagin

This chapter begins by introducing the structure constants of a Lie group. This system of numbers form a tensor, that is, transforms as do the components of a tensor under a change of coordinates in the group. The chapter examines the fundamental properties of structure constants comprising the content of the third theorem of Lie and the corresponding properties of the Lie algebra. The homomorphism is said to be an isomorphism if it is one-to-one and onto. In a local investigation, and the classical theory was local in nature, the functions studied are ordinarily not defined for all values of the variables under consideration, but in each case on one or another open set. Consequently, an accurate statement of results would require a series of provisos indicating in every case the domain of definition of the functions involved.