The Concept Of A Lie Group
The concept of a Lie group, contains in its very definition the requirement of the analyticity, or at the very least, the differentiability of certain functions, viz., the functions determining the operation of group multiplication. The analytic apparatus permits the reduction of the study of Lie groups to certain quite delicate but elementary questions of an algebraic character, these questions being, in essence, certain special problems in the theory of matrices. The classical theory of Lie groups is concerned for the most part with the study of local groups. A local group G is a local Lie group if it is possible to introduce differentiable coordinates in G. A topological group G is a Lie group if it is separable and if it is a local Lie group considered as a local group.