Introduction to Lie Groups and Lie Algebras (Pure & Applied Mathematics)

PREFACE: "This text is intended for the beginning graduate student with minimal preparation. However since Lie groups abstract the analytic properties of matrix groups, the student is expected to have some knowledge of senior level algebra, topology, and analysis as given in some of the references. We ahve not attempted to prove all basic results so the serious student should take the indicated detours to such text as those by Freudenthal and de Vries, Helgason, Jacobson, or Wolf. In particular the student must develop his own taste in this subject and ours is only one point of view. We shall present many familiar concepts of differential calculus in the terminology of linear algebra. Thus, for functions from one Euclidean space to another, derivatives are given as linear transformations, higher order derivatives are given as multilinear forms, and Taylor's series is presented in this terminology. Instead of giving detailed proofs we present many examples involving matrix groups which will be abstracted in later chapters.