This self-contained text is an excellent introduction to Lie groups and their actions on manifolds. The authors start with an elementary discussion of matrix groups, followed by chapters devoted to the basic structure and representation theory of finite dimensinal Lie algebras. They then turn to global issues, demonstrating the key issue of the interplay between differential geometry and Lie theory. Special emphasis is placed on homogeneous spaces and invariant geometric structures. The last section of the book is dedicated to the structure theory of Lie groups. Particularly, they focus on maximal compact subgroups, dense subgroups, complex structures, and linearity.

This text is accessible to a broad range of mathematicians and graduate students; it will be useful both as a graduate textbook and as a research reference.

From the reviews:

“The monograph under review is an introduction to the structure theory and geometry of Lie groups accessible both to a broad range of mathematicians and to graduate students. … The book consists of twenty one chapters divided into five parts. It starts with an introduction which presents the structure of the book, contains various teaching suggestions and introduces some fundamental notation.” (Volodymyr Mazorchuk, Zentralblatt MATH, Vol. 1229, 2012)