Lie groups beyond an introduction. 2nd ed.

*(English)*Zbl 1075.22501
Progress in Mathematics (Boston, Mass.) 140. Boston, MA: Birkhäuser (ISBN 0-8176-4259-5/hbk). xviii, 812 p. (2002).

From the preface: The publication of a second edition is an opportunity to underscore that the subject of Lie groups is important both for its general theory and for its examples. To this end I have added material at both the beginning and the end of the first edition (1996; Zbl 0862.22006).

At the beginning is now an introduction, directly developing some of the elementary theory just for matrix groups, so that the reader at once has a large stock of concrete and useful examples. In addition, the part of Chapter I summarizing the full elementary theory of Lie groups has been expanded to provide greater flexibility where one begins to study the subject. The goal has been to include a long enough summary of the elementary theory so that a reader can proceed in the subject confidently with or without prior knowledge of the detailed foundations of Lie theory.

At the end are two new chapters, IX and X. Partly these explore specific examples and carry the theory closer to some of its applications, especially infinite-dimensional representation theory. Chapter IX is largely about branching theorems, which also have applications to mathematical physics and which relate compact groups to the structure theory of noncompact groups. Chapter X is largely about actions of compact Lie groups on polynomial algebras. It points toward invariant theory and some routes to infinite-dimensional representation theory.

The reader’s attention is drawn to the historical notes near the end of the book. These notes often put the content of the text in a wider perspective; they supply certain details that have been omitted in the text and they try to anticipate and answer questions that the reader might ask.

At the beginning is now an introduction, directly developing some of the elementary theory just for matrix groups, so that the reader at once has a large stock of concrete and useful examples. In addition, the part of Chapter I summarizing the full elementary theory of Lie groups has been expanded to provide greater flexibility where one begins to study the subject. The goal has been to include a long enough summary of the elementary theory so that a reader can proceed in the subject confidently with or without prior knowledge of the detailed foundations of Lie theory.

At the end are two new chapters, IX and X. Partly these explore specific examples and carry the theory closer to some of its applications, especially infinite-dimensional representation theory. Chapter IX is largely about branching theorems, which also have applications to mathematical physics and which relate compact groups to the structure theory of noncompact groups. Chapter X is largely about actions of compact Lie groups on polynomial algebras. It points toward invariant theory and some routes to infinite-dimensional representation theory.

The reader’s attention is drawn to the historical notes near the end of the book. These notes often put the content of the text in a wider perspective; they supply certain details that have been omitted in the text and they try to anticipate and answer questions that the reader might ask.