Lie theory is one of the most wide-reaching areas of mathematics:~from number theory and automorphic forms to the mathematical formulations of quantum physics, Lie groups and Lie algebras take center stage. Anthony Knapp's Lie Groups Beyond an Introduction, 2nd edition, is a beautiful introduction to this area of mathematics, appropriate for a variety different audiences.
Recall that a Lie group G is a topological group endowed with the structure of a smooth manifold, so that multiplication and inversion are smooth. The Lie algebra g of G is the algebra of left-invariant vector fields on G; equivalently, g is the tangent space at the identity element of G. Given these basic facts, the prerequisite mathematics for the book consist of introductory graduate courses in algebra, analysis, and differential topology.
A reader having little or no background in the subject is quickly brought up to speed with a 20-page introduction (a new feature of the second edition), which contains a wealth of concrete examples. Chapters I-III cover the basics of Lie algebras (Lie and Engel's theorems, the universal enveloping algebra, classification of root systems, Serre relations) and some of the analytic, topological, and geometric aspects of Lie group theory. This sets the stage for the rest of the book: chapters IV-VII cover the structure and finite-dimensional representation theory of compact and semisimple Lie groups. Chapters VIII-X are largely independent of one another, and cover integration theory and the beginnings of infinite-dimensional representations.
Knapp's books tend to be thick, and Lie Groups is not an exception. At over 800 pages, locating specific theorems, proofs, and formulas may seem like a daunting task. Fortunately, the book is well-organized with concise, focused introductions to each chapter, a very thorough index of notation, and appendices covering tensor algebras, Ado's theorem, the Levi decomposition, Lie's third theorem, and the Campbell-Baker-Hausdorff formula, plus 35 pages of data for simple Lie algebras. In addition, there are hints to the hundreds of exercises, and a section on historical notes.
Knapp's writing is clear, and he avoids excessive notation. The first few chapters comprise a standard introductory course in Lie theory, while numerous second courses could be taught out of the later chapters. Its breadth of coverage and extensive tables also make the book a valuable reference for researchers in representation theory. Lie Groups Beyond an Introduction will not replace the classics by (among others) Bourbaki, Humphreys, and Helgason, but Knapp's talent for exposition insures its future as a standard text in representation theory.
John Cullinan is Visiting Assistant Professor of Mathematics at Colby College.