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.
Lie Groups Beyond an Introduction
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 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 pages, locating specific theorems, proofs, and formulas may seem like a daunting task.
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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.
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Knapp's writing is clear, and he avoids excessive notation. Matrix Groups. An Introduction To Lie Groups. Lie groups. An introduction through linear groups. Lie groups, physics, and geometry: an introduction.
Analysis on Lie Groups: An introduction. An introduction to Lie groups and Lie algebras. Lie groups, Lie algebras, and representations: an elementary introduction. Introduction to Lie groups and Lie algebras. Introduction to Lie groups and transformation groups.
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