Lie Groups, Lie Algebras, and Representations: An Elementary Introduction

Description

This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject. In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including: a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebrasmotivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C)an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebrasa self-contained construction of the representations of compact groups, independent of Lie-algebraic arguments The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincaré–Birkhoff–Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula. Review of the first edition: This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition to the textbook literature ... it is highly recommended. — The Mathematical Gazette Front Matter....Pages i-xiii Front Matter....Pages 1-1 Matrix Lie Groups....Pages 3-30 The Matrix Exponential....Pages 31-48 Lie Algebras....Pages 49-76 Basic Representation Theory....Pages 77-107 The Baker–Campbell–Hausdorff Formula and Its Consequences....Pages 109-137 Front Matter....Pages 139-139 The Representations of \(\mathsf{sl}(3; \mathbb{C})\) ....Pages 141-167 Semisimple Lie Algebras....Pages 169-196 Root Systems....Pages 197-240 Representations of Semisimple Lie Algebras....Pages 241-264 Further Properties of the Representations....Pages 265-304 Front Matter....Pages 305-305 Compact Lie Groups and Maximal Tori....Pages 307-341 The Compact Group Approach to Representation Theory....Pages 343-370 Fundamental Groups of Compact Lie Groups....Pages 371-405 Erratum....Pages E1-E1 Back Matter....Pages 407-449