Lie Groups: (225 Graduate Texts in Mathematics)

This book is intended for a one-year graduate course on Lie groups and Lie algebras. The book goes beyond the representation theory of compact Lie groups, which is the basis of many texts, and provides a carefully chosen range of material to give the student the bigger picture. The book is organized to allow different paths through the material depending on one's interests. This second edition has substantial new material, including improved discussions of underlying principles, streamlining of some proofs, and many results and topics that were not in the first edition. For compact Lie groups, the book covers the Peter-Weyl theorem, Lie algebra, conjugacy of maximal tori, the Weyl group, roots and weights, Weyl character formula, the fundamental group and more. The book continues with the study of complex analytic groups and general noncompact Lie groups, covering the Bruhat decomposition, Coxeter groups, flag varieties, symmetric spaces, Satake diagrams, embeddings of Lie groups and spin. Other topics that are treated are symmetric function theory, the representation theory of the symmetric group, Frobenius-Schur duality and GL(n) × GL(m) duality with many applications including some in random matrix theory, branching rules, Toeplitz determinants, combinatorics of tableaux, Gelfand pairs, Hecke algebras, the "philosophy of cusp forms" and the cohomology of Grassmannians. An appendix introduces the reader to the use of Sage mathematical software for Lie group computations.

Table of Contents:
Part I: Compact Topological Groups.- 1 Haar Measure.- 2 Schur Orthogonality.- 3 Compact Operators.- 4 The Peter–Weyl Theorem.- Part II: Compact Lie Groups.- 5 Lie Subgroups of GL(n,C).- 6 Vector Fields.- 7 Left-Invariant Vector Fields.- 8 The Exponential Map.- 9 Tensors and Universal Properties.- 10 The Universal Enveloping Algebra.- 11 Extension of Scalars.- 12 Representations of sl(2,C).- 13 The Universal Cover.- 14 The Local Frobenius Theorem.- 15 Tori.- 16 Geodesics and Maximal Tori.- 17 The Weyl Integration Formula.- 18 The Root System.- 19 Examples of Root Systems.- 20 Abstract Weyl Groups.- 21 Highest Weight Vectors.- 22 The Weyl Character Formula.- 23 The Fundamental Group.- Part III: Noncompact Lie Groups.- 24 Complexification.- 25 Coxeter Groups.- 26 The Borel Subgroup.- 27 The Bruhat Decomposition.- 28 Symmetric Spaces.- 29 Relative Root Systems.- 30 Embeddings of Lie Groups.- 31 Spin.- Part IV: Duality and Other Topics.- 32 Mackey Theory.- 33 Characters of GL(n,C).- 34 Duality between Sk and GL(n,C).- 35 The Jacobi–Trudi Identity.- 36 Schur Polynomials and GL(n,C).- 37 Schur Polynomials and Sk.  38 The Cauchy Identity.- 39 Random Matrix Theory.- 40 Symmetric Group Branching Rules and Tableaux.- 41 Unitary Branching Rules and Tableaux.- 42 Minors of Toeplitz Matrices.- 43 The Involution Model for Sk.- 44 Some Symmetric Alegras.- 45 Gelfand Pairs.- 46 Hecke Algebras.- 47 The Philosophy of Cusp Forms.- 48 Cohomology of Grassmannians.- Appendix: Sage.- References.- Index.

About the Author :
Daniel Bump is Professor of Mathematics at Stanford University. His research is in automorphic forms, representation theory and number theory. He is a co-author of GNU Go, a computer program that plays the game of Go. His previous books include Automorphic Forms and Representations (Cambridge University Press 1997) and Algebraic Geometry (World Scientific 1998).

Review :
"This is the second edition of Bump's successful graduate textbook 'Lie Groups' that first appeared in 2004. ... this book is even more highly recommended than its first edition. It deserves a place on the shelves of every physicists, mathematicians and graduate or PhD students of physics or mathematics who are interested in Lie group theory." (Arpad Kurusa, Acta Scientiarum Mathematicarum, Vol. 82 (1-2), 2016) "This is a graduate math level text. Concise with lots of proofs. The chapters are short enough to read in one sitting. ... I was asked to look for books on this topic. It was challenging to search for material with this title. This was the best book that I could find. I look forward to exploring this topic further." (Mary Anne, Cats and Dogs with Data, maryannedata.com, April, 2014) "The book begins with a detailed explanation of the basic facts. ... It contains a discussion of very nontrivial modern applications of Lie group theory in other areas of mathematics. ... The text is very interesting and is superior to other textbooks on Lie group theory." (Dmitri Artamonov, zbMATH, Vol. 1279, 2014) "This second edition of a successful graduate textbook and reference is now divided in four parts. ... the book under review is the one every one of us must have on its desk or night table. ... this is a well-organized book with clear and well-established goals, taking the interested reader to the frontiers of today's research." (Felipe Zaldivar, MAA Reviews, December, 2013)