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Lie Groups

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...

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Main Author: Bump, Daniel. (Author, http://id.loc.gov/vocabulary/relators/aut)
Corporate Author: SpringerLink (Online service)
Language:English
Published: New York, NY : Springer New York : Imprint: Springer, 2013.
Edition:2nd ed. 2013.
Series:Graduate Texts in Mathematics, 225
Subjects:
Topological groups.
Lie groups.
Topological Groups, Lie Groups.
Online Access:https://doi.org/10.1007/978-1-4614-8024-2
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.

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