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Lie Groups

  • Daniel Bump

Part of the Graduate Texts in Mathematics book series (GTM, volume 225)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Compact Groups

    1. Front Matter
      Pages 1-1
    2. Daniel Bump
      Pages 3-5
    3. Daniel Bump
      Pages 7-17
    4. Daniel Bump
      Pages 19-22
    5. Daniel Bump
      Pages 23-28
  3. Compact Lie Groups

    1. Front Matter
      Pages 29-29
    2. Daniel Bump
      Pages 39-43
    3. Daniel Bump
      Pages 45-49
    4. Daniel Bump
      Pages 51-55
    5. Daniel Bump
      Pages 57-60
    6. Daniel Bump
      Pages 61-66
    7. Daniel Bump
      Pages 67-70
    8. Daniel Bump
      Pages 81-91
    9. Daniel Bump
      Pages 93-99
    10. Daniel Bump
      Pages 101-108
    11. Daniel Bump
      Pages 109-121
    12. Daniel Bump
      Pages 123-128
    13. Daniel Bump
      Pages 129-144
    14. Daniel Bump
      Pages 145-155
    15. Daniel Bump
      Pages 157-167
    16. Daniel Bump
      Pages 169-175
    17. Daniel Bump
      Pages 177-190
    18. Daniel Bump
      Pages 191-201
  4. Noncompact Lie Groups

    1. Front Matter
      Pages 203-203
    2. Daniel Bump
      Pages 205-211
    3. Daniel Bump
      Pages 213-226
    4. Daniel Bump
      Pages 227-242
    5. Daniel Bump
      Pages 243-256
    6. Daniel Bump
      Pages 257-280
    7. Daniel Bump
      Pages 281-301
    8. Daniel Bump
      Pages 303-318
    9. Daniel Bump
      Pages 319-334
  5. Duality and Other Topics

    1. Front Matter
      Pages 335-335
    2. Daniel Bump
      Pages 337-347
    3. Daniel Bump
      Pages 365-377
    4. Daniel Bump
      Pages 387-393
    5. Daniel Bump
      Pages 395-406
    6. Daniel Bump
      Pages 407-417
    7. Daniel Bump
      Pages 427-435
    8. Daniel Bump
      Pages 437-444
    9. Daniel Bump
      Pages 445-454
    10. Daniel Bump
      Pages 455-460
    11. Daniel Bump
      Pages 461-469

About this book

Introduction

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.

Keywords

Frobenius-Schur duality Keating-Snaith formula Lie algebras Lie groups complex analytic groups conjugacy of maximal tori random matrix theory representation theory

Authors and affiliations

  • Daniel Bump
    • 1
  1. 1.Stanford University Department of MathematicsStanfordUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-8024-2
  • Copyright Information Springer Science+Business Media New York 2013
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-8023-5
  • Online ISBN 978-1-4614-8024-2
  • Series Print ISSN 0072-5285
  • Series Online ISSN 2197-5612
  • Buy this book on publisher's site