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  • © 2015

Lie Groups, Lie Algebras, and Representations

An Elementary Introduction

Authors:

  • New edition extensively revised and updated

  • Covers the core topics of Lie theory from an elementary point of view

  • Includes many new exercises

Part of the book series: Graduate Texts in Mathematics (GTM, volume 222)

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eBook USD 59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-13467-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 79.99
Price excludes VAT (USA)
Hardcover Book USD 79.99
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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xiii
  2. General Theory

    1. Front Matter

      Pages 1-1
    2. Matrix Lie Groups

      • Brian Hall
      Pages 3-30
    3. The Matrix Exponential

      • Brian Hall
      Pages 31-48
    4. Lie Algebras

      • Brian Hall
      Pages 49-76
    5. Basic Representation Theory

      • Brian C. Hall
      Pages 77-107
  3. Semisimple Lie Algebras

    1. Front Matter

      Pages 139-139
    2. Semisimple Lie Algebras

      • Brian Hall
      Pages 169-196
    3. Root Systems

      • Brian Hall
      Pages 197-240
  4. Compact Lie Groups

    1. Front Matter

      Pages 305-305
    2. Compact Lie Groups and Maximal Tori

      • Brian Hall
      Pages 307-341
    3. Fundamental Groups of Compact Lie Groups

      • Brian Hall
      Pages 371-405
  5. Erratum

    • Brian C. Hall
    Pages E1-E1
  6. Back Matter

    Pages 407-449

About this book

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 algebras
  • motivation 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 algebras
  • a 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

Keywords

  • Baker-Campbell-Hausdorff formula
  • Cartan-Weyl theory
  • Lie algebras
  • Lie groups
  • representation theory

Reviews

“The first edition of this book was very good; the second is even better, and more versatile. This text remains one of the most attractive sources available from which to learn elementary Lie group theory, and is highly recommended.” (Mark Hunacek, The Mathematical Gazette, Vol. 101 (551), July, 2017)

Authors and Affiliations

  • Department of Mathematics, University of Notre Dame, Notre Dame, USA

    Brian C. Hall

About the author

Brian Hall is Professor of Mathematics at the University of Notre Dame, IN.

Bibliographic Information

Buying options

eBook USD 59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-13467-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 79.99
Price excludes VAT (USA)
Hardcover Book USD 79.99
Price excludes VAT (USA)