Lie Groups, Lie Algebras, and Representations

An Elementary Introduction

  • Brian C. Hall

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

Table of contents

  1. Front Matter
    Pages i-xiii
  2. General Theory

    1. Front Matter
      Pages 1-1
    2. Brian C. Hall
      Pages 3-30
    3. Brian C. Hall
      Pages 31-48
    4. Brian C. Hall
      Pages 49-75
    5. Brian C. Hall
      Pages 77-107
  3. Semisimple Lie Algebras

    1. Front Matter
      Pages 139-139
    2. Brian C. Hall
      Pages 169-196
    3. Brian C. Hall
      Pages 197-240
    4. Brian C. Hall
      Pages 241-265
    5. Brian C. Hall
      Pages 265-304
  4. Compact Lie Groups

    1. Front Matter
      Pages 305-305
    2. Brian C. Hall
      Pages 307-341
    3. Brian C. Hall
      Pages 371-405
  5. 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


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

Authors and affiliations

  • Brian C. Hall
    • 1
  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA

Bibliographic information