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Representations of Lie Algebras

Book Description

This bold and refreshing approach to Lie algebras assumes only modest prerequisites (linear algebra up to the Jordan canonical form and a basic familiarity with groups and rings), yet it reaches a major result in representation theory: the highest-weight classification of irreducible modules of the general linear Lie algebra. The author's exposition is focused on this goal rather than aiming at the widest generality and emphasis is placed on explicit calculations with bases and matrices. The book begins with a motivating chapter explaining the context and relevance of Lie algebras and their representations and concludes with a guide to further reading. Numerous examples and exercises with full solutions are included. Based on the author's own introductory course on Lie algebras, this book has been thoroughly road-tested by advanced undergraduate and beginning graduate students and it is also suited to individual readers wanting an introduction to this important area of mathematics.

Table of Contents

  1. Cover Page
  2. Representations of Lie Algebras
  3. AUSTRALIAN MATHEMATICAL SOCIETY LECTURE SERIES
  4. Title Page
  5. Copyright
  6. Contents
  7. Preface
  8. Notational conventions
  9. 1 Motivation: representations of Lie groups
    1. 1.1 Homomorphisms of general linear groups
    2. 1.2 Multilinear algebra
    3. 1.3 Linearization of the problem
    4. 1.4 Lie’s theorem
  10. 2 Definition of a Lie algebra
    1. 2.1 Definition and first examples
    2. 2.2 Classification and isomorphisms
    3. 2.3 Exercises
  11. 3 Basic structure of a Lie algebra
    1. 3.1 Lie subalgebras
    2. 3.2 Ideals
    3. 3.3 Quotients and simple Lie algebras
    4. 3.4 Exercises
  12. 4 Modules over a Lie algebra
    1. 4.1 Definition of a module
    2. 4.2 Isomorphism of modules
    3. 4.3 Submodules and irreducible modules
    4. 4.4 Complete reducibility
    5. 4.5 Exercises
  13. 5 The theory of sl2-modules
    1. 5.1 Classification of irreducibles
    2. 5.2 Complete reducibility
    3. 5.3 Exercises
  14. 6 General theory of modules
    1. 6.1 Duals and tensor products
    2. 6.2 Hom-spaces and bilinear forms
    3. 6.3 Schur’s lemma and the Killing form
    4. 6.4 Casimir operators
    5. 6.5 Exercises
  15. 7 Integral gln-modules
    1. 7.1 Integral weights
    2. 7.2 Highest-weight modules
    3. 7.3 Irreducibility of highest-weight modules
    4. 7.4 Tensor-product construction of irreducibles
    5. 7.5 Complete reducibility
    6. 7.6 Exercises
  16. 8 Guide to further reading
    1. 8.1 Classification of simple Lie algebras
    2. 8.2 Representations of simple Lie algebras
    3. 8.3 Characters and bases of representations
  17. Appendix Solutions to the exercises
    1. Solutions for Chapter 2 exercises
    2. Solutions for Chapter 3 exercises
    3. Solutions for Chapter 4 exercises
    4. Solutions for Chapter 5 exercises
    5. Solutions for Chapter 6 exercises
    6. Solutions for Chapter 7 exercises
  18. References
  19. Index