Introduction to Lie Algebras and Representation Theory

  • James E. Humphreys

Part of the Graduate Texts in Mathematics book series (GTM, volume 9)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. James E. Humphreys
    Pages 1-14
  3. James E. Humphreys
    Pages 15-41
  4. James E. Humphreys
    Pages 42-72
  5. James E. Humphreys
    Pages 73-88
  6. James E. Humphreys
    Pages 89-106
  7. James E. Humphreys
    Pages 107-144
  8. James E. Humphreys
    Pages 145-164
  9. Back Matter
    Pages 165-177

About this book

Introduction

This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor­ porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.

Keywords

Lie algebra algebraic geometry automorphism field homomorphism lie algebra linear algebra matrix polynomial representation theory transformation

Authors and affiliations

  • James E. Humphreys
    • 1
  1. 1.University of MassachusettsAmherstUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-6398-2
  • Copyright Information Springer-Verlag New York 1972
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-90052-0
  • Online ISBN 978-1-4612-6398-2
  • Series Print ISSN 0072-5285
  • About this book