© 2001

Complex Semisimple Lie Algebras


Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages N1-ix
  2. Jean-Pierre Serre
    Pages 5-9
  3. Jean-Pierre Serre
    Pages 10-16
  4. Jean-Pierre Serre
    Pages 24-42
  5. Jean-Pierre Serre
    Pages 43-55
  6. Jean-Pierre Serre
    Pages 66-71
  7. Back Matter
    Pages 72-75

About this book


These short notes, already well-known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers, including classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and linear representations. The last chapter discusses the connection between Lie algebras, complex groups and compact groups; it is intended to guide the reader towards further study.


Lie algebra Lie algebras Matrix Representation theory algebra group theory

Authors and affiliations

  1. 1.Collège de FranceParis Cedex 05France

Bibliographic information

  • Book Title Complex Semisimple Lie Algebras
  • Authors Jean-Pierre Serre
  • Series Title Springer Monographs in Mathematics
  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2001
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-3-540-67827-4
  • Softcover ISBN 978-3-642-63222-8
  • eBook ISBN 978-3-642-56884-8
  • Series ISSN 1439-7382
  • Series E-ISSN 2196-9922
  • Edition Number 1
  • Number of Pages IX, 75
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Additional Information Original French edition published by Benjamin, New York, 1966. Former English edition published as a monograph
  • Topics Topological Groups, Lie Groups
  • Buy this book on publisher's site


From the reviews of the French edition:

"...the book is intended for those who have an acquaintance with the basic parts of the theory, namely, with those general theorems on Lie algebras which do not depend on the notion of Cartan subalgebra. The author begins with a summary of these general theorems and then discusses in detail the structure and representation theory of complex semisimple Lie algebras. One recognizes here a skillful ordering of the material, many simplifications of classical arguments and a new theorem describing fundamental relations between canonical generators of semisimple Lie algebras. The classical theory being thus introduced in such modern form, the reader can quickly reach the essence of the theory through the present book." (Mathematical Reviews)