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Complex Semisimple Lie Algebras

  • Book
  • © 2001

Overview

Authors:
  1. Jean-Pierre Serre

    1. Collège de France, Paris Cedex 05, France

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

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Table of contents (8 chapters)

  1. Front Matter

    Pages N1-ix
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  2. Nilpotent Lie Algebras and Solvable Lie Algebras

    • Jean-Pierre Serre
    Pages 1-4

  3. Semisimple Lie Algebras (General Theorems)

    • Jean-Pierre Serre
    Pages 5-9

  4. Cartan Subalgebras

    • Jean-Pierre Serre
    Pages 10-16

  5. The Algebra \(\mathfrak{s}\mathfrak{l}_2\) and Its Representations

    • Jean-Pierre Serre
    Pages 17-23

  6. Root Systems

    • Jean-Pierre Serre
    Pages 24-42

  7. Structure of Semisimple Lie Algebras

    • Jean-Pierre Serre
    Pages 43-55

  8. Linear Representations of Semisimple Lie Algebras

    • Jean-Pierre Serre
    Pages 56-65

  9. Complex Groups and Compact Groups

    • Jean-Pierre Serre
    Pages 66-71

  10. Back Matter

    Pages 72-75
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Keywords

About this book

These notes are a record of a course given in Algiers from 10th to 21st May, 1965. Their contents are as follows. The first two chapters are a summary, without proofs, of the general properties of nilpotent, solvable, and semisimple Lie algebras. These are well-known results, for which the reader can refer to, for example, Chapter I of Bourbaki or my Harvard notes. The theory of complex semisimple algebras occupies Chapters III and IV. The proofs of the main theorems are essentially complete; however, I have also found it useful to mention some complementary results without proof. These are indicated by an asterisk, and the proofs can be found in Bourbaki, Groupes et Algebres de Lie, Paris, Hermann, 1960-1975, Chapters IV-VIII. A final chapter shows, without proof, how to pass from Lie algebras to Lie groups (complex-and also compact). It is just an introduction, aimed at guiding the reader towards the topology of Lie groups and the theory of algebraic groups. I am happy to thank MM. Pierre Gigord and Daniel Lehmann, who wrote up a first draft of these notes, and also Mlle. Franl(oise Pecha who was responsible for the typing of the manuscript.

Reviews

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)

Authors and Affiliations

  • Collège de France, Paris Cedex 05, France

    Jean-Pierre Serre

Bibliographic Information

  • Book Title: Complex Semisimple Lie Algebras

  • Authors: Jean-Pierre Serre

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-642-56884-8

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2001

  • Hardcover ISBN: 978-3-540-67827-4Published: 12 December 2000

  • Softcover ISBN: 978-3-642-63222-8Published: 21 October 2012

  • eBook ISBN: 978-3-642-56884-8Published: 06 December 2012

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: IX, 75

  • Additional Information: Original French edition published by Benjamin, New York, 1966. Former English edition published as a monograph

  • Topics: Topological Groups, Lie Groups

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