The Classical Groups and K-Theory

  • Alexander J. Hahn
  • O. Timothy O’Meara

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 291)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Alexander J. Hahn, O. Timothy O’Meara
    Pages 1-2
  3. Alexander J. Hahn, O. Timothy O’Meara
    Pages 3-4
  4. Alexander J. Hahn, O. Timothy O’Meara
    Pages 5-67
  5. Alexander J. Hahn, O. Timothy O’Meara
    Pages 68-95
  6. Alexander J. Hahn, O. Timothy O’Meara
    Pages 96-138
  7. Alexander J. Hahn, O. Timothy O’Meara
    Pages 139-182
  8. Alexander J. Hahn, O. Timothy O’Meara
    Pages 183-291
  9. Alexander J. Hahn, O. Timothy O’Meara
    Pages 292-380
  10. Alexander J. Hahn, O. Timothy O’Meara
    Pages 381-440
  11. Alexander J. Hahn, O. Timothy O’Meara
    Pages 441-507
  12. Alexander J. Hahn, O. Timothy O’Meara
    Pages 508-542
  13. Alexander J. Hahn, O. Timothy O’Meara
    Pages 543-544
  14. Back Matter
    Pages 545-578

About this book

Introduction

It is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invariant theory, which his famous book helped to revive. Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of "simplicity" was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for the exceptional Lie algebras G and E • However, a deep reason for this 2 6 parallelism was missing, and it is only Chevalley who, in 1955 and 1961, discovered that to each complex simple Lie algebra corresponds, by a uniform process, a group scheme (fj over the ring Z of integers, from which, for any field K, could be derived a group (fj(K).

Keywords

K-theory algebra clifford algebra commutative ring

Authors and affiliations

  • Alexander J. Hahn
    • 1
  • O. Timothy O’Meara
    • 2
  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA
  2. 2.University of Notre DameNotre DameUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-13152-7
  • Copyright Information Springer-Verlag Berlin Heidelberg 1989
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-05737-3
  • Online ISBN 978-3-662-13152-7
  • Series Print ISSN 0072-7830
  • About this book