Clifford Algebras and Lie Theory

  • Eckhard Meinrenken

Table of contents

  1. Front Matter
    Pages I-XX
  2. Eckhard Meinrenken
    Pages 1-21
  3. Eckhard Meinrenken
    Pages 23-48
  4. Eckhard Meinrenken
    Pages 49-85
  5. Eckhard Meinrenken
    Pages 87-107
  6. Eckhard Meinrenken
    Pages 109-133
  7. Eckhard Meinrenken
    Pages 135-162
  8. Eckhard Meinrenken
    Pages 163-190
  9. Eckhard Meinrenken
    Pages 191-217
  10. Open image in new window as a geometric Dirac operator
    Eckhard Meinrenken
    Pages 219-229
  11. Eckhard Meinrenken
    Pages 231-248
  12. Eckhard Meinrenken
    Pages 249-274
  13. Back Matter
    Pages 275-321

About this book


This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–Witt theorem.

This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra.

Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.


Clifford algebras Dirac operators Lie algebras Lie groups Spinors

Authors and affiliations

  • Eckhard Meinrenken
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

Bibliographic information