Group Theory

Birdtracks, Lie's, and Exceptional Groups

Predrag Cvitanović

• Publisher: Princeton University Press
• Year: 2008
• Print ISBN:
  9780691202983
  9780691118369
• eBook ISBN:
  9781400837670
• Topic: Mathematics
• Subtopic: Group Theory
• Index: Mathematics

If classical Lie groups preserve bilinear vector norms, what Lie groups preserve trilinear, quadrilinear, and higher order invariants? Answering this question from a fresh and original perspective, Predrag Cvitanovic takes the reader on the amazing, four-thousand-diagram journey through the theory of Lie groups. This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional.


The invariant tensors are presented in a somewhat unconventional, but in recent years widely used, "birdtracks" notation inspired by the Feynman diagrams of quantum field theory. Notably, invariant tensor diagrams replace algebraic reasoning in carrying out all group-theoretic computations. The diagrammatic approach is particularly effective in evaluating complicated coefficients and group weights, and revealing symmetries hidden by conventional algebraic or index notations. The book covers most topics needed in applications from this new perspective: permutations, Young projection operators, spinorial representations, Casimir operators, and Dynkin indices. Beyond this well-traveled territory, more exotic vistas open up, such as "negative dimensional" relations between various groups and their representations. The most intriguing result of classifying primitive invariants is the emergence of all exceptional Lie groups in a single family, and the attendant pattern of exceptional and classical Lie groups, the so-called Magic Triangle. Written in a lively and personable style, the book is aimed at researchers and graduate students in theoretical physics and mathematics.

Table of contents

• Cover
• Contents
• Acknowledgments
• Chapter 1. Introduction
• Chapter 2. A preview
• Chapter 3. Invariants and reducibility
• Chapter 4. Diagrammatic notation
• Chapter 5. Recouplings
• Chapter 6. Permutations
• Chapter 7. Casimir operators
• Chapter 8. Group integrals
• Chapter 9. Unitary groups
• Chapter 10. Orthogonal groups
• Chapter 11. Spinors
• Chapter 12. Symplectic groups
• Chapter 13. Negative dimensions
• Chapter 14. Spinors’ symplectic sisters
• Chapter 15. SU(n) family of invariance groups
• Chapter 16. G[sub(2)] family of invariance groups
• Chapter 17. E[sub(8)] family of invariance groups
• Chapter 18. E[sub(6)] family of invariance groups
• Chapter 19. F[sub(4)] family of invariance groups
• Chapter 20. E[sub(7)] family and its negative-dimensional cousins
• Chapter 21. Exceptional magic
• Appendix A. Recursive decomposition
• Appendix B. Properties of Young projections
• Bibliography
• Index