Group Theory: Birdtracks, Lie's, and Exceptional Groups

✍ Scribed by Predrag Cvitanović

• Publisher: Princeton University Press
• Year: 2008
• Tongue: English
• Leaves: 277
• Edition: Course Book
• Category: Library

✦ Synopsis

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

Contents
Acknowledgments
Chapter One. Introduction
Chapter Two. A preview
Chapter Three. Invariants and reducibility
Chapter Four. Diagrammatic notation
Chapter Five. Recouplings
Chapter Six. Permutations
Chapter Seven. Casimir operators
Chapter Eight. Group integrals
Chapter Nine. Unitary groups
Chapter Ten. Orthogonal groups
Chapter Eleven. Spinors
Chapter Twelve. Symplectic groups
Chapter Thirteen. Negative dimensions
Chapter Fourteen. Spinors’ symplectic sisters
Chapter Fifteen. SU(n) family of invariance groups
Chapter Sixteen. G2 family of invariance groups
Chapter Seventeen. E8 family of invariance groups
Chapter Eighteen. E6 family of invariance groups
Chapter Nineteen. F4 family of invariance groups
Chapter Twenty. E7 family and its negative-dimensional cousins
Chapter Twenty-One. Exceptional magic
Appendix A. Recursive decomposition
Appendix B. Properties of Young projections
Bibliography
Index