文档介绍：GroupTheory February 11, 2004
Group Theory
Tracks, Lie’s, and Exceptional Groups
Predrag Cvitanovic´
GONE WITH THE WIND PRESS
ATLANTA AND COPENHAGEN
GroupTheory February 11, 2004
dedicated to the memory of
Boris Weisfeiler and William E. Caswell
GroupTheory February 11, 2004
Contents
Chapter 1. Introduction 1
chapterChapter 2. A preview5
Basic concepts 5
First example: SU(n) 9
Second example: E6 family 12
Chapter 3. Invariants and reducibility 15
Defining space, tensors, reps 18
Invariants 20
Invariance groups 23
Projection operators 24
Further invariants 26
Chapter 4. Diagrammatic notation 29
Clebsch-Gordan coefficients 31
Zero- and one-dimensional subspaces 33
Infinitesimal transformations 34
Lie algebra 37
Other forms of Lie mutators 39
Irrelevancy of clebsches 40
A brief history of birdtracks 41
Chapter 5. Recouplings 43
and
Wigner 3n-j coefficients 46
Wigner-Eckart theorem 47
Chapter 6. Permutations 51
Antisymmetrization 53
Determinants 57
Characteristic equations 59
Fully (anti)symmetric tensors 59
Chapter 7. Casimir operators 61
and Lie
GroupTheory February 11, 2004
ii CONTENTS
Independent casimirs 63
Adjoint rep casimirs 65
Casimir operators 65
Dynkin indices 67
Quadratic, cubic casimirs 71
Quartic casimirs 72
Sundry relations between quartic casimirs 73
Identically vanishing tensors 77
Dynkin labels 77
Chapter 8. Group integrals 81
integrals for arbitrary
Characters 84
Examples of group integrals 85
Chapter 9. Unitary groups 87
-index
Three-index tensors 88
Young