Group Theory. Exceptional Lie groups as invariance groups

Cvitanovic P.

The ultimate birdtracker guide to exceptional Lie groups.

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Table of contents :
Introduction……Page 9
Basic concepts……Page 13
First example: SU(n)……Page 17
Second example: E6 family……Page 20
Groups……Page 23
Vector spaces……Page 24
Algebra……Page 25
Defining space, tensors, representations……Page 26
Invariants……Page 28
Algebra of invariants……Page 30
Invariance groups……Page 31
Projection operators……Page 32
Further invariants……Page 33
Birdtracks……Page 35
Clebsch-Gordan coefficients……Page 36
Zero- and one-dimensional subspaces……Page 39
Infinitesimal transformations……Page 40
Lie algebra……Page 44
Irrelevancy of clebsches……Page 46
Couplings and recouplings……Page 49
Wigner 3n-j coefficients……Page 52
Wigner-Eckart theorem……Page 53
Permutations in birdtracks……Page 57
Symmetrization……Page 58
Antisymmetrization……Page 60
Levi-Civita tensor……Page 61
Determinants……Page 63
Fully (anti)symmetric
tensors……Page 65
Young tableaux, Dynkin labels……Page 66
Casimir operators……Page 67
Dynkin labels……Page 68
Group integrals……Page 71
Examples of group integrals……Page 72
Two-index tensors……Page 73
Three-index tensors……Page 74
Definitions……Page 76
SU(n) Young
tableaux……Page 77
Reduction of direct products……Page 78
Young projection operators……Page 79
A dimension formula……Page 80
Dimension as the number of strand colorings……Page 81
Three- and four-index tensors……Page 82
3-j symbols……Page 83
Application of the negative dimension theorem……Page 85
A sum rule for 3-j’s……Page 86
Mixed two-index tensors……Page 87
Mixed defining adjoint tensors……Page 89
Two-index adjoint tensors……Page 91
Dynkin labels for SU(n)
representations……Page 92
Orthogonal groups……Page 93
Dynkin labels of SO(n)
representations……Page 94
Spinors……Page 97
Kahane algorithm……Page 98
Symplectic groups……Page 99
Two-index tensors……Page 100
Dynkin labels of Sp(n)
representations……Page 101
Negative dimensions……Page 103
SU(n) = SU(-n)……Page 105
SO(n) = Sp(-n)……Page 106
Spinsters……Page 109
Representations of SU(2)……Page 111
SU(4) – SO(6)
isomorphism……Page 113
G2 family of invariance groups……Page 115
Jacobi relation……Page 117
Alternativity and reduction of f-contractions……Page 118
Primitivity implies alternativity……Page 120
Casimirs for G2……Page 123
Hurwitz’s theorem……Page 124
Representations of G2……Page 126
E8 family of invariance groups……Page 127
Two-index tensors……Page 128
Decomposition of Sym3 A……Page 131
Decomposition of |??||??|-.16667em |??|*……Page 133
Generalized Young tableaux for E8……Page 135
Conjectures of Deligne……Page 136
Reduction of two-index tensors……Page 137
Reduction of antisymmetric 3-index tensors……Page 138
Springer’s construction of E6……Page 139
Two-index tensors……Page 141
Jordan algebra and F4(26)……Page 144
E7 family of invariance groups……Page 145
Magic triangle……Page 147
E6 and SU(3)……Page 151
Recursive decomposition……Page 153
Uniqueness of Young projection operators……Page 155
Normalization……Page 156
The dimension formula……Page 157
Literature……Page 159