Group theory: Lie’s, track, and exceptional groups

Cvitanovic P.

Table of contents :
Basic concepts……Page 13
First example: SU(n)……Page 17
Second example: E6 family……Page 20
Preliminaries……Page 23
Invariants……Page 27
Invariance groups……Page 31
Projection operators……Page 32
Further invariants……Page 33
Birdtracks……Page 35
Clebsch-Gordan coefficients……Page 37
Zero- and one-dimensional subspaces……Page 39
Infinitesimal transformations……Page 40
Lie algebra……Page 43
Other forms of Lie algebra commutators……Page 45
Irrelevancy of clebsches……Page 46
A brief history of birdtracks……Page 47
Couplings and recouplings……Page 49
Wigner 3n-j coefficients……Page 52
Wigner-Eckart theorem……Page 53
Symmetrization……Page 57
Antisymmetrization……Page 59
Levi-Civita tensor……Page 61
Determinants……Page 63
Fully (anti)symmetric tensors……Page 65
Casimirs and Lie algebra……Page 68
Independent casimirs……Page 69
Casimir operators……Page 71
Dynkin indices……Page 72
Quadratic, cubic casimirs……Page 77
Quartic casimirs……Page 78
Sundry relations between quartic casimirs……Page 80
Dynkin labels……Page 83
Group integrals for arbitrary reps……Page 88
Characters……Page 90
Examples of group integrals……Page 91
Two-index tensors……Page 93
Three-index tensors……Page 94
Young tableaux……Page 95
Young projection operators……Page 98
Reduction of tensor products……Page 101
3-j symbols……Page 102
Mixed two-index tensors……Page 105
Mixed defining adjoint tensors……Page 107
Two-index adjoint tensors……Page 109
Casimirs for the fully symmetric reps of SU(n)……Page 113
SU(n), U(n) equivalence in adjoint rep……Page 115
Two-index tensors……Page 124
Mixed adjoint defining rep tensors……Page 125
Two-index adjoint tensors……Page 126
Three-index tensors……Page 129
Gravity tensors……Page 132
SO(n) Dynkin labels……Page 135
Spinography……Page 138
Fierzing around……Page 141
Fierz coefficients……Page 145
6-j coefficients……Page 146
Invariance of -matrices……Page 148
Handedness……Page 150
Kahane algorithm……Page 151
Two-index tensors……Page 154
SU(n) = SU(-n)……Page 158
SO(n) = Sp(-n)……Page 159
Spinsters……Page 161
Racah coefficients……Page 166
Heisenberg algebras……Page 167
Representations of SU(2)……Page 169
SU(3) as invariance group of a cubic invariant……Page 171
Levi-Civita tensors and SU(n)……Page 174
SU(4) – SO(6) isomorphism……Page 175
Alternativity and reduction of f-contractions……Page 179
Primitivity implies alternativity……Page 182
Sextonians……Page 184
Casimirs for G2……Page 186
Hurwitz’s theorem……Page 187
Representations of G2……Page 188
Two-index tensors……Page 190
Decomposition of Sym3 A……Page 194
Decomposition of a18.1x1box2 a18.1x1box.filled2……Page 196
Recent progress……Page 198
Reduction of two-index tensors……Page 201
Mixed two-index tensors……Page 202
Diophantine conditions and the E6 family……Page 204
Three-index tensors……Page 205
Defining adjoint tensors……Page 207
Two-index adjoint tensors……Page 210
Dynkin labels and Young tableaux for E6……Page 214
Casimirs for E6……Page 215
Subgroups of E6……Page 218
Springer relation……Page 219
Two-index tensors……Page 221
Defining adjoint tensors……Page 224
Two-index adjoint tensors……Page 226
Jordan algebra and F4(26)……Page 227
The antisymmetric quartic invariant……Page 230
Further Diophantine conditions……Page 232
Lie algebra identification……Page 233
Symmetric quartic invariant……Page 236
The extended supergravities and the magic triangle……Page 237
Magic triangle……Page 243
Epilogue……Page 246
E6 and SU(3)……Page 251
Uniqueness of Young projection operators……Page 255
Normalization……Page 256
The dimension formula……Page 257
Literature……Page 259
Evaluation rules for G2……Page 261
G2, further calculations……Page 263
Index……Page 267