نظریه گروه. گروه‌های لی استثنایی به عنوان گروه‌های invariance

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 Introduction 9 A preview 13 Basic concepts 13 First example: SU(n) 17 Second example: E6 family 20 Invariants and reducibility 23 Preliminaries 23 Groups 23 Vector spaces 24 Algebra 25 Defining space, tensors, representations 26 Invariants 28 Algebra of invariants 30 Invariance groups 31 Projection operators 32 Further invariants 33 Birdtracks 35 Clebsch-Gordan coefficients 36 Zero- and one-dimensional subspaces 39 Infinitesimal transformations 40 Lie algebra 44 Other forms of Lie algebra commutators 46 Irrelevancy of clebsches 46 Recouplings 49 Couplings and recouplings 49 Wigner 3n-j coefficients 52 Wigner-Eckart theorem 53 Permutations 57 Permutations in birdtracks 57 Symmetrization 58 Antisymmetrization 60 Levi-Civita tensor 61 Determinants 63 Characteristic equations 65 Fully (anti)symmetric tensors 65 Young tableaux, Dynkin labels 66 Casimir operators 67 Casimirs and Lie algebra 68 Independent casimirs 68 Casimir operators 68 Dynkin indices 68 Quadratic, cubic casimirs 68 Quartic casimirs 68 Sundry relations between quartic casimirs 68 Identically vanishing tensors 68 Dynkin labels 68 Group integrals 71 Group integrals for arbitrary representations 72 Characters 72 Examples of group integrals 72 Unitary groups 73 Two-index tensors 73 Three-index tensors 74 Young tableaux 76 Definitions 76 SU(n) Young tableaux 77 Reduction of direct products 78 Young projection operators 79 A dimension formula 80 Dimension as the number of strand colorings 81 Reduction of tensor products 82 Three- and four-index tensors 82 Basis vectors 83 3-j symbols 83 Evaluation by direct expansion 85 Application of the negative dimension theorem 85 A sum rule for 3-j's 86 Characters 87 Mixed two-index tensors 87 Mixed defining adjoint tensors 89 Two-index adjoint tensors 91 Casimirs for the fully symmetric representations of SU(n) 92 SU(n), U(n) equivalence in adjoint representation 92 Dynkin labels for SU(n) representations 92 Orthogonal groups 93 Two-index tensors 94 Three-index tensors 94 Mixed defining adjoint tensors 94 Two-index adjoint tensors 94 Gravity tensors 94 Dynkin labels of SO(n) representations 94 Spinors 97 Spinograpy 98 Fierzing around 98 Fierz coefficients 98 6j coefficients 98 Exemplary evaluations 98 Invariance of -matrices 98 Handedness 98 Kahane algorithm 98 Symplectic groups 99 Two-index tensors 100 Mixed defining adjoint tensors 101 Dynkin labels of Sp(n) representations 101 Negative dimensions 103 SU(n) = SU(-n) 105 SO(n) = Sp(-n) 106 Spinsters 109 SU(n) family of invariance groups 111 Representations of SU(2) 111 SU(3) as invariance group of a cubic invariant 113 Levi-Civita tensors and SU(n) 113 SU(4) - SO(6) isomorphism 113 G2 family of invariance groups 115 Jacobi relation 117 Alternativity and reduction of f-contractions 118 Primitivity implies alternativity 120 Casimirs for G2 123 Hurwitz's theorem 124 Representations of G2 126 E8 family of invariance groups 127 Two-index tensors 128 Decomposition of Sym3 A 131 Decomposition of |??||??|-.16667em |??|* 133 Diophantine conditions 135 Generalized Young tableaux for E8 135 Conjectures of Deligne 136 E6 family of invariance groups 137 Reduction of two-index tensors 137 Mixed two-index tensors 138 Diophantine conditions and the E6 family 138 Three-index tensors 138 Fully symmetric V3 tensors 138 Mixed symmetry V3 tensors 138 Fully antisymmetric V3 tensors 138 Defining adjoint tensors 138 Two-index adjoint tensors 138 Reduction of antisymmetric 3-index tensors 138 Dynkin labels and Young tableaux for E6 139 Casimirs for E6 139 Subgroups of E6 139 Springer relation 139 Springer's construction of E6 139 F4 family of invariance groups 141 Two-index tensors 141 Defining adjoint tensors 144 Two-index adjoint tensors 144 Jordan algebra and F4(26) 144 E7 family of invariance groups 145 Exceptional magic 147 Magic triangle 147 Magic negative dimensions 151 E7 and SO(4) 151 E6 and SU(3) 151 Recursive decomposition 153 Properties of Young Projections 155 Uniqueness of Young projection operators 155 Normalization 156 Orthogonality 157 The dimension formula 157 Literature 159