Table of Contents
Preface v
1 Lie superalgebras 1
1.1 Generalities on Lie superalgebras 1
1.2 Classification of finite-dimensional Lie superalgebras 3
1.3 Root systems 7
1.3.1 Classical Lie superalgebras 7
1.3.2 Basic classical Lie superalgebras 8
1.3.3 Affine basic classical superalgebras 17
1.4 Representations of simple Lie superalgebras 17
1.5 Real forms of the basic classical Lie superalgebras 20
2 Conformal supersymmetry in 4D 23
2.1 Representations of D = 4 conformal supersymmetry 24
2.1.1 The setting 24
2.1.2 Verma modules 24
2.1.3 Singular vectors and invariant submodules at the unitary reduction points 30
2.1.4 Structure of single-reducibility-condition Verma modules and UIRs 34
2.1.5 Structure of double-reducibility-condition Verma modules and UIRs 37
2.2 Character formulas of positive energy UIRs 41
2.2.1 Character formulas: generalities 41
2.2.2 Character formulas for the long UIRs 44
2.2.3 Character formulas of SRC UIRs 48
2.2.4 Character formulas of DRC UIRs 52
2.2.5 Summary and discussion 58
2.3 Explicit character formulas for N = 1,2,4 59
2.3.1 Characters of the even subalgebra 59
2.3.2 N = 1 64
2.3.3 N = 2 72
2.3.4 N = 4 80
2.4 BPS states for D = 4 87
2.4.1 R-symmetry scalars 87
2.4.2 R-symmetry non-scalars 89
2.4.3 PSU(2,2/4) 90
2.4.4 SU(2,2/N),N ≤ 3 94
2.4.5 SU(2,2/N),N ≤ 5 97
2.4.6 1/N-BPS states 102
3 Examples of conformal supersymmetry for D > 4 105
3.1 Positive energy UIRs of D = 6 conformal supersymmetry 105
3.1.1 Representations of D = 6 conformal supersymmetry 106
3.1.2 Unitarity 112
3.1.3 Unitarity at the reduction points 123
3.2 Positive energy UIRS of osp(1|2n, R) 129
3.2.1 Representations of osp(1|2n) and osp(1|2n,R) 129
3.2.2 Unitarity 137
3.2.3 Character formulas 142
3.2.4 The case n = 3 145
3.2.5 Conformal superalgebras for D = 9, 10, 11 150
4 Quantum superalgebras 155
4.1 Defining relations for q-deformed superalgebras 155
4.1.1 Example of Uq(sl(M/N; C) 156
4.1.2 The cases of sl(2/2; C) and D(2,1; σ) 157
4.1.3 q-deformed conformal superalgebras Uq(su(2,2/N)) 159
4.2 Duality for multiparameter quantum deformation of the supergroup GL(m/n) 160
4.2.1 Multiparameter quantum deformation of GL(m/n) 160
4.2.2 Hopf structure of the superalgebra Uuq 166
4.2.3 Drinfeld-Jimbo form of the superalgebra Uuq 167
4.2.4 Multiparameter superalgebra deformation of U(s\(m/n)) 171
4.3 Induced representations of the multiparameter Hopf superalgebras Uuq(gl(m/n)) and Uuq(sl(m/n)) 173
4.3.1 Left and right actions of U and U' 174
4.3.2 Basis via Gauss decomposition 178
4.3.3 Representations of U and U' 184
Bibliography 193
Author Index 215
Subject Index 217
