Title Page
Abstract
Contents
Chapter 1. Introduction 9
1.1. Quantum affine superalgebras 10
1.2. Super duality 11
1.3. Main results 12
1.3.1. Generalized quantum groups 13
1.3.2. Super duality for polynomial representations 14
1.3.3. Oscillator representations and super duality 16
1.4. Organization 18
Chapter 2. Preliminaries 20
2.1. General linear Lie superalgebra glM|N[이미지참조] 21
2.2. Quantum affine algebra 25
2.2.1. Affine Lie algebras and quantum affine algebras 25
2.2.2. Finite-dimensional representations of quantum affine algebras 28
2.3. Quiver Hecke algebra 33
2.3.1. Quiver Hecke algebra 34
2.3.2. Quiver Hecke algebra of type A and their simple modules 38
Chapter 3. Generalized Quantum Groups of type A 41
3.1. Generalized quantum group of affine type A 42
3.1.1. Definition 42
3.1.2. Quantum affine superalgebra and algebra isomorphism 44
3.1.3. Universal R-matrix 47
3.2. Finite type subalgebra and its polynomial representations 51
Chapter 4. Super duality for polynomial representations 55
4.1. Super duality for polynomial representations of glₙ 56
4.2. Finite-dimensional representations of U(ϵ) 60
4.2.1. Fundamental representations 60
4.2.2. R-matrix 63
4.2.3. Fusion construction of irreducible polynomial representations 67
4.2.4. Generalized quantum affine Schur-Weyl duality 70
4.3. Super duality 74
4.3.1. Truncation 75
4.3.2. Equivalence of duality functors 80
4.3.3. Inverse limit category 85
4.3.4. Super duality 89
Chapter 5. Oscillator representations of Uq(glₙ)[이미지참조] 93
5.1. Howe duality and oscillator representations of glₙ 94
5.2. Oscillator representations of Uq(glₙ)[이미지참조] 99
5.2.1. Fock space and fundamental q-oscillator representations 100
5.2.2. Oscillator representations of Uq(glₙ)[이미지참조] 103
5.3. Oscillator representations of Uq(glₙ)[이미지참조] 110
5.3.1. Category Oosc,ϵ[이미지참조] 110
5.3.2. R-matrix and spectral decomposition 112
5.3.3. Fusion construction of irreducible q-oscillator representations 118
5.3.4. Correspondence of irreducibles and super duality 120
Chapter 6. Proofs 123
6.1. Chapter 4 123
6.1.1. Proof of Lemma 4.2.6 123
6.1.2. Proof of Theorem 4.3.13 128
6.2. Chapter 5 134
6.2.1. Proof of Proposition 5.3.2 134
6.2.2. Proof of Conjecture 5.3.16 for s=1 136
Bibliography 143
국문초록 150