q-deformed Howe duality for orthosymplectic Lie superalgebras = 직교사교 리 초대수에 대한 Howe 쌍대성의 q-변형

국문 초록 (Abstract)

우리는 직교군 $G=O_\ell$ 또는 사교군 $Sp_{2\ell}$과 직교사교 리 초대수인 $\mathfrak{g}=\mathfrak{osp}_X(\epsilon)$에 대응하는 직교사교 Howe 쌍대성의 $q$-아날로그를 제시한다. 우리는 $\mathfrak{g}$의 $q$-변형으로서 Kuniba-Okado-Sergeev가 도입한 D형과 C형의 일반화된 양자군을 사용하고, $G$의 $q$-변형으로서는 AI형 대칭쌍 $(\mathfrak{sl}_\ell, \mathfrak{so}_\ell)$과 AII형 대칭쌍 $(\mathfrak{sl}_{2\ell}, \mathfrak{sp}_{2\ell})$에 연관된 $\imath$양자군을 사용한다.

또한 우리는 $q$-변형 초대칭 대수의 반단순 분해를 제시하고 이 분해가 고전 극한에서 기존의 직교사교 Howe 쌍대성을 복원함을 보인다.

다국어 초록 (Multilingual Abstract)

We give a $q$-analogue of the orthosymplectic Howe duality associated to the orthogonal or symplectic Lie group $G=O_\ell$ or $Sp_{2\ell}$ and orthosymplectic Lie superalgebra $\mathfrak{g}=\mathfrak{osp}_X(\epsilon)$. As a $q$-deformation of $\mathfrak{g}$, we use the generalized quantum groups of types D and C introduced by Kuniba-Okado-Sergeev. As a $q$-deformation of $G$, we use $\imath$quantum group associated with the symmetric pairs of types AI $(\mathfrak{sl}_\ell, \mathfrak{so}_\ell)$ and AII $(\mathfrak{sl}_{2\ell}, \mathfrak{sp}_{2\ell})$.

We further present a semisimple decomposition of the $q$-deformed supersymmetric algebra, which recovers the classical orthosymplectic Howe duality in the classical limit.

목차 (Table of Contents)

• 1 Introduction 1
• 1.1 Howe duality 1
• 1.2 Main results 1
• 1.3 Organization 3
• 2 Lie superalgebras and Howe duality 5
• 2.1 Lie superalgebras 5
• 2.1.1 Orthosymplectic Lie superalgebras osp(V ) 5
• 2.1.2 Fundamental systems of osp(V ) 6
• 2.2 Howe duality 9
• 3 Generalized quantum groups UX (ϵ) 11
• 3.1 Generalized quantum groups of type D and C 11
• 3.2 Quantum superalgebras of type D and C 14
• 3.3 Classical limit of modules of UX (ϵ) 16
• 4 q-oscillator representations of UX (ϵ) 20
• 4.1 q-deformed supersymmetric space and q-oscillator representations 20
• 4.1.1 q-oscillator representation W of UX (ϵ) 21
• 4.2 Classical limit and semisimplicity of W ⊗ℓ 26
• 5 Quantum symmetric pairs (U(g), Uı(k)) 30
• 5.1 Quantum symmetric pairs 30
• 5.2 Quantum symmetric pairs of types AI and AII 33
• 5.3 Classical weight modules 34
• 6 q-deformed Howe duality on W ⊗ℓ 38
• 6.1 Commutativity of actions of UX (ϵ) and Uı(k) on W ⊗ℓ 38
• 6.2 The space of multiplicity of Vλ in W ⊗ℓ 40
• 6.3 Decomposition of W ⊗ℓ as a (UX (ϵ), Uı(k))-module 42
• A Proofs 45
• A.1 Proof of Proposition 6.1.1 45
• A.2 Proof of Proposition 6.1.3 47
• A.2.1 Proof of Proposition 6.1.3(1) 47
• A.2.2 Proof of Proposition 6.1.3(2) 50
• B List of Notations 64
• Abstract (in Korean) 67
• Acknowledgement (in Korean) 68