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Softcover ISBN:  9781470441753 
Product Code:  MEMO/265/1285 
List Price:  $85.00 
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Product Code:  MEMO/265/1285.E 
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Softcover ISBN:  9781470441753 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 265; 2020; 123 ppMSC: Primary 17; 20; 14
The quantum groups of finite and affine type \(A\) admit geometric realizations in terms of partial flag varieties of finite and affine type \(A\). Recently, the quantum group associated to partial flag varieties of finite type \(B/C\) is shown to be a coideal subalgebra of the quantum group of finite type \(A\). In this paper the authors study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type \(C\). The authors show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine \(\mathfrak{sl}\) and \(\mathfrak{gl}\) types, respectively. In this way, the authors provide geometric realizations of eight quantum symmetric pairs of affine types. The authors construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine \(\mathfrak{sl}\) type, the authors establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, the authors obtain a new and geometric construction of the idempotented quantum affine \(\mathfrak{gl}\) and its canonical basis.

Table of Contents

Chapters

1. Introduction

1. Affine flag varieties, Schur algebras, and Lusztig algebras

2. Constructions in affine type $A$

3. Lattice presentation of affine flag varieties of type $C$

4. Multiplication formulas for Chevalley generators

5. Coideal algebra type structures of Schur algebras and Lusztig algebras

2. Lusztig algebras and coideal subalgebras of $\mathbf {U}(\widehat {\mathfrak {sl}}_n)$

6. Realization of the idempotented coideal subalgebra $\dot {\mathbf {U}}^{\mathfrak {c}}_n$ of $\mathbf {U}(\widehat {\mathfrak {sl}}_n)$

7. A second coideal subalgebra of quantum affine $\mathfrak {sl}_\mathfrak {n}$

8. More variants of coideal subalgebras of quantum affine $\mathfrak {sl}_n$

3. Schur algebras and coideal subalgebras of $\mathbf {U}(\widehat {\mathfrak {gl}}_n)$

9. The stabilization algebra $\dot {\mathbf K}^{\mathfrak {c}}_n$ arising from Schur algebras

10. Stabilization algebras arising from other Schur algebras

A. Constructions in finite type $C$


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The quantum groups of finite and affine type \(A\) admit geometric realizations in terms of partial flag varieties of finite and affine type \(A\). Recently, the quantum group associated to partial flag varieties of finite type \(B/C\) is shown to be a coideal subalgebra of the quantum group of finite type \(A\). In this paper the authors study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type \(C\). The authors show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine \(\mathfrak{sl}\) and \(\mathfrak{gl}\) types, respectively. In this way, the authors provide geometric realizations of eight quantum symmetric pairs of affine types. The authors construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine \(\mathfrak{sl}\) type, the authors establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, the authors obtain a new and geometric construction of the idempotented quantum affine \(\mathfrak{gl}\) and its canonical basis.

Chapters

1. Introduction

1. Affine flag varieties, Schur algebras, and Lusztig algebras

2. Constructions in affine type $A$

3. Lattice presentation of affine flag varieties of type $C$

4. Multiplication formulas for Chevalley generators

5. Coideal algebra type structures of Schur algebras and Lusztig algebras

2. Lusztig algebras and coideal subalgebras of $\mathbf {U}(\widehat {\mathfrak {sl}}_n)$

6. Realization of the idempotented coideal subalgebra $\dot {\mathbf {U}}^{\mathfrak {c}}_n$ of $\mathbf {U}(\widehat {\mathfrak {sl}}_n)$

7. A second coideal subalgebra of quantum affine $\mathfrak {sl}_\mathfrak {n}$

8. More variants of coideal subalgebras of quantum affine $\mathfrak {sl}_n$

3. Schur algebras and coideal subalgebras of $\mathbf {U}(\widehat {\mathfrak {gl}}_n)$

9. The stabilization algebra $\dot {\mathbf K}^{\mathfrak {c}}_n$ arising from Schur algebras

10. Stabilization algebras arising from other Schur algebras

A. Constructions in finite type $C$