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Affine Flag Varieties and Quantum Symmetric Pairs
 
Zhaobing Fan Harbin Engineering University, Harbin, China
Chun-Ju Lai University of Georgia, Athens, GA
Yiqiang Li University of Buffalo, Buffalo, NY
Li Luo East China Normal University, Shanghai, China
Weiqiang Wang East China Normal University, Shanghai, China
Affine Flag Varieties and Quantum Symmetric Pairs
Softcover ISBN:  978-1-4704-4175-3
Product Code:  MEMO/265/1285
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-6138-6
Product Code:  MEMO/265/1285.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-4175-3
eBook: ISBN:  978-1-4704-6138-6
Product Code:  MEMO/265/1285.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $136.00 $102.00
Affine Flag Varieties and Quantum Symmetric Pairs
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Affine Flag Varieties and Quantum Symmetric Pairs
Zhaobing Fan Harbin Engineering University, Harbin, China
Chun-Ju Lai University of Georgia, Athens, GA
Yiqiang Li University of Buffalo, Buffalo, NY
Li Luo East China Normal University, Shanghai, China
Weiqiang Wang East China Normal University, Shanghai, China
Softcover ISBN:  978-1-4704-4175-3
Product Code:  MEMO/265/1285
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-6138-6
Product Code:  MEMO/265/1285.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-4175-3
eBook ISBN:  978-1-4704-6138-6
Product Code:  MEMO/265/1285.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $136.00 $102.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2652020; 123 pp
    MSC: 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 $C$
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2652020; 123 pp
MSC: 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.

  • 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 $C$
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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