Affine flag varieties and quantum symmetric pairs / Zhaobing Fan, Chun-Ju Lai, Yiqiang Li, Li Luo, Weiqiang Wang

By:

Fan, Zhaobing [VerfasserIn]

Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Memoirs of the American Mathematical Society ; number 1285Publisher: Providence, Rhode Island : American Mathematical Society, [2020]Description: 1 Online-Ressource (v, 123 pages)ISBN:

1470461382
9781470461386

Subject(s):

Electronic books

Additional physical formats: 9781470441753 | 1470441756 Online resources:

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Summary: 7.4. The coideal subalgebra of type \ji -- Chapter 8. More variants of coideal subalgebras of quantum affine _{ } -- 8.1. The Schur algebras of type \ijw -- 8.2. Comultiplication and transfer map of type \ijw -- 8.3. Quantum symmetric pair (\bU(\slh_{\nn}),\mbf ^{\ij}_{\nn}) and canonical basis on \mbf ^{\ij}_{\nn} -- 8.4. The Schur algebras of type \ii -- 8.5. Realization of a new coideal subalgebra \mbf ^{\ii}_{\mm} -- Part 3 . Schur algebras and coideal subalgebras of \bU(̂ _{ }) -- Chapter 9. The stabilization algebra \K^{\C}_{ } arising from Schur algebrasSummary: 9.1. Monomial bases for Schur algebras -- 9.2. Stabilization of the Schur algebras -- 9.3. Comultiplication and stabilization -- 9.4. The algebra \K^{\C}_{ } and its stably canonical basis -- 9.5. The algebra \K_{ } of affine type and its comultiplication -- 9.6. The comultiplication on \K^{\C}_{ } -- 9.7. A homomorphism from \K^{\C}_{ } to \Sj -- 9.8. The algebra \K^{\C}_{ } as a subquotient of \K^{\C}_{ } -- Chapter 10. Stabilization algebras arising from other Schur algebras -- 10.1. A monomial basis for Schur algebra \Sji^{\ji}_{\nn, }Summary: Cover -- Title page -- Chapter 1. Introduction -- 1.1. Background -- 1.2. The goal: affine type -- 1.3. An overview -- 1.4. The organization -- Part 1 . Affine flag varieties, Schur algebras, and Lusztig algebras -- Chapter 2. Constructions in affine type -- 2.1. Lattice presentation of affine flag varieties of type -- 2.2. Monomial basis for quantum affine _{ } -- 2.3. Algebras \bU_{ } and \bU_{ } -- Chapter 3. Lattice presentation of affine flag varieties of type -- 3.1. Affine complete flag varieties of type -- 3.2. Affine partial flag varieties of type -- 3.3. Local property at ₀Summary: Chapter 6. Realization of the idempotented coideal subalgebra \bU^{\fc}_{ } of \bU(\slh_{ }) -- 6.1. The coideal subalgebra \bU^{\fc}_{ } of \bU_{ } -- 6.2. The algebra \bU^{\fc}_{ } and its monomial basis -- 6.3. Bilinear form on \bU^{\fc}_{ } -- 6.4. The canonical basis of \bU^{\fc}_{ } and positivity -- 6.5. Another presentation of the algebra \bU^{\fc}_{ } -- Chapter 7. A second coideal subalgebra of quantum affine _{\nn} -- 7.1. The Schur algebras of type \ji -- 7.2. The comultiplication -- 7.3. The monomial basis of \mbf ^{\ji}_{\nn, }Summary: The quantum groups of finite and affine type admit geometric realizations in terms of partial flag varieties of finite and affine type . Recently, the quantum group associated to partial flag varieties of finite type is shown to be a coideal subalgebra of the quantum group of finite type . In this paper we study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type . We show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine and types, respectively. In this way, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotented quantum affine and its canonical basisPPN: PPN: 1755157444Package identifier: Produktsigel: BSZ-4-NLEBK-KAUB | ZDB-4-NLEBK

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