Canonical bases arising from quantum symmetric pairs of Kac–Moody type

Bao, Huanchen; Wang, Weiqiang

doi:10.1112/s0010437x2100734x

Cited by 27 publications

(25 citation statements)

References 26 publications

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“…One may hope that these new braid group symmetries preserve the integral Z[q, q −1 ]form on (modified) ıquantum groups in [BW18b,BW21]. (This will be highly nontrivial to verify, as the ıdivided powers are much more sophisticated than the divided powers.)…”

Section: ) the Symmetries T ′mentioning

confidence: 99%

An intrinsic approach to relative braid group symmetries on $\imath$quantum groups

Wang¹,

Zhang²

2022

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We initiate a general approach to the relative braid group symmetries on (universal) ıquantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi K-matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on ıquantum groups are obtained. We establish a number of fundamental properties for these symmetries on ıquantum groups, strikingly parallel to their well-known quantum group counterparts. We apply these symmetries to fully establish rank one factorizations of quasi K-matrices, and this factorization property in turn helps to show that the new symmetries satisfy relative braid relations. Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time. WEIQIANG WANG AND WEINAN ZHANG 4.1. Rescaled braid group action on U 4.2. Symmetries T ′′ j,+1 , for j ∈ I • 4.3. Characterization of T ′ i,−1 4.4. Quantum symmetric pairs of diagonal type 4.5. Action of T ′ i,−1 on U ı0 U • 4.6. Integrality of T ′ i,−1 4.7. A uniform formula for T ′ i,−1 (B i ) 5. Rank two formulas for T ′ i,−1 (B j ) 5.1. Some commutator relations with Υ 5.2. Motivating examples: types BI, DI, DIII 4 5.3. Formulation for T ′ i,−1 (B j ) 5.4. Proof of Theorem 5.5 5.5. A comparison with earlier results 6. New symmetries T ′′ i,+1 on U ı 6.1. Characterization of T ′′ i,+1 6.2. Action of T ′′ i,+1 on U ı0 U • 6.3. Rank one formula for T ′′ i,+1 (B i ) 6.4. Rank two formulas for T ′′ i,+1 (B j ) 6.5. T ′ i,e and T ′′ i,−e as inverses 7. A basic property of new symmetries 7.1. Rank 2 cases with ℓ • (w • ) = 3 7.2. Rank 2 cases with ℓ • (w • ) = 4 7.3. Rank 2 case with ℓ • (w • ) = 6 7.4. The general identity T w (B i ) = B wi 8. Factorization of quasi K-matrices 8.1. Factorization of Υ 8.2. Reduction to rank 2 8.3. Factorizations in rank 2 9. Relative braid group actions on ıquantum groups 9.1. Braid group relations among T i 9.2. Action of the braid group Br(W • ) ⋊ Br(W • ) on U ı 9.3. Intertwining properties of T ′ i,+1 , T ′′ i,−1 9.4. Braid group action on U ı ς 10. Relative braid group actions on U-modules 10.1. Intertwining relations on U ı ς 10.2. Compatible actions of T ′ i,e , T ′′ i,e on U-modules 10.3. Relative braid relations on U-modules Appendix A. Proofs of Proposition 5.11 and Table 3 A.1. Some preparatory lemmas A.2. Split types of rank 2 A.3. Type AII

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Section: ) the Symmetries T ′mentioning

confidence: 99%

An intrinsic approach to relative braid group symmetries on $\imath$quantum groups

Wang¹,

Zhang²

2022

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“…of cyclotomic Birman-Murakami-Wenzl algebras and cyclotomic Nazarov-Wenzl algebras over the complex field C to the Kazhdan-Lusztig theory (ı-canonical basis, etc.) of certain g ♯ -modules studied in [6][7][8][9][10]. This is a project in progress.…”

Section: Introductionmentioning

confidence: 96%

Representations of weakly triangular categories

Gao¹,

Rui²,

Song³

2020

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A new class of locally unital and locally finite dimensional algebras A over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of an upper finite triangular decomposition. Any locally unital algebra which admits an upper finite Cartan decomposition is Morita equivalent to some special locally unital algebra A which admits an upper finite weakly triangular decomposition. It is established that the category A-lfdmod of locally finite dimensional left A-modules is an upper finite fully stratified category in the sense of Brundan-Stroppel. Moreover, A is semisimple if and only if its centralizer subalgebras associated to certain idempotent elements are semisimple. Furthermore, certain endofunctors are defined and give categorical actions of some Lie algebras on the subcategory of A-lfdmod consisting of all objects which have a finite standard filtration. In the case A is the locally unital algebra associated to one of cyclotomic oriented Brauer categories, cyclotomic Brauer categories and cyclotomic Kauffman categories, A admits an upper finite weakly triangular decomposition. This leads to categorifications of representations of the classical limits of coideal algebras, which come from all integrable highest weight modules of sl∞ or ŝle. Finally, we study representations of A associated to either cyclotomic Brauer categories or cyclotomic Kauffman categories in details, including explicit criteria on the semisimplicity of A over an arbitrary field, and on A-lfdmod being upper finite highest weight category in the sense of Brundan-Stroppel, and on Morita equivalence between A and direct sum of infinitely many (degenerate) cyclotomic Hecke algebras.

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“…In recent years, Bao and Wang have generalized Lusztig's approach on canonical bases in [Lu94] and developed a general theory of the canonical basis for ıquantum groups arising from quantum symmetric pairs of arbitrary finite type in [BW18b] and of Kac-Moody type in [BW21]. They showed that any based module of a quantum group of finite type (cf.…”

Section: Introductionmentioning

confidence: 99%

Canonical bases of $q$-Brauer algebras and $\imath$Schur dualities

Cui¹,

Yaolong²

2022

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Expanding the classical work of Kazhdan-Lusztig, we construct a bar involution and canonical bases on the q-Brauer algebra introduced by Wenzl. We define explicit actions of the q-Brauer algebra on the tensor spaces, and formulate ıSchur dualities between the q-Brauer algebra and the ıquantum groups of type AI and AII respectively. CONTENTS 1. Introduction 1 2. q-Brauer algebras 3 3. A bar involution and canonical bases 6 4. ıSchur duality of type AI 12 5. ıSchur duality of type AII 17 References 22

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Canonical bases arising from quantum symmetric pairs of Kac–Moody type

Cited by 27 publications

References 26 publications

An intrinsic approach to relative braid group symmetries on $\imath$quantum groups

An intrinsic approach to relative braid group symmetries on $\imath$quantum groups

Representations of weakly triangular categories

Canonical bases of $q$-Brauer algebras and $\imath$Schur dualities

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