Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality

Ehrig, Michael; Stroppel, Catharina

doi:10.48550/arxiv.1310.1972

Cited by 25 publications

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“…The construction in the present paper is significantly inspired by the examples (sl 2N , s(gl N × gl N )) and (sl 2N +1 , s(gl N × gl N +1 )) considered by Bao and Wang in [BW13]. The papers [BW13] and [ES13] both observed the existence of a bar involution for quantum symmetric pair coideal subalgebras B c,s in this special case. Bao and Wang then constructed an intertwiner Υ ∈ U between the new bar involution and Lusztig's bar involution.…”

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confidence: 89%

“…Bao and Wang developed the theory for these two examples in astonishing similarity to Lusztig's exposition of quantized enveloping algebras in [Lus94]. In a closely related program M. Ehrig and C. Stroppel showed that quantum symmetric pairs for (gl 2N , gl N × gl N ) and (gl 2N +1 , gl N × gl N +1 ) appear via categorification using parabolic category O of type D [ES13]. The recent developments as well as the previously known results suggest that quantum symmetric pairs allow as deep a theory as quantized enveloping algebras themselves.…”

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confidence: 94%

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Universal K-matrix for quantum symmetric pairs

Balagović

Kolb

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

213

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Let g be a symmetrizable Kac-Moody algebra and let Uq(g) denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras Bc,s of Uq(g) have a universal K-matrix if g is of finite type. By a universal K-matrix for Bc,s we mean an element in a completion of Uq(g) which commutes with Bc,s and provides solutions of the reflection equation in all integrable Uq(g)-modules in category O. The construction of the universal K-matrix for Bc,s bears significant resemblance to the construction of the universal R-matrix for Uq(g). Most steps in the construction of the universal K-matrix are performed in the general Kac-Moody setting.In the late nineties T. tom Dieck and R. Häring-Oldenburg developed a program of representations of categories of ribbons in a cylinder. Our results show that quantum symmetric pairs provide a large class of examples for this program.2010 Mathematics Subject Classification. 17B37; 81R50.For the symmetric pairs (sl 2N , s(gl N × gl N )) and (sl 2N +1 , s(gl N × gl N +1 )) the construction of K coincides with the construction of the B c,s -module homomorphisms T M in [BW13] up to conventions. The longest element w 0 induces a diagram automorphism τ 0 of g and of U q (g). Any U q (g)-module M can be twisted by an algebra automorphism ϕ : U q (g) → U q (g) if we define u⊲m = ϕ(u)m for all u ∈ U q (g), m ∈ M . We denote the resulting twisted module by M ϕ . We show in Corollary 7.7 that the element K defines a B c,s -module isomorphismfor all finite-dimensional U q (g)-modules M . Alternatively, this can be written asThe construction of the bar involution for B c,s , the intertwiner X, and the B c,smodule homomorphism K are three expected key steps in the wider program of canonical bases for quantum symmetric pairs proposed in [BW13]. The existence of the bar involution was explicitly stated without proof and reference to the parameters in [BW13, 0.5] and worked out in detail in [BK15]. Weiqiang Wang has informed us that he and Huanchen Bao have constructed X and K ′ M independently in the case X = ∅, see [BW15].In the final Section 9 we address the crucial problem to determine the coproduct ∆(K) in U (2) . The main step to this end is to determine the coproduct of the quasi K-matrix X in Theorem 9.4. Even for the symmetric pairs (sl 2N , s(gl N × gl N )) and (sl 2N +1 , s(gl N × gl N +1 )), this calculation goes beyond what is contained in [BW13]. It turns out that if τ τ 0 = id then the coproduct ∆(K) is given by formula (1.1). Hence, in this case K is a universal K-matrix as defined above for the coideal subalgebra B c,s . If τ τ 0 = id then we obtain a slight generalization of the properties (1) and (2) of a universal K-matrix. Motivated by this observation we introduce the notion of a ϕ-universal K-matrix for B if ϕ is an automorphism of a braided bialgebra H and B is a right coideal subalgebra, see Section 4.3. With this terminology it hence turns out in Theorem 9.5 that in general K is a τ τ 0 -universal Kmatrix for B c,s . The f...

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confidence: 89%

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confidence: 94%

Universal K-matrix for quantum symmetric pairs

Balagović

Kolb

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

213

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“…We start by introducing a diagram combinatorics which will later be used to describe the geometry of the topological Springer fibres. It generalizes the well-known approaches from [Kho04], [Rus11], [SW12] to the more involved type D setting and is motivated by its connections to the type D highest weight Lie theory [ES13b] and [LS13].…”

Section: Diagram Combinatoricsmentioning

confidence: 99%

“…This paper is part of a series of four quite different papers, [ES13a], [ES13b], [ES15] dealing with type D generalizations of Khovanov's arc algebra. We develop in detail the geometric background of this algebra using the geometry of topological and algebraic Springer fibres and explain connections to category O for the orthogonal Lie algebra and to categories of finite dimensional representations of the associated W-algebras.…”

Section: Introductionmentioning

confidence: 99%

“…The type D Khovanov algebra is defined in [ES13a] using Braden's description of the category of perverse sheaves on isotropic Grassmannians, hence it gives an elementary description of parabolic category O of type (D k , A k−1 ) with all its nice properties like Koszulness, quasi-hereditary, cellularity, etc. Based on [CDV11] we introduce in [ES13b] suitable idempotent truncations of the type D Khovanov algebra and show in [ES15] that these algebras are indeed graded analogues of Brauer algebras Br d (δ) for arbitrary integral parameter δ. These idempotent truncations can also be viewed as a kind of limit, in which case this is very much in the spirit of a similar result obtained in [BS12] for the walled Brauer algebra using a limit version of generalized Khovanov algebras of type A.…”

Section: Introductionmentioning

confidence: 99%

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2-row Springer Fibres and Khovanov Diagram Algebras for Type D

Ehrig¹,

Stroppel²

2016

Can. j. math.

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Abstract. We study in detail two row Springer bres of even orthogonal type from an algebraic as well as topological point of view. We show that the irreducible components and their pairwise intersections are iterated P -bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group.e main tool is a type D diagram calculus labelling the irreducible components in a convenient way which relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. e diagram calculus generalizes Khovanov's arc algebra to the type D setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type A to other types.

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Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

Ehrig

Stroppel

2016

Sel. Math. New Ser.

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For each integer k ≥ 4, we describe diagrammatically a positively graded Koszul algebra D k such that the category of finite dimensional D k -modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type D k or B k−1 , constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) "folding" procedure from a generalized Khovanov arc algebra. Properties such as graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.Mathematics Subject Classification 05E10 · 14M15 · 17B10 · 17B45 · 55N91 · 20C08

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Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality

Cited by 25 publications

References 0 publications

Universal K-matrix for quantum symmetric pairs

Universal K-matrix for quantum symmetric pairs

2-row Springer Fibres and Khovanov Diagram Algebras for Type D

Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

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