Quiver Hecke algebras and categorification

Brundan, Jonathan

doi:10.4171/125-1/4

Cited by 7 publications

(2 citation statements)

References 47 publications

(74 reference statements)

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“…for M ∈ R(β)-gmod and N ∈ R(γ)-gmod (see, for example, [Bru13]). Let L q be the set of the classes of self-dual simple objects in R-gmod, and P q the set of the classes of projective covers of self-dual simple objects in R-Mod.…”

Section: R-modmentioning

confidence: 99%

Wilson lines and their Laurent positivity

Ishibashi¹,

Oya²

2020

Preprint

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For a marked surface Σ and a semisimple algebraic group G of adjoint type, we study the Wilson line function g [c] : P G,Σ → G associated with the homotopy class of an arc c connecting boundary intervals of Σ. We show that g [c] defines a morphism of algebraic stacks with respect to the algebraic structure on P G,Σ investigated by Shen [She20]. Combining with Shen's result, we show that the cluster Poisson algebra with respect to the natural cluster Poisson structure on P G,Σ coincides with the ring of global functions with respect to the Betti structure. Moreover we show that the matrix coefficients c) give Laurent polynomials with positive integral coefficients in the Goncharov-Shen coordinate system associated with any decorated triangulation of Σ, for suitable f and v. Pt G,Σ after Claudius Ptolemy, is isomorphic to the cluster Poisson algebra [She20, Theorem 1.1]. Here is our first result: Theorem 1 (Theorem 3.18). The Wilson line along an arc class [c] : E in → E out defines a morphism g [c] : P Pt G,Σ → G of Artin stacks. Given a free loop |γ| on Σ, by cutting the surface along an edge that intersects |γ| we get another marked surface and an arc class [c] obtained from |γ|. Based on this obsevation, one can deduce: Theorem 2. The Wilson loop along a free loop |γ| ∈ π(Σ) defines a morphism ρ |γ| : P Pt G,Σ → [G/AdG] of Artin stacks. Here [G/AdG] denotes the quotient stack of G with respect to the conjugation action on itself. As a direct consequence of Theorem 1, we can compare the two stack structures on P G,Σ : Theorem 3 (Theorem 3.25). We have an open embedding P Pt

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Section: R-modmentioning

confidence: 99%

Wilson lines and their Laurent positivity

Ishibashi¹,

Oya²

2020

Preprint

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“…We follow the approach and definitions of Rouquier [Rou08], [Rou12]. We also refer to [Bru13] for a survey of the subject. At the end of the section, we prove some elementary computational results about KLR algebras that will be useful in the proofs of our main results.…”

Section: Klr Algebrasmentioning

confidence: 99%

Categorification of the adjoint action of quantum groups

Laurent¹

2020

Preprint

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Let U be a quantized enveloping algebra. We consider the adjoint action of an sl 2 -subalgebra of U on a subalgebra of U + that is maximal integrable for this action. We categorify this representation in the context of quiver Hecke algebras. We obtain an action of the 2-category associated with sl 2 on a category of modules over certain quotients of quiver Hecke algebras. Our approach is similar to that of Kang-Kashiwara [KK12] for categorifications of highest weight modules via cyclotomic quiver Hecke algebras. One of the main new features is a compatibility of the categorical action with the monoidal structure, categorifying the notion of derivation on an algebra. As an application of some of our results, we categorify the higher order quantum Serre relations, extending results of Stošić [Sto15] to the non simply-laced case.

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Dualities in quantum integrable many-body systems and integrable probabilities. Part I

Gorsky

Vasilyev

Zotov

2022

J. High Energ. Phys.

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In this study we map the dualities observed in the framework of integrable probabilities into the dualities familiar in a realm of integrable many-body systems. The dualities between the pairs of stochastic processes involve one representative from Macdonald-Schur family, while the second representative is from stochastic higher spin six-vertex model of TASEP family. We argue that these dualities are counterparts and generalizations of the familiar quantum-quantum (QQ) dualities between pairs of integrable systems. One integrable system from QQ dual pair belongs to the family of inhomogeneous XXZ spin chains, while the second to the Calogero-Moser-Ruijsenaars-Schneider (CM-RS) family. The wave functions of the Hamiltonian system from CM-RS family are known to be related to solutions to (q)KZ equations at the inhomogeneous spin chain side. When the wave function gets substituted by the measure, bilinear in wave functions, a similar correspondence holds true. As an example, we have elaborated in some details a new duality between the discrete-time inhomogeneous multispecies TASEP model on the circle and the quantum Goldfish model from the RS family. We present the precise map of the inhomogeneous multispecies TASEP and 5-vertex model to the trigonometric and rational Goldfish models respectively, where the TASEP local jump rates get identified as the coordinates in the Goldfish model. Some comments concerning the relation of dualities in the stochastic processes with the dualities in SUSY gauge models with surface operators included are made.

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Quiver Hecke algebras and categorification

Abstract: This is a brief introduction to the quiver Hecke algebras of Khovanov, Lauda and Rouquier, emphasizing their application to the categorification of quantum groups. The text is based on lectures given by the author at the ICRA workshop in Bielefeld in August, 2012.

Cited by 7 publications

References 47 publications

Wilson lines and their Laurent positivity

Wilson lines and their Laurent positivity

Categorification of the adjoint action of quantum groups

Dualities in quantum integrable many-body systems and integrable probabilities. Part I

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