Diagrammatics for Coxeter groups and their braid groups

Elias, Ben; Williamson, Geordie

doi:10.4171/qt/94

Cited by 18 publications

(27 citation statements)

References 30 publications

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“…Proof. By work of Elias and Williamson [EW,1.18], it suffices to show that we have isomorphisms lifting the braid relations which satisfy the Zamolodchikov tetrahedral equations. This will hold since we have defined a canonical functor not just for braid generators, but for all positive lifts of permutations.…”

Section: Our Induction Proceeds By Showingmentioning

confidence: 99%

Knot invariants and higher representation theory

2017

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We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl 2 and sl 3 and by Mazorchuk-Stroppel and Sussan for sl n .Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is sl n , we show that these categories agree with certain subcategories of parabolic category O for gl k .We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory.The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius.In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.

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Section: Our Induction Proceeds By Showingmentioning

confidence: 99%

Knot invariants and higher representation theory

2017

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“…This was introduced in [KT] where it is called the marked Brauer category, motivated by Schur-Weyl duality for the Lie superalgebra p n (C [EW2,§1.3]). We adopt the following language for such structures:…”

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confidence: 99%

Monoidal Supercategories

Brundan

Ellis

2017

Commun. Math. Phys.

114

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This work is a companion to our article "Super Kac-Moody 2categories," which introduces super analogs of the Kac-Moody 2-categories of Khovanov-Lauda and Rouquier. In the case of sl 2 , the super Kac-Moody 2-category was constructed already in [A. Ellis and A. Lauda, "An odd categorification of Uq(sl 2 )"], but we found that the formalism adopted there became too cumbersome in the general case. Instead, it is better to work with 2-supercategories (roughly, 2-categories enriched in vector superspaces). Then the Ellis-Lauda 2-category, which we call here a Π-2-category (roughly, a 2category equipped with a distinguished involution in its Drinfeld center), can be recovered by taking the superadditive envelope then passing to the underlying 2-category. The main goal of this article is to develop this language and the related formal constructions, in the hope that these foundations may prove useful in other contexts.

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“…We will not repeat the definition of the Zamolodchikov relations here, and instead refer the reader to [6, §1.4.3]. (See [7] for topological background).…”

Section: Zamolodchikov Relationsmentioning

confidence: 99%