HOMFLYPT homology for links in handlebodies via type A Soergel bimodules

Rose, David E. V.; Tubbenhauer, Daniel

doi:10.48550/arxiv.1908.06878

Cited by 3 publications

(3 citation statements)

References 27 publications

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“…Another motivation for the results presented here resides in the potential applications to low-dimensional topology, as indicated in [18]. We find that it would be also interesting to investigate the use of several Verma modules in a tensor product as suggested in [10].…”

Section: Introductionmentioning

confidence: 93%

Schur-Weyl duality, Verma modules, and row quotients of Ariki-Koike algebras

Lacabanne,

Vaz

2020

Preprint

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We prove a Schur-Weyl duality between the quantum enveloping algebra of gl m and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natural representation of gl m . We also give a new presentation by generators and relations of the generalized blob algebras of Martin and Woodcock as well as an interpretation in terms of Schur-Weyl duality by showing they occur as a particular case of our algebras.

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Section: Introductionmentioning

confidence: 93%

Schur-Weyl duality, Verma modules, and row quotients of Ariki-Koike algebras

Lacabanne,

Vaz

2020

Preprint

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“…Partial traces. We now recall the partial Hochschild (co)homology functors from [RT21], which generalize the (uncolored) partial trace functors first introduced in [Hog18]. These functors refine the Hochschild (co)homology functors from §5.1, and allow them to be applied to the complex C(β b ) "one strand at a time."…”

Section: Deformed Colored Link Homologymentioning

confidence: 99%

Link splitting deformation of colored Khovanov--Rozansky homology

Hogancamp,

Rose,

Wedrich

2021

Preprint

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We introduce a multi-parameter deformation of the triply-graded Khovanov-Rozansky homology of links colored by one-column Young diagrams, generalizing the "y-ified" link homology of Gorsky-Hogancamp and work of Cautis-Lauda-Sussan. For each link component, the natural set of deformation parameters corresponds to interpolation coordinates on the Hilbert scheme of the plane. We extend our deformed link homology theory to braids by introducing a monoidal dg 2-category of curved complexes of type A singular Soergel bimodules. Using this framework, we promote to the curved setting the categorical colored skein relation from [HRW21] and also the notion of splitting map for the colored full twists on two strands. As applications, we compute the invariants of colored Hopf links in terms of ideals generated by Haiman determinants and use these results to establish general link splitting properties for our deformed, colored, triply-graded link homology. Informed by this, we formulate several conjectures that have implications for the relation between (colored) Khovanov-Rozansky homology and Hilbert schemes.

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“…We expect that by introducing braiding functors as in [49], we obtain a type B link homology, yielding invariants of links in the annulus akin to ones introduced by Asaeda-Przytycki-Sikora [2] (see also [4,9,36]). In a different direction, one could try to extend our results to construct a Khovanov invariant for links in handlebodies, in the spirit of the handlebody HOMFLY-PT-link homology of Rose-Tubbenhauer in [37].…”

Section: The Blob 2-category (Sections 5 and 6)mentioning

confidence: 99%

Tensor product categorifications, Verma modules and the blob 2-category

Lacabanne¹,

Naisse²,

Vaz³

2020

Preprint

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We construct a dg-enhancement of Webster's tensor product algebras that categorifies the tensor product of a universal sl 2 Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are twodimensional. This action intertwines with the categorical action of sl 2 . From the above we derive a categorification of the blob algebra.

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HOMFLYPT homology for links in handlebodies via type A Soergel bimodules

Cited by 3 publications

References 27 publications

Schur-Weyl duality, Verma modules, and row quotients of Ariki-Koike algebras

Schur-Weyl duality, Verma modules, and row quotients of Ariki-Koike algebras

Link splitting deformation of colored Khovanov--Rozansky homology

Tensor product categorifications, Verma modules and the blob 2-category

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