2-Verma modules and the Khovanov–Rozansky link homologies

Naisse, Grégoire; Vaz, Pedro

doi:10.1007/s00209-020-02658-7

Cited by 5 publications

(4 citation statements)

References 25 publications

(57 reference statements)

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“…and on the other hand `λ´1 q k`1 `hλqrk `1s q ˘ˆλq ´2k rk `1s q rβ `1 ´ks h q ´hλq 2 rk `1s q ˆλq ´2k rβs h q " q 1´k rk `1s q rβ ´k `1s h q `hλ 2 q 1´2k rk `1s q rk `1q srβ ´k `1s h q ´hλ 2 q 2´2k rk `1s q rβs h q " q ´krβ ´ks h q rk `1s q ´hλq 1´2k rk `1s q `hλ 2 q 1´2k rk `1s q `rks q rβ ´ks h q `q´1´k rβ ´ks h q ´hλq ´krk `1s q hλ 2 q 2´2k rk `1s q rβs h q , using (31) and (32). We remark that the first and third terms coincide with (33). We gather the remaining terms, putting hλq ´2k rk `1s q in evidence, so that we obtain ´q `λq ´krβ ´ks h q ´hλ 2 q 1´k rk `1s q ´λq 2 rβs h q " 1 q ´1 ´q `´qpq ´1 ´qq `λq ´kpλ ´1q k `hλq ´kq ´hλ 2 q 1´k pq ´k´1 ´qk`1 q ´λq 2 pλ ´1 `hλq " 0.…”

Section: ˘mentioning

confidence: 72%

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

Lacabanne¹,

Naisse²,

Vaz³

2020

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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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Section: ˘mentioning

confidence: 72%

“…This can be interpreted as a categorification of the projection of a universal Verma module onto an integrable irreducible module. Categorification of Verma modules was used by the second and third authors in [33] to give a quantum group higher representation theory construction of Khovanov-Rozansky's HOMFLY-PT link homology.…”

Section: Introductionmentioning

confidence: 99%

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

Lacabanne¹,

Naisse²,

Vaz³

2020

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“…The second author and Vaz constructed categorifications of universal Verma modules for sl 2 in [35,36], and extended it to any generic parabolic Verma module for any quantum Kac-Moody algebra in [34]. They also showed in [37] that their construction is related to Khovanov-Rozansky triply-graded link homology [26]. Moreover, in a collaboration [27] with Lacabanne, they gave a categorification of the tensor product of a Verma module with multiple integrable modules for quantum sl 2 .…”

Section: Introductionmentioning

confidence: 99%

Categorification of infinite-dimensional $\mathfrak{sl}_2$-modules and braid group 2-actions I: tensor products

Dupont,

Naisse

2021

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This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence-Krammer-Bigelow representations. In this part, we categorify all tensor products of Verma modules and integrable modules for quantum sl 2 . The categorification is given by derived categories of dg versions of KLRW algebras which generalize both the tensor product algebras of Webster, and the dg-algebras used by Lacabanne, the second author and Vaz. We compute a basis for these dgKLRW algebras by using rewriting methods modulo braid-like isotopy, which we develop in an Appendix.

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“…These issues motivated the introduction with Antonio Sartori of the notion of doubled Schur algebra [28], which is based on skew Howe duality with duals, and is closely related to the Brauer category at generic parameter β, thus giving a quantum description of the HOMFLY-PT polynomial. This approach was later categorified by Naisse and Vaz [26].…”

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confidence: 99%

Polynomial link invariants and quantum algebras

Queffelec¹

2021

Winter Braids Lecture Notes

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The definition of the Jones polynomial in the 80's gave rise to a large family of so-called quantum link invariants, based on quantum groups. These quantum invariants are all controlled by the same two-variable invariant (the HOMFLY-PT polynomial), which also specializes to the older Alexander polynomial. Building upon quantum Schur-Weyl duality and variants of this phenomenon, I will explain an algebraic setup that allows for global definitions of these quantum polynomials, and discuss extensions of these quantum objects designed to encompass all of the mentioned invariants, including the HOMFLY-PT polynomial. IV-1Hoel Queffelec words better replacements, for the categories of representations. Furthermore, these objects, contrarily to the categories of representations, can be extended to encompass the HOMFLY-PT polynomial as well, which I'll briefly illustrate at the end of these notes.Organization: Section 1 is devoted to brief definitions of the notions of knots, links and tangles, and to their invariants. These invariants are defined using the so-called quantized oriented Brauer category. Section 2 relates this to the original Reshetikhin-Turaev approach by defining a functor between the quantized oriented Brauer category and the category of representations of the quantum group U q (gl m|n ). Finally, Section 3 is devoted to Schur-Weyl and skew Howe dualities and the role they recently played in knot theory.The presentation given in these notes is far from being historically accurate: I have rather tried to define all invariants in a unified way and use this definition to recover several quantum constructions. In many situations, the story actually went the other way around. However, I hope that the presentation I chose to follow will make the relations and the central role that quantum algebras play in this picture more apparent, and that these notes can serve as a motivation and a general illustration before reading more involved references.Acknowledgements: These lecture notes follow a lecture series given at Winterbraids IX in Reims in March 2019. I would like to warmly thank the organizers (Paolo Bellingeri, Vincent Florens, Jean-Baptiste Meilhan, Loïc Poulain d'Andecy, Emmanuel Wagner) for their invitation, their support, and more generally for putting together these great winter schools year after year. Many thanks also to Antonio Sartori for his comments on a preliminary version of these notes and for teaching me most of what's now in these notes, to Baptiste Loreau Unger for carefully reading them, and to the anonymous referee for her/his helpful comments.

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2-Verma modules and the Khovanov–Rozansky link homologies

Cited by 5 publications

References 25 publications

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

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

Categorification of infinite-dimensional $\mathfrak{sl}_2$-modules and braid group 2-actions I: tensor products

Polynomial link invariants and quantum algebras

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