2-Verma modules

Naisse, Grégoire; Vaz, Pedro

doi:10.1515/crelle-2021-0046

Cited by 5 publications

(31 citation statements)

References 36 publications

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“…After the introduction of the diagrammatic version of the KLR algebra [KhLa09] (see also [Ro08]) they have become quite popular, and might admit handlebody extensions. For example, alongside with KLR algebras Webster's tensor product algebras [Web17], algebras related to Verma categorifications [NV18], [NV17], [MV19], [LNV20], Soergel diagrammatics [EW16], potentially admit handlebody versions, just to name a few.…”

Section: (B)mentioning

confidence: 99%

Handlebody diagram algebras

Tubbenhauer,

Vaz

2021

Preprint

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In this paper we study handlebody versions of classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer/BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory. Contents1. Introduction 1 2. A generalization of cellularity 4 3. Handlebody braid and Coxeter groups 11 4. Handlebody Temperley-Lieb and blob algebras 16 5. Handlebody Brauer and BMW algebras 28 6. Handlebody Hecke and Ariki-Koike algebras 36 References 41

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Section: (B)mentioning

confidence: 99%

Handlebody diagram algebras

Tubbenhauer,

Vaz

2021

Preprint

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“…( 33) and Eq. (34). Moreover, without these relations we sould still have a terminating rewriting system, but some normal forms would not be basis elements.…”

Section: Basis Theoremmentioning

confidence: 99%

“…One can also verify by hand that all the regular critical branchings modulo of r T µ b pδq are confluent modulo braid-like isotopies. However, indexed critical branchings given by overlappings of the rewriting rules (32), (33) and (34) produce infinitely many cases to check, which can be unwieldy in practice. We show that they are confluent in the case of tensor products of Verma modules (i.e.…”

Section: Basis Theoremmentioning

confidence: 99%

“…In this section, we explain how derived categories of pT µ , d µ q-modules categorify Lpµq. Since the categorical action is similar to [36] and [34], we will rely heavily on the references for the details.…”

Section: Categorification Of Lpµqmentioning

confidence: 99%

“…[47,29]), the categorification of infinite dimensional (in the sense non-integrable) representations is still quite new and not so well understood. 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].…”

Section: Introductionmentioning

confidence: 99%

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

Dupont,

Naisse

2021

Preprint

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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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DG-Enhanced Hecke and KLR Algebras

Maksimau,

Vaz

2023

SIGMA

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We construct DG-enhanced versions of the degenerate affine Hecke algebra and of the affine Hecke algebra. We extend Brundan-Kleshchev and Rouquier's isomorphism and prove that after completion DG-enhanced versions of affine Hecke algebras (degenerate or nondegenerate) are isomorphic to completed DG-enhanced versions of KLR algebras for suitably defined quivers. As a byproduct, we deduce that these DG-algebras have homologies concentrated in degree zero. These homologies are isomorphic respectively to the degenerate cyclotomic Hecke algebra and the cyclotomic Hecke algebra.

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2-Verma modules

Cited by 5 publications

References 36 publications

Handlebody diagram algebras

Handlebody diagram algebras

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

DG-Enhanced Hecke and KLR Algebras

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