Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groups

Dupont, Benjamin

doi:10.1016/j.aim.2020.107524

Cited by 6 publications

(7 citation statements)

References 21 publications

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“…For this, we declare that the generators are in degree given by the monomial written below them in Eq. (11), where the monomial h a q b`cβ :" h a q b λ c means the element is in homological degree a, q-degree b and λ-degree c.…”

Section: Dg-enhanced Klrw Algebrasmentioning

confidence: 99%

“…One proves following the proof of convergence for the KLR algebras of [11], that all these critical branchings are confluent modulo braid-like isotopies. As a consequence, P is a convergent presentation of NH n and the monomials in normal form with respect to P yield a linear basis of NH n , recovering the usual basis for the nilHecke algebra (see for example given by setting δ " 0.…”

mentioning

confidence: 94%

“…for any monomial D. Following [11], it suffices to check the confluence of these indexed critical branchings for D " ‚ p , ‚ p for any p P N.…”

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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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Section: Dg-enhanced Klrw Algebrasmentioning

confidence: 99%

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confidence: 94%

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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 was formulated originally as the nondegeneracy condition by Khovanov and Lauda in [27, §3.2.3]. It was proved by them in finite type A, and it was proved in general in [41]; see also [18] for a completely different approach.…”

Section: Kac-moody Gcqsmentioning

confidence: 99%

“…Lemma 5. 18 The module P belongs to mod lfd -H Z (m|n) κ . Moreover, under the categorical action of U(g) just defined, we have isomorphisms…”

Section: Lemma 511mentioning

confidence: 99%

Heisenberg and Kac–Moody categorification

Brundan

Savage

Webster

2020

Sel. Math. New Ser.

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We show that any Abelian module category over the (degenerate or quantum) Heisenberg category satisfying suitable finiteness conditions may be viewed as a 2representation over a corresponding Kac-Moody 2-category (and vice versa). This gives a way to construct Kac-Moody actions in many representation-theoretic examples which is independent of Rouquier's original approach via "control by K 0 ." As an application, we prove an isomorphism theorem for generalized cyclotomic quotients of these categories, extending the known isomorphism between cyclotomic quotients of type A affine Hecke algebras and quiver Hecke algebras.

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Rewriting modulo isotopies in pivotal linear (2,2)-categories

Dupont

2022

Journal of Algebra

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Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groups

Cited by 6 publications

References 21 publications

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

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

Heisenberg and Kac–Moody categorification

Rewriting modulo isotopies in pivotal linear (2,2)-categories

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