Super Kac–Moody 2‐categories

Brundan, Jonathan; Ellis, Alexander P.

doi:10.1112/plms.12055

Cited by 10 publications

(14 citation statements)

References 26 publications

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“…see [W2,Proposition 2.8] or [BHLW, §3.2]. When d ij > 1, the proof of the final two bubble slides above is not as straightforward as those references may suggest as it requires also an application of the deformed braid relation; we refer to [BE2,Proposition 7.3] for the detailed argument. (K8) Alternating crossings.…”

Section: Kac-moody 2-categoriesmentioning

confidence: 99%

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Categorical actions and crystals

Brundan¹,

Davidson²

2017

Categorification and Higher Representation Theory

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Section: Kac-moody 2-categoriesmentioning

confidence: 99%

“…As usual with objects defined by generators and relations, one then needs to play the game of deriving consequences from the defining relations. Here we record some which were established in [B3]; we also cite below the more recent exposition in [BE2] since that uses exactly the same normalization as here.…”

Section: Next We Have the Right Adjunction Relationsmentioning

confidence: 99%

Categorical actions and crystals

Brundan¹,

Davidson²

2017

Categorification and Higher Representation Theory

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“…To name a few examples, they appear explicitly or implicitly in the categorification of Heisenberg algebras [RS], "odd" categorifications of Kac-Moody (super)algebras (e.g. [EL,KKO1,KKO2]), the definition of super Kac-Moody categories [BE1], and in various Schur-Weyl dualities in the Z 2 -graded setting (e.g. [KT]).…”

Section: Introductionmentioning

confidence: 99%

A Basis Theorem for the Degenerate Affine Oriented Brauer–Clifford Supercategory

Brundan¹,

Comes²,

Kujawa³

2019

Can. J. Math.-J. Can. Math.

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We introduce the oriented Brauer-Clifford and degenerate affine oriented Brauer-Clifford supercategories. These are diagrammatically defined monoidal supercategories which provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q. 1

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“…The current paper further illustrates the wide applicability of this approach. It will also be used in [BSW] to prove a basis theorem for quantum Frobenius Heisenberg categories built from the quantum affine wreath product algebras of [RS20], and we expect it can be used to prove basis theorems for the super Kac-Moody 2-categories of [BE17b].…”

Section: Introductionmentioning

confidence: 99%

Foundations of Frobenius Heisenberg categories

Brundan,

Savage,

Webster

2020

Preprint

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We describe bases for the morphism spaces of the Frobenius Heisenberg categories associated to a symmetric graded Frobenius algebra, proving several open conjectures. Our proof uses a categorical comultiplication and generalized cyclotomic quotients of the category. We use our basis theorem to prove that the Grothendieck ring of the Karoubi envelope of the Frobenius Heisenberg category recovers the lattice Heisenberg algebra associated to the Frobenius algebra.Heis −1 (k) [Kho14], and then also to give a basis theorem for Heis k (k) [MS18] for all k 0. Unfortunately, the above method for proving linear independence fails for general A. The natural action of Heis k (A) is on modules for the cyclotomic wreath product algebras studied in [Sav20] (see also [KM19]). However, one can show by degree considerations that certain morphisms, expected to be nonzero in Heis k (A), must act as zero in any such module category; see [RS17, Rem. 8.11]. Therefore, until now, basis theorems in this general setting have remained conjectural, even in the important case where A is a zigzag algebra as in Example 9.1, when the Frobenius Heisenberg category is related to the geometry of Hilbert schemes [CL12] and categorical vertex operators [CL11].

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Super Kac–Moody 2‐categories

Abstract: Abstract. We introduce generalizations of Kac-Moody 2-categories in which the quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced by the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka.

Cited by 10 publications

References 26 publications

Categorical actions and crystals

Categorical actions and crystals

A Basis Theorem for the Degenerate Affine Oriented Brauer–Clifford Supercategory

Foundations of Frobenius Heisenberg categories

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