Supercategorification of quantum Kac–Moody algebras

Kang, Seok Jin; Kashiwara, Masaki; Oh, Se Jin

doi:10.1016/j.aim.2013.04.008

Cited by 29 publications

(60 citation statements)

References 29 publications

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“…Their quiver Hecke superalgebras become isomorphic to affine Hecke-Clifford superalgebras or affine Sergeev superalgebras after a suitable completion, and the sl 2 case of their construction is isomorphic to the odd nilHecke algebra. Cyclotomic quotients of quiver Hecke superalgebras supercategorify certain irreducible representations of Kac-Moody algebras [33,34]. A closely related spin Hecke algebra associated to the affine Hecke-Clifford superalgebra appeared in earlier work of Wang [72] and many of the essential features of the odd nilHecke algebra including skew-polynomials appears in this and related works [38][39][40]71].…”

Section: Introductionmentioning

confidence: 85%

An odd categorification of $U_q (\mathfrak{sl}_2)$

Ellis¹,

Lauda²

2016

Quantum Topol.

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ABSTRACT. We define a 2-category that categorifies the covering Kac-Moody algebra for sl 2 introduced by Clark and Wang. This categorification forms the structure of a super-2-category as formulated by Kang, Kashiwara, and Oh. The super-2-category structure introduces a × 2 -grading giving its Grothendieck group the structure of a free module over the group algebra of × 2 . By specializing the 2 -action to +1 or to −1, the construction specializes to an "odd" categorification of sl 2 and to a supercategorification of osp 1|2 , respectively.

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Section: Introductionmentioning

confidence: 85%

An odd categorification of $U_q (\mathfrak{sl}_2)$

Ellis¹,

Lauda²

2016

Quantum Topol.

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“…are isomorphisms of supermodules. All the above are presented with a more categorical flavor in [12,21] (see also [22]), showing that the supermodules and superbimodules give supercategories. Of course, all the above extend to the case when R has additional gradings, making it a multigraded superring.…”

Section: Superbimodulesmentioning

confidence: 99%

An approach to categorification of Verma modules

Naisse

Vaz

2018

Proc. London Math. Soc.

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We give a geometric categorification of the Verma modules M (λ) for quantum sl2. ContentsFrom the work of Lauda in [34,35] adjusted to our context it follows that for each n 0 there is an action of the nilHecke algebra NH n on F n and on E n . As a matter of fact, there is an enlargement of NH n , which we denote as A n , acting on F n and E n , also admitting a nice diagrammatic description (see § 1.1.6 for a sketch).We define M as the direct sum of all the categories M k and functors F, E and Q in the obvious way. One of the main results in this paper is the following. † All our functors are, in fact, superfunctors which we tend to see as functors between categories endowed with a Z/2Z-action, whence the use of the terminology functor. ‡ We thank Aaron Lauda for explaining this to us.

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“…Since Clifford algebras (which are Z 2 -graded) play a fundamental role, superbicategories are the natural language. Our main references for the foundations are [17,22,23], but we give the full definitions below as these references work only with strict bicategories.…”

Section: Supercategoriesmentioning

confidence: 99%

The cut operation on matrix factorisations

Murfet

2018

Journal of Pure and Applied Algebra

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The bicategory LG of Landau-Ginzburg models has polynomials as objects and matrix factorisations as 1-morphisms. The composition of these 1-morphisms produces infinite rank matrix factorisations, which is a nuisance. In this paper we define a bicategory C which is equivalent to LG in which composition of 1-morphisms produces finite rank matrix factorisations equipped with the action of a Clifford algebra. This amounts to a finite model of composition in LG.

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Supercategorification of quantum Kac–Moody algebras

Abstract: We show that the quiver Hecke superalgebras and their cyclotomic quotients provide a supercategorification of quantum Kac-Moody algebras and their integrable highest weight modules.

Cited by 29 publications

References 29 publications

An odd categorification of $U_q (\mathfrak{sl}_2)$

An odd categorification of $U_q (\mathfrak{sl}_2)$

An approach to categorification of Verma modules

The cut operation on matrix factorisations

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