Khovanov homology is a skew Howe 2–representation of categorified quantum 𝔰𝔩<sub><i>m</i></sub>

Lauda, Aaron D.; Queffelec, Hoel; Rose, David E. V.

doi:10.2140/agt.2015.15.2517

Cited by 45 publications

(84 citation statements)

References 105 publications

(295 reference statements)

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“…An interesting feature of this relationship is that the natural system of symmetries in L m U q .gl m / actually translates into the braiding on Rep.sl n / which is crucial for the definition of the knot polynomials. This relationship was particular exploited in Lauda, Queffelec and Rose [13] and Queffelec and Rose [17], which categorify the features of the above technique and allows Lauda, Queffelec and Rose to provide a way that homology theories defined by Khovanov [10] and Khovanov and Rozansky [11] arise from categorified quantum groups.…”

Section: Introductionmentioning

confidence: 99%

“…The intention of this work is to set a possible grounding for a categorification in the style of [13] and [17], or of Cautis, Kamnitzer and Licata [3] and Cautis [2] which would provide a homology theory categorifying the Alexander polynomial that arises from representation theory, rather than from the categorifications arising from Floer homology. There is ongoing research into the connection between knot Floer homology and quantum sl n homology (see, for instance, Manolescu [15]), and a quantum gl.1j1/ homology theory may give a helpful intermediary.…”

Section: Introductionmentioning

confidence: 99%

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A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

Grant

2016

Algebr. Geom. Topol.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

Grant

2016

Algebr. Geom. Topol.

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“…The facets in front of the drawing surface are colored purple and those behind golden (and later also: cyan ). For readers familiar with the relationship between foams and categorified quantum groups (as developed in ), we emphasize that the orientation conventions in phase diagrams are not identical to those used in the string diagrams of the skew Howe dual categorified quantum group. The first equations in Lemma are simply

The following phase diagrams represent associativity and MP relations on foams, which hold in

S bold-italic Foam

by Proposition :

The relations obtained be reversing the orientation on all seams in also hold.…”

Section: Gln‐equivariant Foamsmentioning

confidence: 99%

“…The facets in front of the drawing surface are colored purple and those behind golden (and later also: cyan ). For readers familiar with the relationship between foams and categorified quantum groups (as developed in [31,34,35,42]), we emphasize that the orientation conventions in phase diagrams are not identical to those used in the string diagrams of the skew Howe dual categorified quantum group.…”

Section: )mentioning

confidence: 99%

Functoriality of colored link homologies

Ehrig

Tubbenhauer

Wedrich

2018

Proc. London Math. Soc.

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“…It is clear that the 2-category U cyc Q (g) introduced here is the most natural version of the categorified quantum group for a general choice of scalars Q. Indeed, the 2-isomorphism defined in section 2 removes some of the inconvenient signs that have appeared in the study of various 2-representations [9,12,11]. …”

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confidence: 99%

Cyclicity for categorified quantum groups

et al. 2016

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Abstract. We equip the categorified quantum group attached KLR algebra and an arbitrary choice of scalars with duality functor which is cyclic, that is, such that f = f * * for all 2-morphisms f . This is accomplished via a modified diagrammatic formalism.Consider the 2-category U Q (g) (as defined in [3,5,13]) which categorifies a Kac-Moody Lie algebra g. In this 2-category, every 1-morphism possesses a left dual and a right dual. Standard arguments show that the left and right duals (which are isomorphic) are well-defined up to unique isomorphism; however they are not well-defined as 1-morphisms. That is, the operations of left and right dual are anafunctors, not functors. In particular, every 1-morphism is isomorphic to its double-dual, but there is no fixed isomorphism intrinsic to the 2-category structure.However, we can choose a right duality functor, which allows us to fix a left and right dual. In fact, this duality will be strict: any 1-morphism will be equal to its double dual. One such functor is supplied by the functor τ defined in [6, 3.46], that is by rightward rotation by 180 • . However, in the graphical calculus defined in [5], this functor lacks one of the basic properties we expect from a duality: while a 1-morphism is equal to its double dual, for a 2-morphism f : u → v, we will not necessarily have that f and f * * are equal.If the equality f = f * * does hold, we call the resulting duality cyclic. In this case, the functor of double dual is isomorphic to the identity; thus, the identity map defines a (strictly) pivotal structure on this 2-category. This property is also equivalent to the induced biadjunction on the objects (u, u * ) being cyclic. In diagrammatic terms, this means that a morphism is unchanged by a full 360 • rotation. It follows that the string diagram calculus which uses a cyclic duality functor enjoys a topological invariance that greatly simplifies computations. Furthermore, a pivotal structure is necessary for defining convolution in the Hochschild cohomology of a representation of this 2-category; in particular, it plays a key role in the authors' previous work on traces of categorified quantum groups, and related work of Shan, Varagnolo and Vasserot [1,14,17].Our work is motivated by work of Brundan [3], which shows that the 2-category introduced in [5] can be defined as in [13], where the functor F i is the right dual of E i (by definition), but there is no a priori connection between F i and the left dual of E i . In order to define a duality functor, we must thus choose an isomorphism of E i to the right dual of F i (that is, of F i with the left dual of E i ).In [3, 1.2], one such isomorphism is defined, which matches the choice implicit in [5]. Unfortunately, as [5, (2.5)] shows, this duality is usually not cyclic; in particular, it is not for a generic choice of parameters, or for the choice which is most important in geometric and representation theoretic applications, such as [4,15].In this paper, we introduce a 2-category U cyc Q (g) for an arbitrary choice of...

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Khovanov homology is a skew Howe 2–representation of categorified quantum 𝔰𝔩_m

Cited by 45 publications

References 105 publications

A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

Functoriality of colored link homologies

Cyclicity for categorified quantum groups

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Khovanov homology is a skew Howe 2–representation of categorified quantum 𝔰𝔩m

Cited by 45 publications

References 105 publications

A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

Functoriality of colored link homologies

Cyclicity for categorified quantum groups

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Khovanov homology is a skew Howe 2–representation of categorified quantum 𝔰𝔩_m