Supercategorification of quantum Kac–Moody algebras II

Kang, Seok-Jίn; Kashiwara, Masaki; Oh, Se‐jin

doi:10.1016/j.aim.2014.07.036

Cited by 24 publications

(37 citation statements)

References 21 publications

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“…We would like to say finally that many of the general definitions in this article can be found in some equivalent form in many places in the literature. We were influenced especially by the work of Kang, Kashiwara and Oh in [KKO,Section 7]; see also [EL, Section 2]. Our choice of terminology is different.…”

mentioning

confidence: 99%

Monoidal Supercategories

Brundan

Ellis

2017

Commun. Math. Phys.

114

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This work is a companion to our article "Super Kac-Moody 2categories," which introduces super analogs of the Kac-Moody 2-categories of Khovanov-Lauda and Rouquier. In the case of sl 2 , the super Kac-Moody 2-category was constructed already in [A. Ellis and A. Lauda, "An odd categorification of Uq(sl 2 )"], but we found that the formalism adopted there became too cumbersome in the general case. Instead, it is better to work with 2-supercategories (roughly, 2-categories enriched in vector superspaces). Then the Ellis-Lauda 2-category, which we call here a Π-2-category (roughly, a 2category equipped with a distinguished involution in its Drinfeld center), can be recovered by taking the superadditive envelope then passing to the underlying 2-category. The main goal of this article is to develop this language and the related formal constructions, in the hope that these foundations may prove useful in other contexts.

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mentioning

confidence: 99%

Monoidal Supercategories

Brundan

Ellis

2017

Commun. Math. Phys.

114

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“…Because and the characters of V (λ) 1 and V (λ) −1 coincide for dominant weights (cf. [KKO14], [CHW14, Remark 2.5]), we have dim R π f ± = dim R π R V (λ) = dim R V (λ) 1 = dim R f ± 1 = ℓ n 2 gcd(2, ℓ) n 2 −n where f 1 is the (non-super) half small quantum group, i.e., f specialized at π = 1. The last equality is due to [Lu90b,Theorem 8.3(iv)].…”

Section: Letmentioning

confidence: 99%

Quantum supergroups VI: roots of 1

2019

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A quantum covering group is an algebra with parameters q and π subject to π 2 = 1 and it admits an integral form; it specializes to the usual quantum group at π = 1 and to a quantum supergroup of anisotropic type at π = −1. In this paper we establish the Frobenius-Lusztig homomorphism and Lusztig-Steinberg tensor product theorem in the setting of quantum covering groups at roots of 1. The specialization of these constructions at π = 1 recovers Lusztig's constructions for quantum groups at roots of 1.2010 Mathematics Subject Classification. Primary 17B37.

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“…We begin by defining various additional 2‐morphisms in

U (g)

. Definition We have the downward dots and crossings , which are the right mates of the upward dots and crossings:

The sign in is easily checked using the diagrammatics; see also [, Proposition 7.14]. Using and , we deduce:

…”

Section: More Generatorsmentioning

confidence: 99%

“…The sign in (2.2) is easily checked using the diagrammatics; see also [17,Proposition 7.14]. Using (1.10) and (1.11), we deduce: Sometimes we will use the following convenient shorthand for dotted bubbles for any n ∈ Z:…”

Section: More Generatorsmentioning

confidence: 99%

“…The only additional finite type possibilities come from odd

b_{n}

, that is, type

b_{n}

with the element of

I

corresponding to the short simple root chosen to be odd. For these, the Decategorification Conjecture may be deduced from . We hope that Webster's methods from can be extended to the super case to prove the Nondegeneracy Conjecture in general, but there is a great deal of work still to be done in order to see this through.…”

Section: Introductionmentioning

confidence: 99%

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