Implicit structure in 2-representations of quantum groups

By Sabin Cautis, Aaron D. Lauda. 2011

73 citations recorded by the source corpus.

Given a strong 2-representation of a Kac–Moody Lie algebra (in the sense of Rouquier), we show how to extend it to a 2-representation of categorified quantum groups (in the sense of Khovanov–Lauda). This involves checking certain extra 2-relations, which are explicit in the definition by Khovanov–Lauda and, as it turns out, implicit in Rouquier’s definition. Some applications are also discussed.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c2a7a4370a2480f6f214cb13c4d143e7d9b9fcc1

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Citations in the corpus

Transitive 2-representations of finitary 2-categories (2014, methodology): “l order on h-weights. Let U be the 2-category categorifying the idempotent version U˙ of the universal enveloping algebra of g as deﬁned in [We, Deﬁnition 2.4] (the origins of this 2- category are in [CL], see also [KL, Ro1] for other variations). The categoriﬁcation statement is justiﬁed by [We, Th”
Canonical bases and higher representation theory (2012, background): “ degree 2d i and v degree 2d j. We will always assume that the leading order of Q ij in u is −c ij, and that Q ij(u,v) = Q ji(v,u). We let t ij = Q (−cij,0) ij = Q ij(1,0); by convention t ii = 1. In [CL], the coeﬃcients of this polynomial are denoted Q ij(u,v) = t iju−cij +t jiv−cji + X pdi+qdj=dic”
On uniqueness of tensor products of irreducible categorifications (2013, background): “eting deﬁnitions ofcategorical g-action, andit’snotclear thatthey give the same result. The classiﬁcation mentioned above in terms of minimal polynomials is known for the Cautis-Lauda 2-category from [CL] by [Weba, 1.12]; Rouquier has announced the same result for his 2-category, but the proof has y”
Quantizations of conical symplectic resolutions I: local and global structure (2012, methodology): “eﬁne a 2-categories HCg(µ) by replacing the diagonal and its vanishing ideal with that of the stable diagonal. The third author [Web, Theorem A] relates this construction to works by Cautis and Lauda [CL] and Nakajima [Nak98]. Proposition 6.20 (Webster) There is a 2-functor from the version of the 2”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, methodology): “WW i λ : FiEi1λ → 1λ{1 + (λ, αi)} GG i λ : EiFi1λ → 1λ{1− (λ, αi)} Here we follow the grading conventions in [12] which are opposite to those from [27] but line up nicely with the gradings on foams used later in the paper.”
On the definition of Kac–Moody 2-category (2015, methodology): “7) i λ := i λ •−hhi,λi if hhi,λi ≤ 0, (1.18) j i λ := j i λ −1 if i 6= j. (1.19) Now to prove the Main Theorem we must show that all of the deﬁning relations for the Khovanov-Lauda 2-category from [CL] are satisﬁed by the 2-morphisms in U(g) just introduced. First we will show that Lauda’s inﬁnite ”
Tensor product categorifications and the super Kazhdan-Lusztig conjecture (2013, methodology): “ta of an slI-categoriﬁcation as above is equivalent to the data of an integrable 2- representation of the Kac-Moody 2-category associated to slI in the sense of [R, Deﬁnition 5.1]; see also [KL2] and [CaL] for closely related notions. (We point out 10 JONATHAN BRUNDAN, IVAN LOSEV AND BEN WEBSTER als”