2-Kac-Moody algebras

By Raphaël Rouquier. 2008

552 citations recorded by the source corpus.

We construct a 2-category associated with a Kac-Moody algebra and we study its 2-representations. This generalizes earlier work with Chuang for type A. We relate categorifications relying on K_0 properties and 2-representations.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7c43dcb01854186b4db5cffbdbafc90ea256c331

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Higher Dimensional Algebra VII: Groupoidification (2009, unknown): “algebra Uqgknown as a ‘quantum group’. There has been a lot of work attempting to categorify quantum groups, from the early ideas of Crane and Frenkel [11], to the work of Khovanov, Lauda and Rouqier [26, 27, 37], and beyond. Here we sketch how to groupoidify, not the whole quantum group, but only i”
Homological properties of finite type Khovanov-Lauda-Rouquier algebras (2012, background): “1i in degree −αi k ·αi k+1 . There’s also an anti-automorphism T : Hα →Hα which ﬁxes all the generators. Here are a fewotherbasic facts aboutthe structureofthese algebras established in [KL1, KL2] or [R1]. Fix once and for all a reduced expression for each w ∈Sn and let τw be the corresponding produ”
Additive versus abelian 2-representations of fiat 2-categories (2011, methodology): “ry. In [MM] we worked with abelian 2-representations of C. In the present paper we would like to slightly change the approach and start with additive 2-representations (as was originally suggested in [Ro2]). We also formulate a direct 2-functorial link between these two classes of representations. A”
An approach to categorification of some small quantum groups II (2012, methodology): “ universal enveloping algebra u+(sl 2) ∼=Fp[E]/(Ep). 4 The p-DG KLR algebra 4.1 p-derivations on the KLR algebra The KLR algebras. The KLR algebras associated with any Cartan datum were introduced in [20, 21, 41] as a generalization of nilHecke algebras to categorify one-half of the corresponding qu”
Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words (2009, unknown): “R T ] 1 0 Se p”
Erratum to: Hecke algebras, finite general linear groups, and Heisenberg categorification (2011, background): “functors, that is, functors that are both left and right adjoint to each other. The importance of such functors in low dimensional topology was emphasized in [12] and more recently in subsequent work [3, 13, 14, 16] on categoriﬁed quantum groups. In the last section of this paper, we give some examp”
Deformations of colored sl(N) link homologies via foams (2015, methodology): “d direct summand. Our main tool is the slN foam 2-category NFoam constructed in [QR], as well as its relation to the Khovanov-Lauda[KhL3]diagrammaticcategoriﬁcationofquantumslm (see alsoworkofRouquier[Rou]). The former can be viewed as the universal framework for the deﬁnition of categoriﬁed slN Res”
Highest weight categories arising from Khovanov's diagram algebra III: category O (2008, methodology): “ (5.3)–(5.4) and theadjunction (Fi,Ei◦D −1 i Di+1h1i) fromLemma 3.3 make the graded abelian category Rep(K(m,n;I)) into an integrable representation of the 2-Kac-Moody algebra A(slI+) in the sense of [R] (except that we have interchanged the roles of Ei and Fi). This is easy to verify using Theorem ”
Crystals from categorified quantum groups (2009, background): “Kang and Kashiwara show that functors lifting the action of Ei and Fi in Uq(g) are biadjoint, showing that cyclotomic quotients categorify V (Λ) as Uq(g)-modules and give a 2-representation in the sense of Rouquier [53].”
Heisenberg categorification and Hilbert schemes (2010, result): “KL1, KL2, KL3] deﬁne a 2-category U(g) whose Grothendieck group is isomorphic to (an integral version of) the quantized enveloping algebra. There is also independent and very similar work of Rouquier [R] in the same direction. In [CKL2] the authors, together with Joel Kamnitzer, construct a “geometr”
Affine highest weight categories and affine quasihereditary algebras (2014, background): “so (9.10) follows from the deﬁnition of µ, since h1eu′ 1e ′u′ 2e= h1e. 10. Examples Let P be the class of graded polynomial algebras, cf. §5.2. 10.1. KLR algebras. Khovanov-Lauda [27,28] and Rouquier [45] have introduced a family of algebras Rα(C) labeled by a generalized Cartan matrix Cand an eleme”