Quiver Hecke Algebras and 2-Lie Algebras

By Raphaël Rouquier. 2011

192 citations recorded by the source corpus.

We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f892c7a2832e4ee40d4bf9743e71761825bc54c

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Citations in the corpus

Finite Dimensional Representations of Khovanov-Lauda-Rouquier algebras I: Finite Type (2012, background): “in [KL2] shows that the choice of total ordering on I is irrelevant (and indeed, it is included only to match up with the geometric picture in the simply-laced case presented in [VV] and [R2]).”
Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras (2013, background): “tes the nondegenerate bilinear form on U− q (g) deﬁned in [16, § 3]. The Khovanov-Lauda-Rouquier algebras also categorify the global bases. For the deﬁnition of global bases, see [16]. Theorem 1.2.3 ([32, 30]). Assume that Ais symmetric and that Qi,j(u,v) is a polynomial in u−v. Then under the isomo”
On uniqueness of tensor products of irreducible categorifications (2013, unknown): “By [Rou12, 4.25], this universal categoriﬁcation is the category of projective modules over a deformed cyclotomic quotient B ( ν ), and the base change precisely consists of killing the deformation parameters z i to obtain the category of projective modules over R ν .”
Homological properties of finite type Khovanov-Lauda-Rouquier algebras (2012, background): “ a unique algebra isomorphism γ: f →∼ Q(q) ⊗ Z[q,q−1][Proj(H)], θi→[Hαi], where θi is the generator of f corresponding to simple root αi. Assuming the ground ﬁeld Kis of characteristic zero, Rouquier [R2] and Varagnolo and Vasserot [VV] have shown further that this algebraic categoriﬁcation of f is ”