Related papers: Quiver Hecke algebras and categorification
In this paper we continue to study Belavin-Drinfeld cohomology introduced in arXiv:1303.4046 [math.QA] and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra. Here we compute…
We introduce a new family of superalgebras which should be considered as a super version of the Khovanov-Lauda-Rouquier algebras. Let $I$ be the set of vertices of a Dynkin diagram with parity. To this data, we associate a family of graded…
We propose a categorification of the cyclotomic Hecke algebra in terms of the equivariant K-theory of the framed matrix factorizations. The construction generalizes the earlier construction of the authors for a categorification of the…
The elements of the wide class of quantum universal enveloping algebras are prooved to be Hopf algebras $H$ with spectrum $Q(H)$ in the category of groups. Such quantum algebras are quantum groups for simply connected solvable Lie groups…
Lead by examples we introduce the notions of Hopf algebra and quantum group. We study their geometry and in particular their Lie algebra (of left invariant vectorfields). The examples of the quantum sl(2) Lie algebra and of the quantum…
Homology theories categorifying quantum group link invariants are known to be governed by the representation theory of quiver Hecke algebras, also called KLRW algebras. Here we show that certain cylindrical KLRW algebras, relevant in…
We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid --- the category of permutation representations of a finite group. As an immediate consequence, we obtain a…
The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke…
We prove that the quotient of the group algebra of the braid group on 5 strands by a generic cubic relation has finite rank. This was conjectured in 1998 by Brou\'e, Malle and Rouquier and has for consequence that this algebra is a flat…
We prove some recent conjectures of Khovanov-Lauda concerning the categorification of one-half of the quantum group associated with a simply laced Cartan datum.
We introduce generalizations of Kac-Moody 2-categories in which the quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced by the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka.
We determine explicitly the center of the twisted graded Hecke algebras associated to homocyclic groups. Our results are a generalization of formulas by M. Douglas and B. Fiol in [J. High Energy Phys. 2005 (2005), no. 9, 053, 22 pages,…
We discuss some consequences of the invertibility of Rickard complexes in a categorified quantum group. Results include a description of reflection functors for quiver Hecke algebras and a theory of restricting categorical representations…
This paper is a survey on the representation theory of Hecke algebras, Ariki-Koike algebras and connections with quantum group.
We introduce Hecke algebras associated to discrete quantum groups with commensurated quantum subgroups. We study their modular properties and the associated Hecke operators. In order to investigate their analytic properties we adapt the…
This article is an extended version of the minicourse given by the second author at the summer school of the conference "Interactions of quantum affine algebras with cluster algebras, current algebras and categorification", held in June…
The so called quantized algebras of functions on affine Hecke algebras of type A and the corresponding q-Schur algebras are defined and their irreducible unitarizable representations are classified.
This survey article, is written as an extended note and supplement of my lectures in the current developments in mathematics conference in 2015. We discuss some recent developments on the conjugacy classes of affine Weyl groups and $p$-adic…
These are notes for a minicourse on Hall algebras given at the ICTP in Trieste in January 2006. After giving the definition and first properties of Hall algebras, we study in some details the classical Hall algebra, the Hall algebra of…
We categorify the Hecke algebra with parameters 1 and v using a variation of the category of Soergel bimodules.