Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also [16]) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel [29], whose triangulated structure goes back to Auslander-Reiten’s work [6], [44], [7].

/pdf/on-triangulated-orbit-categories-4k4m3u3a5b.pdf

On triangulated orbit categories.

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.

2-Kac-Moody algebras

/pdf/a-diagrammatic-approach-to-categorification-of-quantum-28fc3fedas.pdf

A diagrammatic approach to categorification of quantum groups II

To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ℙ
 r
 , where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=?⊕?*, then the algebras Hκ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let $Γ=S

 n
 , the Weyl group of ?=??
 n
 . We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)?, the algebra of invariant polynomial differential operators on ??
 n
 , to the algebra of S

 n
 -invariant differential operators with rational coefficients on the space ℂ
 n
 of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂ
 n
 , with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)? ↠ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ=S

 n
 . This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of ‘quantum’ Hamiltonian reduction. In the ‘classical’ limit κ→∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of S

 n
 . Moreover, we prove that the algebra $H∞ is isomorphic to the endomorphism algebra of that vector bundle.

Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

We define and study sl2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show

http://annals.math.princeton.edu/wp-content/uploads/annals-v167-n1-p05.pdf

Derived equivalences for symmetric groups and sl2-categorification

We define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.

/pdf/dimensions-of-triangulated-categories-4asjsvvmwo.pdf

Dimensions of triangulated categories

Presentations a la Coxeter are given for all irreducible nite com plex re ection groups They provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite Coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra Hecke algebras We introduce monodromy representations of the braid groups which factorize through the Hecke algebras extending results of Cherednik Opdam Kohno and others Summary Introduction Complex re ection groups and their presentations A Background from complex re ection groups B Presentations Braid groups and their diagrams A Generalities about hyperplane complements B Generalities about the braid groups C The braid diagrams Proofs of the main theorems for the braid groups B de e r A Notation and prerequisites B Computation of B de e r and of its center for d C Computation of B e e r and of its center Hecke algebras A Background from di erential equations and monodromy B A family of monodromy representations of the braid group C Hecke algebras Diagrams and tables Appendix Generators of the monodromy around a divisor Appendix tables to Mathematics Subject Classi cation Primary G We thank Jean Michel Peter Orlik Pierre Vogel for useful conversations and the Isaac Newton Institute for its hospitality while the last version of this manuscript was written up The second named author gratefully acknowledges nancial support by the Fondation Alexander von Humboldt for his stays in Paris Michel Brou e Gunter Malle Raphael Rouquier

/pdf/complex-reflection-groups-braid-groups-hecke-algebras-4uuj7frs6k.pdf

Complex reflection groups, braid groups, Hecke algebras

We study the category O of representations of the rational Cherednik algebra AW attached to a complex reflection group W.W e construct an exact functor, called Knizhnik-Zamolodchikov functor: O → HW -mod, where HW is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equiv- alence between O/Otor, the quotient of O by the subcategory of AW -modules supported on the discriminant, and the category of finite-dimensional HW - modules. The standard AW -modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of "cells", provided W is a Weyl group and the Hecke algebra HW has equal parameters. We prove that the category O is equivalent to the module category over a finite dimensional algebra, a generalized "q-Schur algebra" associated to W.

/pdf/on-the-category-o-for-rational-cherednik-algebras-9h43pkdef3.pdf

Raphaël Rouquier

Papers

2-Kac-Moody algebras

Derived equivalences for symmetric groups and sl2-categorification

Dimensions of triangulated categories

Complex reflection groups, braid groups, Hecke algebras

On the category O for rational Cherednik algebras