• Corpus ID: 17066586

Categorification of the braid groups

  title={Categorification of the braid groups},
  author={Raphael Rouquier},
  journal={arXiv: Representation Theory},
  • R. Rouquier
  • Published 30 September 2004
  • Mathematics
  • arXiv: Representation Theory
We construct a categorification of the braid groups associated with Coxeter groups inside the homotopy category of Soergel's bimodules. Classical actions of braid groups on triangulated categories should come from an action of this monoidal category. We construct representations of this monoidal category on category O of a complex semi-simple Lie algebra and on constructible sheaves over flag varieties. We also consider general constructions of self-equivalences as reflections around another… 

Braid cobordisms, triangulated categories, and flag varieties

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also

Braid group actions via categorified Heisenberg complexes

Abstract We construct categorical braid group actions from 2-representations of a Heisenberg algebra. These actions are induced by certain complexes which generalize spherical (Seidel–Thomas) twists

Braid groups and Kleinian singularities

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 2-spherical objects in a derived category. Our major tool is the Garside structure on braid groups

Diagrammatics for Soergel Categories

This paper presents the monoidal category of Soergel bimodules, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.

A Diagrammatic Temperley-Lieb Categorification

  • Ben Elias
  • Mathematics
    Int. J. Math. Math. Sci.
  • 2010
The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group, and it is demonstrated how further subquotients of this category will categorify the irreducible modules of the Temperley-Lieb algebra.

Diagrammatics for Coxeter groups and their braid groups

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter

Curves in the disc, the type B braid group, and the type B zigzag algebra

We construct a faithful categorical action of the type B braid group on the bounded homotopy category of finitely generated projective module over a finite dimensional algebra which we call the type

Equivariant coherent sheaves, Soergel bimodules, and categorification of affine Hecke algebras

We give a description of certain categories of equivariant coherent sheaves on Grothendieck's resolution in terms of the categorical affine Hecke algebra of Soergel. As an application, we deduce a

Categorical diagonalization of full twists

We conjecture that the complex of Soergel bimodules associated with the full twist braid is categorically diagonalizable, for any finite Coxeter group. This utilizes the theory of categorical

Knot homology via derived categories of coherent sheaves I, sl(2) case

Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to sl(2) and its standard representation. Our construction is related to



Braid group actions on derived categories of coherent sheaves

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is Kontsevich's homological mirror conjecture, together

Quivers, Floer cohomology, and braid group actions

We consider the derived categories of modules over a certain family A_m of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of

Action du groupe des tresses sur une catégorie

Abstract. We describe how to define by generators and relations an action of the braid group on a category. More generally, how to define a functor from a generalized positive braids monoid,

Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors

To each generic tangle projection from the three-dimensional real vector space onto the plane, we associate a derived endofunctor on a graded parabolic version of the Bernstein-Gel'fand category

Combinatorics of Harish-Chandra modules

These lectures survey recent work on the combinatorics of certain infinite dimensional representations of complex semisimple Lie algebras. Their focus is not on understanding the irreducible objects

Twisting Functors on O

This paper studies twisting functors on the main block of the Bernstein-Gelfand-Gelfand category O and describes what happens to (dual) Verma modules. We consider properties of the right adjoint

Translation Functors and Equivalences of Derived Categories for Blocks of Algebraic Groups

We prove the existence of many self-equivalences of the derived categories of blocks of reductive groups in prime characteristic that are not induced by self-equivalences of the module categories of

Algebraic construction of contragradient quasi-Verma modules in positive characteristic

In the present paper we investigate a new class of infinite-dimensional modules over the hyperalgebra of a semi-simple algebraic group in positive chararacteristic called quasi-Verma modules. We

Wall-crossing functors and -modules

We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D-modules. This functorial machinery is then used to prove