Singular Soergel Bimodules and Their Diagrammatics

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of Harish–Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.

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Koszul duality and Soergel bimodules for dihedral groups

Transactions of the American Mathematical Society ◽

10.1090/tran/7014 ◽

2017 ◽

Vol 370 (2) ◽

pp. 1251-1283

Author(s):

Marc Sauerwein

Keyword(s):

Koszul Duality ◽

Dihedral Groups ◽

Soergel Bimodules

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A Diagrammatic Temperley-Lieb Categorification

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2010/530808 ◽

2010 ◽

Vol 2010 ◽

pp. 1-47 ◽

Cited By ~ 1

Author(s):

Ben Elias

Keyword(s):

Symmetric Group ◽

Weyl Group ◽

Planar Graphs ◽

Monoidal Category ◽

Hecke Algebra ◽

Coxeter Complex ◽

Quotient Category ◽

Irreducible Modules ◽

Soergel Bimodules

The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We demonstrate how further subquotients of this category will categorify the irreducible modules of the Temperley-Lieb algebra.

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