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

We construct a finite dimensional quiver algebra from the non-simply laced type \(B\) Dynkin diagram, which we call the type \(B\) zigzag algebra. This leads to a faithful categorical action of the type \(B\) braid group \(\mathcal{A}(B)\), acting on the homotopy category of its projective modules....

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Bibliographic Details
Published in:arXiv.org 2023-02
Main Authors: Heng, Edmund, Nge, Kie Seng
Format: Article
Language:English
Subjects:
Algebra
Braid theory
Braiding
Topology
Online Access:Get full text
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Summary:We construct a finite dimensional quiver algebra from the non-simply laced type \(B\) Dynkin diagram, which we call the type \(B\) zigzag algebra. This leads to a faithful categorical action of the type \(B\) braid group \(\mathcal{A}(B)\), acting on the homotopy category of its projective modules. This categorical action is also closely related to the topological action of \(\mathcal{A}(B)\), viewed as mapping class group of the punctured disc -- hence our exposition can be seen as a type \(B\) analogue of Khovanov-Seidel's work in arXiv:math/0006056v2. Moreover, we show that certain category of bimodules over our type \(B\) zigzag algebra is a quotient category of Soergel bimodules, resulting in an alternative proof to Rouquier's conjecture on the faithfulness of the 2-braid groups for type \(B\).
ISSN:2331-8422