Categorical action of the braid group of the cylinder

Abstract: Using the various definitions of the classical braid group (diagrammatic presentation, mapping class group, fundamental group of configuration space), Khovanov and Seidel constructed in their seminal article of 2000 an action of the braid group on a category of algebraic nature that categorifies the Burau representation. They proved the faithfulness of this action through the study of curves in a punctured disk (while Burau representation is not faithful for braids with five strands or more). In an article with Anne-Laure Thiel and Emmanuel Wagner, we extended this result to the braid group of the cylinder. The work of Khovanov and Seidel also had a faithful symplectic categorical action that we generalise to our cylindrical case in a joint work in progress. In this talk, I will explain some ingredients and strategy involved to obtain these categorical actions and faithfulness results.

Mathematics

Audience: researchers in the topic

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