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

Abstract

We construct a finite-dimensional quiver algebra from the non-simply laced type $B$ Dynkin diagram, which we call type $B$ zigzag algebra. This leads to a faithful categorical action of the type $B$ Artin (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.