From Homological Algebra To Topology via Type B Zigzag Algebra and Heisenberg Algebra
We construct a faithful categorical action of the type $B$ braid group on the bounded homotopy category of finitely generated projective modules over a finite dimensional algebra which we call the type $B$ zigzag algebra. This categorical action is closely related to the action of the type $B$ braid group on curves on the disc. Thus, our exposition can be seen as a type $B$ analogue of the work of Khovanov-Seidel. Moreover, we relate our topological (respectively categorical) action of the type $B$ Artin braid group to their topological (respectively categorical) action of the type $A$ Artin braid group. Then, we prove Rouquier's conjecture \cite[Conjecture 9.8]{Rouq} on the faithfulness of Type $B$ $2$-braid group on Soergel category following the strategy used by Jensen's master with the diagrammatic tools from Elias-Williamson. In the final part of the thesis, we produce a graded Fock vector in the Laurent ring $\Z[t,t^{-1}]$ for a crossingless matching using Heisenberg algebra. We conjecture that the span of such vectors forms a Temperley-Lieb representation, and hence, a new presentation of Jones polynomial can be obtained.
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