arXiv subjects
Publications and source records attributed to Kie Seng Nge.
We show that certain category of bimodules over a finite dimensional quiver algebra known as type B zigzag algebra is a quotient category of the category of type B Soergel bimodules. This leads to an alternate proof of Rouquier's conjecture on the faithfulness of the 2-braid groups for type B.
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 $B$.
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 in arXiv:math/0006056. 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, that is Conjecture 3.8 in arXiv:math/0409593 on the faithfulness of Type $B$ $2$-braid group on Soergel category following the strategy used by Jensen's master's thesis with the diagrammatic tools from arXiv:1309.0865. In the final part of the thesis, we produce a graded Fock vector in the Laurent ring $\mathcal{Z}[t,t^{-1}]$ 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.