From Homological Algebra to Topology via Type B Zigzag Algebra and Heisenberg Algebra

TL;DR

This paper constructs a categorical action of the type B braid group on a homotopy category related to a new algebra, proving a conjecture on faithfulness and linking topology, algebra, and knot invariants.

Contribution

It introduces the type B zigzag algebra, establishes a faithful categorical action of the type B braid group, and relates it to topological actions and knot invariants.

Findings

Proves Rouquier's conjecture on faithfulness of the type B 2-braid group.

Constructs a graded Fock vector related to crossingless matchings.

Suggests a new Temperley-Lieb representation for Jones polynomial.

Abstract

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.