Extriangulated structures from stability conditions and application to braid representations

TL;DR

This paper explores how Bridgeland stability conditions induce extriangulated structures in triangulated categories and applies this framework to relate braid group representations, including the Burau and Lawrence-Krammer-Bigelow representations.

Contribution

It introduces a novel approach to constructing extriangulated structures from stability conditions and connects these to braid group representations.

Findings

Established a link between stability conditions and extriangulated structures.

Derived relations between different braid group representations.

Provided a new categorical perspective on braid representations.

Abstract

We use the notion of Bridgeland stability condition and its associated metric to endow triangulated categories with extriangulated structures and study their extriangulated Grothendieck groups. This study is motivated by Khovanov-Seidel's categorification of the Burau representation, from which can be extracted the Lawrence-Krammer-Bigelow representation, providing a relation between these two faithful representations of the braid groups.