The Hall Algebras of Surfaces I

TL;DR

This paper explores the structure of Hall algebras associated with surfaces, providing explicit descriptions for disks with marked intervals and conjectures for general surfaces, linking to skein relations in knot theory.

Contribution

It offers the first explicit description of the Hall algebra for disks with marked intervals and proposes a conjectural framework for all surfaces, connecting to skein relations.

Findings

Explicit Hall algebra description for disks with marked intervals

Conjectural descriptions for general surfaces' Hall algebras

Graded HOMFLY-PT skein relation among arcs in Hall algebras

Abstract

We study the derived Hall algebra of the partially wrapped Fukaya category of a surface. We give an explicit description of the Hall algebra for the disk with m marked intervals and we give a conjectural description of the Hall algebras of all surfaces with enough marked intervals. Then we use a functoriality result to show that a graded version of the HOMFLY-PT skein relation holds among certain arcs in the Hall algebras of general surfaces.