The Hall module of an exact category with duality

TL;DR

This paper constructs a Hall module from a finitary exact category with duality, linking it to quantum Kac-Moody algebras and providing explicit decompositions for finite type cases.

Contribution

It introduces the Hall module for categories with duality and establishes its structure as a module over a quantum algebra, with explicit decompositions for finite type quivers.

Findings

Hall module encodes self-dual extension structure

Hall module is a module over a quantum Kac-Moody algebra

Explicit decomposition for finite type quivers

Abstract

We construct from a finitary exact category with duality a module over its Hall algebra, called the Hall module, encoding the first order self-dual extension structure of the category. We study in detail Hall modules arising from the representation theory of a quiver with involution. In this case we show that the Hall module is naturally a module over the specialized reduced sigma-analogue of the quantum Kac-Moody algebra attached to the quiver. For finite type quivers, we explicitly determine the decomposition of the Hall module into irreducible highest weight modules.