The Hall module of an exact category with duality
Young, Matthew B.
Mathematics - Representation Theory | Mathematics - Quantum Algebra
arxiv: Mathematics::Representation Theory | Condensed Matter::Mesoscopic Systems and Quantum Hall Effect
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.