  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.
