Self-dual Hall modules

of the Dissertation Self-dual Hall Modules by Matthew Bruce Young Doctor of Philosophy in Mathematics Stony Brook University 2013 In the past twenty years Hall algebras have played an important role in many areas of mathematics and physics, including the theory of quantum groups and string theory. In its original setting the Hall algebra is constructed from a finitary exact category, the multiplication encoding the extension structure of the category. In this dissertation we introduce the Hall module of a finitary exact category with duality. The duality structure allows objects to carry sesquilinear forms and the action of the Hall algebra provides a categorical generalization of parabolic induction for the classical groups preserving a sesquilinear form. After developing the basic theory of Hall modules we study in detail Hall modules arising from the representation theory of a quiver with involution. In this setting we find a connection with twisted versions of quantum groups. We also study arithmetic properties of the Hall module structure constants.