The Hall Module of an Exact Category with Duality

We construct from a finitary exact category with duality A a module over its Hall algebra, called the Hall module, encoding the first order self-dual extension structure ofA. 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 σ-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. Introduction Let A be an abelian category with finite Hom and Ext sets, called finitary below. In [21] Ringel defined the Hall algebra HA, an associative algebra whose multiplication encodes the first order extension structure of A. There is also a coalgebra structure on HA which, if A is hereditary, makes HA into a (twisted) bialgebra [11]. The category RepFq(Q) of representations of a quiver over a finite field is a finitary hereditary category. The corresponding Hall algebra HQ contains a subalgebra isomorphic to the positive part of the quantum Kac-Moody algebra associated to Q, specialized at √ q [22], [11]. A second example of a finitary hereditary category is the category of coherent sheaves over a smooth projective curve X defined over Fq. In the simplest case, X = P , the Hall algebra contains a subalgebra isomorphic to a positive part of the quantum affine algebra U√q(ŝl2) [14]. More generally, Hall algebras can be defined for exact categories [13] and often behave similarly to quantum nilpotent groups [3]. In this paper we introduce an analogue of the Hall algebra when objects of A are allowed to carry non-degenerate quadratic forms. To do this, we work with exact categories with duality. In this setting, a self-dual object is an object of A together with a symmetric isomorphism with its dual. Instead of using extensions to define an algebra we use the self-dual extension structure of A to define a HA-module, called the Hall module and denoted by MA; see Theorem 2.4. More precisely, a self-dual exact sequence is a diagram 0 → U → M 99K N → 0 presenting U ∈ A as an isotropic subobject of the self-dual object M and presenting the self-dual object N as the isotropic reduction of M by U . We also show that MA is naturally a HA-comodule. It is important to be able to twist the Hall module so as to obtain modules over the Ringel-twisted Hall algebra. For example, the connection between quantum groups and Hall modules described below is most clear when using this twist. In Theorem 2.6 we construct such a module twist using an integer valued function E on the Grothendieck group of A. The function E plays the role of the Euler form for categories with duality and is therefore of independent Date: July 14, 2014. 2010 Mathematics Subject Classification. Primary: 16G20 ; Secondary 17B37.