The Hall module of an exact category with duality

Abstract

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 of A. 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.Date: November 9, 2018. 2010 Mathematics Subject Classification. Primary: 16G20 ; Secondary 17B37. Key words and phrases. Representations of quivers, Hall algebras, quantum groups. 1 2 M. B. YOUNGinterest. In Theorem 2.9, we prove an identity relating E, the Euler form and the stacky number of self-dual extensions in the case that A is hereditary. The proof develops some basic self-dual homological algebra and uses the combinatorics of self-dual analogues of Grothendieck's extensions panachées [12], [4].In Section 3 we study Hall modules arising from the representation theory of a quiver with contravariant involution (Q, σ). From the involution and a choice of signs we define a duality structure on Rep Fq (Q), with q odd. For particular signs, the self-dual objects coincide with the orthogonal and symplectic representations of Derksen and Weyman [7]. The module and comodule structures are incompatible in that M Q is not a Hopf module, even in a twisted sense. In Theorem 3.5, we instead show that the action and coaction of the simple representationsThe proof is combinatorial in nature and involves counting configurations of pairs of self-dual exact sequences, in the spirit of Green's proof of the bialgebra structure of the Hall algebra [11]. We describe in Theorem 3.10 the decomposition of M Q into irreducible highest weight B σ (g Q )-modules. The generators are cuspidal elements of M Q , i.e. elements that are annihilated by the coaction of each [S i ]. The proof relies on a canonically defined non-degenerate bilinear form on the Hall module and a characterization of irreducible highest weight modules due to Enomoto and Kashiwara [9].In Section 4 we restrict attention to finite type quivers. Unlike ordinary quiver representations, self-dual representations in general have non-trivial F q /F q -forms. We extend results of [7] (over algebraically closed fields) to explicitly describe all such forms and classify the indecomposable self-dual F q -representations. The classification is summarized in Theorem 4.2 where a partial root theoretic interpretation of the indecomposables is given. The main application of this result is to the explicit decomposition of Hall modules of finite type quivers into irreducible highest weight B σ (g Q )-modules; see Theorems 4.4 and 4.6. The generators are written as alternating sums of the F q /F q -forms of self-dual indecomposables.In [8], Enomoto proved a result related to Theorems 3.5 and 3.10, showing that induction and restr...

Self-dual quiver moduli and orientifold Donaldson-Thomas invariants

“…It was shown in [40,Proposition 3.3] that E(U ) depends only on dim U and so defines a function E : Λ Q → Z. Explicitly, from loc.…”

“…We want to construct a lift of the homomorphism H to the self-dual setting. To do this, we first recall the definition of the Hall module M Q associated to the category Rep Fq (Q) with fixed duality structure [40]. It is the Q-vector space generated by symbols [M ] indexed by isometry classes of self-dual F q -representations of Q.…”

“…The analogue of the Hall algebra for self-dual representations was introduced in [40]. There the self-dual extension structure of the representation category, controlling three term sequences consisting of a self-dual representation, an isotropic subrepresentation and the resulting self-dual quotient, was used to construct a module over the Hall algebra, called the Hall module.…”

Relative 2–Segal spaces

“…The input for the R • -construction is a proto-exact category with duality which satisfies a reduction assumption. In the case of exact categories the R • -construction categorifies the Hall algebra representations of [43], [10], [46], [47] while for the proto-exact category Rep F1 (Q) of representations of a quiver over F 1 we obtain new modules over Szczesny's combinatorial Hall algebras [41]. The latter modules will be the subject of future work.…”

Degenerate versions of Green's theorem for Hall modules

“…While such representations do not form an abelian category in any natural way, there is a modification of the Hall algebra construction which produces a module over the Hall algebra, called the Hall module [30]. In various settings, Hall modules have been shown to be related to canonical bases [5], [27], representations of quantum groups [28] and Donaldson-Thomas theory with classical structure groups [29], [7]. While the Hall module M Q,Fq of Rep Fq (Q) has a natural comodule structure, the most naive analogue of Green's theorem does not hold: M Q,Fq is not a Hopf module over H Q,Fq .…”