Shard modules

Abstract

Motivated by the goal of studying cluster algebras in infinite type, we study the stability domains of modules for the preprojective algebra in the corresponding infinite types. Specifically, we study real bricks: those modules whose endomorphism algebra is a division ring and which have no self-extensions. We define "shard modules" to be those real bricks whose stability domain is as large as possible (meaning, of dimension one less than the rank of the preprojective algebra). We show that all real bricks are obtained by applying the Baumann-Kamnitzer reflection functors to simple modules, and we give a recursive formula for the stability domain of a real brick. We show that shard modules are in bijection with Nathan Reading's "shards", and that their stability domains are the shards; we also establish many foundational results about shards in infinite type which have not previously appeared in print. With an eye toward applications to cluster algebras, our paper is written to handle skew-symmetrizable as well as skew-symmetric exchange matrices, and we therefore discuss the basics of the theory of species for preprojective algebras. We also give some counterexamples to show ways in which infinite type is more subtle than the well-studied finite type cases.