Quotient closed subcategories of quiver representations

Abstract

Let Q be a finite quiver without oriented cycles, and let k be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in the associated Coxeter group W_Q and the cofinite additive quotient-closed subcategories of the category of finite dimensional right modules over kQ. We prove this correspondence by linking these subcategories to certain ideals in the preprojective algebra associated to Q, which are also indexed by elements of W_Q.

35 pages; v2: added a section showing how the Le-diagram condition arises naturally from our viewpoint; v3: treat the case of hereditary algebras over a finite field