Quiver representations and idempotent loops

Abstract

We study ${\mathbb{C}}^{\times }$-actions on quiver representations by adding idempotent loops. We construct a category mod $\hat{W}(\mathrm{Q})$ and relate it to $\mathbb{C}Q$ modules with ${\mathbb{C}}^{\times }$-actions. We further describe a subcategory ${\hat{W}}_T(\mathrm{Q})$ equivalent to the category of equivariant modules where ${\mathbb{C}}^{\times }$ acts on $\mathbb{C}Q$ via a character $T\in {\mathbb{Z}}^{Q_1}.$ This allows us to study ${\mathbb{C}}^{\times }$-actions on quiver Grassmannians and we can recover known combinatorial descriptions of their Euler characteristics via the category mod $\hat{W}(\mathrm{Q}).$ We directly relate morphisms of quivers to full subcategories of mod $\hat{W}(\mathrm{Q}).$ Finally we show that the representation theory of quivers with idempotents can be applied in a useful way to preprojective algebras. We show that this gives constructions of Galois covers and cluster characters used by Geiss, Leclerc, and Schröer.

38 pages. Part of the author's PhD thesis