Quiver representations and idempotent loops

Abstract

We study $\mathbb{C}^{\times}$-actions on quiver representations by adding idempotent loops. We construct a category mod $\widehat{\operatorname{W}}(\mathrm{Q})$ and relate it to $\mathbb{C}Q$ modules with $\mathbb{C}^{\times}$-actions. We further describe a subcategory $\widehat{\operatorname{W}}_{T}(\mathrm{Q})$ 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 $\widehat{\operatorname{W}}(\mathrm{Q}).$ We directly relate morphisms of quivers to full subcategories of mod $\widehat{\operatorname{W}}(\mathrm{Q}).$ 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 Schr\"oer.