Quiver grassmannians, quiver varieties and the preprojective algebra

TL;DR

This paper explores quiver grassmannians related to preprojective algebras, establishing their homeomorphisms with various Nakajima quiver varieties and connecting these geometric objects to algebraic categories.

Contribution

It identifies new homeomorphisms between quiver grassmannians and Nakajima's quiver varieties, including Demazure and graded types, enriching the geometric understanding of preprojective algebra modules.

Findings

Quiver grassmannians of certain injective modules are homeomorphic to Nakajima's lagrangian quiver varieties.

Demazure quiver grassmannians describe injective objects in the category of locally nilpotent modules.

Connections are established between these geometric objects and Lusztig's constructions using projective modules.

Abstract

Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassmannians which are homeomorphic to the Demazure quiver varieties introduced by the first author, and others which are homeomorphic to the graded/cyclic quiver varieties defined by Nakajima. The Demazure quiver grassmannians allow us to describe injective objects in the category of locally nilpotent modules of the preprojective algebra. We conclude by relating our construction to a similar one of Lusztig using projectives in place of injectives.