# Quiver grassmannians, quiver varieties and the preprojective algebra

*Algebra/Geometry/Topology Seminar*

*by Alistair Savage*

*Institution:*Ottawa

*Date: Mon 31st May 2010*

*Time: 1:00 PM*

*Location: Babel Middle Theatre*

*Abstract*: Quivers play an important role in the representation theory of algebras with key ingredients of the theory 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 this talk, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are isomorphic 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 isomorphic to the Demazure quiver varieties, and others which are isomorphic to the graded/cyclic quiver varieties defined by Nakajima. The Demazure quiver grassmannians also allow us to construct 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. This is joint work with Peter Tingley.

*For More Information:* craigw@ms.unimelb.edu.au