Quiver grassmannians, quiver varieties and the preprojective algebra

Alistair Savage and Peter Tingley

Vol. 251 (2011), No. 2, 393–429

DOI: 10.2140/pjm.2011.251.393

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. In an appendix added after the first version of the current paper was released, we show how subsequent results of Shipman imply that the above homeomorphisms are in fact isomorphisms of algebraic varieties.

Keywords

quiver, preprojective algebra, quiver grassmannian, quiver variety, Kac–Moody algebra, Demazure module

Mathematical Subject Classification 2010

Primary: 16G20

Secondary: 17B10

Milestones

Received: 23 June 2010

Revised: 21 January 2011

Accepted: 31 January 2011

Published: 3 June 2011

Correction: 15 May 2015