Quivers and the Euclidean group

TL;DR

This paper establishes a deep connection between the representation theory of the Euclidean group and quiver varieties, revealing its wild nature but also conditions for finiteness of indecomposables.

Contribution

It demonstrates an equivalence between Euclidean group representations and preprojective algebra representations, and links their moduli spaces to Nakajima's quiver varieties.

Findings

Euclidean group representations are equivalent to preprojective algebra representations.

Moduli spaces of these representations are Nakajima quiver varieties.

The Euclidean group has wild representation type, but restrictions lead to finitely many indecomposables.

Abstract

We show that the category of representations of the Euclidean group of orientation-preserving isometries of two-dimensional Euclidean space is equivalent to the category of representations of the preprojective algebra of infinite type A. We also consider the moduli space of representations of the Euclidean group along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. Using these identifications, we prove various results about the representation theory of the Euclidean group. In particular, we prove it is of wild representation type but that if we impose certain restrictions on weight decompositions, we obtain only a finite number of indecomposable representations.