Character Varieties and the Moduli of Quiver Representations

TL;DR

This paper introduces a framework for G-valued quiver representations, explores their quotient spaces, and provides criteria for their moduli spaces to admit deformation retractions and embeddings into classical moduli spaces.

Contribution

It generalizes G-valued representations of finitely generated groups to quivers, establishes conditions for deformation retractions, and links quiver moduli spaces to traditional representations.

Findings

Quiver moduli spaces admit strong deformation retractions to compact quotients.

Conditions are identified for embedding into classical moduli spaces.

Quiver representations can be generalized to G-valued cases with relations.

Abstract

Let G be a Lie group and Q a quiver with relations. In this paper, we define G-valued representations of Q which directly generalize G-valued representations of finitely generated groups. Although as G-spaces, the G-valued quiver representations are more general than G-valued representations of finitely generated groups, we show by collapsing arrows that their quotient spaces are equivalent. We then establish a general criterion for the moduli of G-valued quiver representations with relations to admit a strong deformation retraction to a compact quotient by pinching vertices on the quiver. This provides two different generalizations of main results in our previous work. Lastly, we establish quiver theoretic conditions for the moduli spaces of GL(n,C) and SL(n,C)-valued quiver representations to embed into traditional moduli spaces of quiver representations having constant dimension vector.