Qurves and Quivers

In this paper we associate to a qurve A (formerly known as a quasi-free or formally smooth algebra) the one-quiver Q(A) and dimension vector a(A). This pair contains enough information to reconstruct for all natural numbers n the GL(n)-etale local structure of the representation scheme rep(n,A). In an appendix we indicate how one might extend this to qurves over non-algebraically closed fields. Further, we classify all finitely generated groups G such that the group algebra lG is an l-qurve. If char(l)=0 these are exactly the virtually free groups. We determine the one-quiver setting in this case and indicate how it can be used to study the finite dimensional representations of virtually free groups. As this approach also applies to fundamental algbras of graphs of separable l-algebras, we state the results in this more general setting.