Adjoint Action of Automorphism Groups on Radical Endomorphisms, Generic Equivalence and Dynkin Quivers

Abstract

Let Q be a connected quiver with no oriented cycles, k the field of complex numbers and P a projective representation of Q. We study the adjoint action of the automorphism group Aut kQ P on the space of radical endomorphisms radEnd kQ P . Using generic equivalence, we show that the quiver Q has the property that there exists a dense open Aut kQ P -orbit in radEnd kQ P , for all projective representations P , if and only if Q is a Dynkin quiver. This gives a new characterisation of Dynkin quivers.