Degenerations for representations of extended Dynkin quivers

Abstract

Let A be the path algebra of a quiver of extended Dynkin type ${\tilde{\mathbb{A}}}_n,{\tilde{\mathbb{D}}}_n,{\tilde{\mathbb{E}}}_6,{\tilde{\mathbb{E}}}_7$ or ${\tilde{\mathbb{E}}}_8$ . We show that a finite dimensional A-module M degenerates to another A-module N if and only if there are short exact sequences $0\to U_i\to M_i\to V_i\to 0$ of A-modules such that $M=M_1$ , $M_{i+1}=U_i\oplus V_i$ for $1\le i\le s$ and $N=M_{s+1}$ are true for some natural number s.