The Gabriel-Roiter submodules of simple homogeneous modules

Abstract

Let Λ \Lambda be a connected tame hereditary algebra over an algebraically closed field. We show that if Λ = k Q \Lambda =kQ is of type A ~ n \widetilde {\mathbb {A}}_n , D ~ n \widetilde {\mathbb {D}}_n , E ~ 6 \widetilde {\mathbb {E}}_6 or E ~ 7 \widetilde {\mathbb {E}}_7 , then every Gabriel-Roiter submodule of a quasi-simple module of rank 1 1 (i.e. a simple homogeneous module) has defect − 1 -1 . In particular, any Gabriel-Roiter submodule of a simple homogeneous module yields a Kronecker pair, and thus induces a full exact embedding of the category mod ⁡ k A ~ 1 \operatorname {mod} k\widetilde {\mathbb {A}}_1 into mod ⁡ Λ \operatorname {mod}\Lambda , where A ~ 1 \widetilde {\mathbb {A}}_1 is the Kronecker quiver. Consequently, we obtain that all quasi-simple modules are Gabriel-Roiter factor modules.