On indecomposable modules over directed algebras

Abstract

Generalizing a result of Bongartz we show that any nonsimple indecomposable module over a finite-dimensional k k -algebra A A is an extension of an indecomposable and a simple module provided k k is a field with more than two elements and A A is representation directed. Our proof is based on fibre sums over simple modules and some known classification results on socle projective modules over peak algebras. In case the global dimension of A A is at most 2 our methods also yield a description of the dimension vectors of the indecomposable A A -modules by the roots of the associated quadratic form.