Gabriel’s Theorem

Abstract

We consider representation of a quiver without oriented cycles, then its path algebra is finite-dimensional. We determine when the path algebra, or the quiver, has finite representation type, this is answered completely by Gabriel’s theorem. Namely the quiver has finite representation type if and only if its underlying graph is the disjoint union of Dynkin diagrams of types A, D and E. In particular the representation type does not depend on the orientation of the arrows, and not on the coefficient field. Moreover, if a quiver has finite representation type, then the indecomposable representations are parametrized by the set of positive roots associated to the underlying graph. We give a complete proof, which is elementary, using only the tools we have developed so far.