On the classification of algebraic structures using quivers

Abstract

There are several ways used to classify algebraic structures, however in this project we classify path algebras of a connected quiver using Gabriel’s Theorem. We also show that the isomorphic classes of the indecomposable representations are in bijection with the set of positive roots of the root system of the Dynkin diagrams. We construct the Auslander-Reiten quiver of type An, Dn, and En using the knitting algorithm from which we obtain the indecomposable representations, irreducible morphisms and Auslander-Reiten sequences which are also used to classify path algebras. These are also used as building blocks for any arbitrary representation, morphisms and short exact sequences.