ncts

In this series of three lectures, we shall introduce basic knowledge on representations of finite-dimensional algebras. First, we give a description of algebras and representations by quivers with relations, and then describe representation-finite algebras by Dynkin diagrams. This is the famous Gabriel classification theorem and closely related to root systems of simple Lie algebras. Finally, we discuss a special class of representations of algebras, called tilting modules, and mention some of their homological properties. Tilting modules provide a class of derived equivalences and appear often in categories of highest weight categories of representations of Lie algebras, while the bridge between tilting modules and highest weight categories is the finite-dimensional quasi-hereditary algebras introduced by Cline-Parshall-Scott.