Representation–finite hereditary algebras

Abstract

As we saw in Chapter II, any basic and connected finite dimensional algebra A over an algebraically closed field K admits a presentation as a bound quiver algebra A ≅ KQ / I , where Q is a finite connected quiver and I is an admissible ideal of KQ . It is thus natural to study the representation theory of the algebras of the form A ≅ KQ , that is, of the path algebras of finite, connected, and acyclic quivers. It turns out that an algebra A is of this form if and only if it is hereditary, that is, every submodule of a projective A -module is projective. We are thus interested in the representation theory of hereditary algebras. In, Gabriel showed that a connected hereditary algebra is representation–finite if and only if the underlying graph of its quiver is one of the Dynkin diagrams m with m ≥ 1; n with n ≥ 4; and 6 , 7 , 8 , that appear also in Lie theory (see, for instance,). Later, Bernstein, Gelfand, and Ponomarev gave a very elegant and conceptual proof underlining the links between the two theories, by applying the nice concept of reflection functors. In this chapter, using reflection functors (which may now be thought of as tilting functors), we prove Gabriel's theorem and show how to compute all the (isomorphism classes of) indecomposable modules over a representation–finite hereditary algebra.