Gabriel's theorem

In mathematics, Gabriel's theorem, proved by Pierre Gabriel, classifies the quivers of finite type in terms of Dynkin diagrams.

A quiver is of finite type if it has only finitely many isomorphism classes of indecomposable representations.

Gabriel (1972) classified all quivers of finite type, and also their indecomposable representations.

More precisely, Gabriel's theorem states that: Dlab & Ringel (1973) found a generalization of Gabriel's theorem in which all Dynkin diagrams of finite-dimensional semisimple Lie algebras occur.

Victor Kac extended these results to all quivers, not only of Dynkin type, relating their indecomposable representations to the roots of Kac–Moody algebras.