Gabriel's theorem

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

Statement

A quiver is of finite type if it has only finitely many isomorphism classes of indecomposable representations. Template:Harvtxt classified all quivers of finite type, and also their indecomposable representations. More precisely, Gabriel's theorem states that:

1. A (connected) quiver is of finite type if and only if its underlying graph (when the directions of the arrows are ignored) is one of the ADE Dynkin diagrams: $A_n$, $D_n$, $E_6$, $E_7$, $E_8$.
2. The indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.

Template:Harvtxt 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.

References

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• Victor Kac, "Root systems, representations of quivers and invariant theory". Invariant theory (Montecatini, 1982), pp. 74–108, Lecture Notes in Math. 996, Springer-Verlag, Berlin 1983. ISBN 3-540-12319-9.