Gabriel's theorem

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. Gabriel (1972) 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.

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.

References

• Bernšteĭn, I. N.; Gelfand, I. M.; Ponomarev, V. A. (1973), "Coxeter functors, and Gabriel's theorem", Russian Mathematical Surveys, 28 (2): 17–32, Bibcode:1973RuMaS..28...17B, CiteSeerX 10.1.1.642.2527, doi:10.1070/RM1973v028n02ABEH001526, ISSN 0042-1316, MR 0393065, S2CID 250762677
• Dlab, Vlastimil; Ringel, Claus Michael (1973), On algebras of finite representation type, Carleton mathematical lecture notes, vol. 2, Department of Mathematics, Carleton Univ., Ottawa, Ont., MR 0347907
• Gabriel, Peter (1972), "Unzerlegbare Darstellungen. I", Manuscripta Mathematica, 6: 71–103, doi:10.1007/BF01298413, ISSN 0025-2611, MR 0332887, S2CID 119425731
• 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.