Gabriel's theorem

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: ${\displaystyle A_{n}}$, ${\displaystyle D_{n}}$, ${\displaystyle E_{6}}$, ${\displaystyle E_{7}}$, ${\displaystyle E_{8}}$.
2. The indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram. Specifically, each positive root is a positive combination of simple roots, and the coefficients of its expansion match the dimensions of the vector spaces that the indecomposable has at the vertices of the quiver (those vertices being in correspondence with the simple roots).

Dlab & Ringel (1973) found a generalization of Gabriel's theorem in which all Dynkin diagrams of finite-dimensional semisimple Lie algebras occur. Victor Kac (Kac (1983)) extended these results to all quivers, not only of Dynkin type, relating their indecomposable representations to the roots of Kac–Moody algebras.

• 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
• Kac, Victor (1983), "Root systems, representations of quivers and invariant theory", in Gherardelli, Francesco (ed.), Invariant theory: proceedings of the 1st 1982 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), held at Montecatini, Italy, June 10-18, 1982, Lecture Notes in Mathematics 996, Springer, pp. 74–108, ISBN 978-3-540-12319-4