Gabriel's theorem

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

{{harvtxt|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.

{{Reflist}}

• {{Citation | last1=Bernšteĭn | first1=I. N. | last2=Gelfand | first2=I. M. | last3=Ponomarev | first3=V. A. | title=Coxeter functors, and Gabriel's theorem | doi=10.1070/RM1973v028n02ABEH001526 | mr=0393065 | year=1973 | journal=Russian Mathematical Surveys | issn=0042-1316 | volume=28 | issue=2 | pages=17–32| bibcode=1973RuMaS..28...17B | citeseerx=10.1.1.642.2527 | s2cid=250762677 }}
• {{Citation | last1=Dlab | first1=Vlastimil | last2=Ringel | first2=Claus Michael | title=On algebras of finite representation type | url=https://books.google.com/books?id=_JrnAAAAMAAJ | publisher=Department of Mathematics, Carleton Univ., Ottawa, Ont. | series=Carleton mathematical lecture notes | mr=0347907 | year=1973 | volume=2}}
• {{Citation | last1=Gabriel | first1=Peter | title=Unzerlegbare Darstellungen. I | doi=10.1007/BF01298413 | mr=0332887 | year=1972 | journal=Manuscripta Mathematica | issn=0025-2611 | volume=6 | pages=71–103| 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.{{Cite book |title=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 |date=1983 |publisher=Springer |isbn=978-3-540-12319-4 |editor-last=Gherardelli |editor-first=Francesco |series=Lecture notes in mathematics |location=Berlin Heidelberg |editor-last2=Centro Internazionale Matematico Estivo}}

Category:Theorems in representation theory