mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Gabriel's theorem, proved by

Pierre Gabriel Pierre Gabriel (1 August 1933 – 24 November 2015), also known as Peter Gabriel, was a French mathematician at the University of Strasbourg (1962–1970), University of Bonn (1970–1974) and University of Zürich (1974–1998) who worked on cat ...

, classifies the quivers of finite type in terms of

Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...

Statement

A quiver is of finite type if it has only finitely many

isomorphism class In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them ...

es of indecomposable representations. classified all quivers of finite type, and also their indecomposable representations. More precisely, Gabriel's theorem states that: # A (

connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...

) quiver is of finite type if and only if its underlying

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

(when the directions of the arrows are ignored) is one of the ADE

A_n

D_n

E_6

E_7

E_8

. # The indecomposable representations are in a one-to-one correspondence with the

positive roots In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation ...

of the

root system In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and ...

of the Dynkin diagram. found a generalization of Gabriel's theorem in which all Dynkin diagrams of finite-dimensional

semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

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

Kac–Moody algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a g ...

References

* * * *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 Theorems in representation theory