In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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. classified all quivers of finite type, and also their indecomposable representations. More precisely, Gabriel's theorem states that:
# 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:
,
,
,
,
.
# 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 of the Dynkin diagram.
found a generalization of Gabriel's theorem in which all Dynkin diagrams of finite-dimensional
semisimple Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s occur.
References
*
*
*{{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
Theorems in representation theory