nLab
Gabriel's theorem

Statement

Gabriel's theorem (Gabriel 72) says that connected quivers with a finite number of indecomposable quiver representations over an algebraically closed field are precisely the Dynkin quivers: those whose underlying undirected graph is a Dynkin diagram in the ADE series.

Moreover, the indecomposable quiver representations in this case are in bijection with the positive roots in the root system of the Dynkin diagram.

Related concepts

• ADE classification

• Dynkin quiver, McKay quiver

References

The result is due to

• Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Mathematica 6: 71–103, (1972) (doi:10.1007/BF01298413)

Lecture notes include

• Quiver representations and Gabriel’s theorem (pdf)

• Wikipedia, Gabriel’s theorem