Gabriel's theorem

statement

thm (Gabriel)

A connected quiver Q has finite type iff the underlying graph is a Dynkin diagram of (A,D,E) type. Moreoever there is a bijection between {indecomposable kQ-modules} and {positive roots} $$M \to \dim M$$ where $\dim$ is dimension vector

 

 

idea of proof

• define tilting functor
• get Coxeter element

 

related items

• Quiver representations
• Coxeter functor and transformation

expositions

• Carroll, Gabriel's Theorem