---
tag: 92RQG73SC5
kind: concept
name: Dynkin quiver
msc: [16G20, 16G60]
specializes: [RE9YFFS3TS]
sources:
  - url: https://en.wikipedia.org/wiki/Gabriel%27s_theorem
---

## Summary

A Dynkin quiver is a [[quiver|RE9YFFS3TS]] whose underlying undirected graph is one of the simply laced Dynkin diagrams $A_n$, $D_n$, $E_6$, $E_7$ or $E_8$.
By [[Gabriel's theorem|FC78F2E87E]], these are exactly the connected quivers with only finitely many indecomposable representations up to isomorphism.
The orientation of the arrows does not matter for this property.

## Explanations

### L1 human

Forget the directions of the arrows of a quiver and look at the graph that remains.
If that graph is a line, a line with a fork at one end, or one of three special trees with a long tail, the quiver is a Dynkin quiver.
For example, $1 \to 2 \leftarrow 3$ is a Dynkin quiver of type $A_3$, since its graph is a line with three vertices.
Dynkin quivers are the quivers whose representations can be listed completely.
The same diagrams classify simple Lie algebras, which is a striking coincidence explained by root systems.