---
tag: FC78F2E87E
kind: statement
role: theorem
name: Gabriel's theorem
msc: [16G20, 16G60]
uses: [RE9YFFS3TS, TXVWFWP6VQ, E2P29GCD9T, 92RQG73SC5]
about: [TXVWFWP6VQ, 92RQG73SC5]
sources:
  - url: https://en.wikipedia.org/wiki/Gabriel%27s_theorem
same_as:
  wikidata: Q5515505
  wikipedia: Gabriel's theorem
---

## Statement

Let $k$ be an algebraically closed field and let $Q$ be a finite connected [[quiver|RE9YFFS3TS]].
For $x \in \Z^{Q_0}$ let $q(x) = \sum_{i \in Q_0} x_i^2 - \sum_{a \in Q_1} x_{s(a)} x_{t(a)}$.

1. $Q$ has only finitely many isomorphism classes of finite-dimensional [[indecomposable|E2P29GCD9T]] [[representations|TXVWFWP6VQ]] over $k$ if and only if $Q$ is a [[Dynkin quiver|92RQG73SC5]].
2. If $Q$ is a Dynkin quiver, then the dimension vector gives a bijection from the isomorphism classes of indecomposable representations to the set $\Phi_+ = \set{x \in \N^{Q_0} : x \ne 0, \ q(x) = 1}$.
The set $\Phi_+$ is the set of positive roots of the root system of the underlying Dynkin diagram.

## Remarks

Bernstein, Gelfand and Ponomarev gave a proof with [[reflection functors|MG4RNMA8AR]].
A reflection functor at a sink or source changes the orientation at one vertex, and acts on dimension vectors by the simple reflection of the root system.
Applying them repeatedly brings every indecomposable representation to a simple one, whose dimension vector is a simple root.
Example: for $1 \to 2$, of type $A_2$, the positive roots are $(1, 0)$, $(0, 1)$ and $(1, 1)$, matching the indecomposables $k \to 0$, $0 \to k$ and $k \xrightarrow{\id} k$.
The Dynkin hypothesis is necessary: for the Kronecker quiver, with two arrows $a, b \colon 1 \to 2$, the representations $V_1 = V_2 = k$, $V_a = \id$, $V_b = \lambda \id$ are indecomposable and pairwise non-isomorphic for $\lambda \in k$.
The proof by reflection functors works over every field.

## Explanations

### L1 human

The theorem answers the question: for which quivers can all the representations be listed?
The answer is exactly the quivers whose shape is one of the Dynkin diagrams $A_n$, $D_n$, $E_6$, $E_7$, $E_8$.
For those quivers, the building blocks are in one-to-one correspondence with the positive roots of a root system, so they can be counted.
For type $A_n$ there are $n(n+1)/2$ building blocks, whatever the directions of the arrows.

### L2 human

The quadratic form $q$ is the Tits form of the quiver; it does not depend on the orientation, and it is positive definite exactly for the Dynkin graphs.
This links representation theory of quivers with the classification of simple Lie algebras, where the same diagrams appear.
[[Kac's theorem|59QJC3ZXA1]] extends the description of dimension vectors to all quivers without loops, through the roots of Kac–Moody algebras.
For path algebras of Dynkin quivers, the Auslander–Reiten quiver can be built from the projective modules by the Auslander–Reiten translation.