# nLab Gabriel's theorem

Contents

### Context

#### Graph theory

graph theory

graph

category of simple graphs

### Extra structure

#### Representation theory

representation theory

geometric representation theory

# Contents

## 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.

## Examples

In the following we write $\mathbb{K}$ for the given ground field (typically $\mathbb{K} = \mathbb{C}$ the complex numbers, but much of the following works for general fields, eg. $\mathbb{K} = \mathbb{R}$ the real numbers or $\mathbb{K} = \mathbb{F}_p$ a finite field).

Given a quiver $Q$ and a quiver representation $\rho$, we denote for any edge $v \xrightarrow{\;e\;} v'$ in $Q_1$ the corresponding value of $\rho$ by

### A-type quivers

For $n \in \mathbb{N}_+$ consider the $\mathbb{A}_{n}$-quiver, hence with this underlying undirected graph:

###### Example

The indecomposable quiver representations of $\mathbb{A}_n$ are labeled by pairs $a,b \in \mathbb{N}^2$ with $1 \leq a \lt b \leq n$ and are given as follows (see also Carlsson & de Silva 2010, reviewed in Oudot 15, p. 17):

for any given orientation of the edges.

So for instance if the quiver is the linear $\mathbb{A}_n$-quiver

then its indecomposable representations are of this form:

###### Remark

Example is of key relevance in the discussion of persistent homology (notably in topological data analysis): Here an $\mathbb{A}_n$-quiver representation is called a zig-zag persistence module and the statement of Ex. is interpreted as saying that every persistence module is spanned by elements which

• appear at some resolution $a$,

• persist across a range $b - a$,

• disappear at some resolution $b$.

The multiset of these pairs $(a,b)$ is then called the barcode or persistence diagram of the persistence module.

This relation between quiver representation theory and persistent homolology was originally highlighted by Carlsson & de Silva 2010.

The result is due to

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

Lecture notes:

• Quiver representations and Gabriel’s theorem (pdf)

$[$ISBN:978-1-4704-3443-4, pdf$]$