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 $1 \leq a \leq b \leq n$ 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.

References

The result is due to

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

Lecture notes:

• Quiver representations and Gabrielβs theorem (pdf)

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