nLab Gabriel's theorem

Contents

Context

Graph theory

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

Given a quiver QQ and a quiver representation ρ\rho, we denote for any edge vβ†’evβ€²v \xrightarrow{\;e\;} v' in Q 1Q_1 the corresponding value of ρ\rho by

A-type quivers

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

Example

The indecomposable quiver representations of 𝔸 n\mathbb{A}_n are labeled by pairs a,bβˆˆβ„• 2a,b \in \mathbb{N}^2 with 1≀a<b≀n1 \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 𝔸 n\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 𝔸 n\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 aa,

  • persist across a range bβˆ’ab - a,

  • disappear at some resolution bb.

The multiset of these pairs (a,b)(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

Lecture notes:

  • Quiver representations and Gabriel’s theorem (pdf)

See also:

Re-proof of Gabriel’s theorem in the case of A-type quivers and relation to persistent homology:

reviewed in:

Last revised on May 20, 2022 at 03:09:05. See the history of this page for a list of all contributions to it.