# nLab Dynkin quiver

Contents

### Context

#### Graph theory

graph theory

graph

category of simple graphs

### Extra structure

#### Representation theory

representation theory

geometric representation theory

# Contents

## Definition

A Dynkin quiver is a quiver whose underlying undirected graph is a one of the Dynkin diagrams in the ADE series.

graphics grabbed from Qiu 15

See e.g. Qiu 15, Def. 2.1

## Properties

### Characterization as finite-type quivers

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 bijection with the positive roots in the root system of the Dynkin diagram.

### McKay correspondence

Dynkin diagram/
Dynkin quiver
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
$A_{n \geq 1}$cyclic group
$\mathbb{Z}_{n+1}$
cyclic group
$\mathbb{Z}_{n+1}$
special unitary group
$SU(n+1)$
D4Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group
$SO(2n)$
$E_6$tetrahedrontetrahedral group
$T$
binary tetrahedral group
$2T$
E6
$E_7$cube,
octahedron
octahedral group
$O$
binary octahedral group
$2O$
E7
$E_8$dodecahedron,
icosahedron
icosahedral group
$I$
binary icosahedral group
$2I$
E8

## References

Gabriel's theorem is due to

• Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Mathematica 6: 71–103, (1972)

Discussion of Bridgeland stability conditions on quiver representations over Dynkin Quivers includes

Last revised on October 2, 2018 at 11:47:04. See the history of this page for a list of all contributions to it.