Associative rings and algebras Definition

Dynkin quiver

Builds on Quiver

Definition

Let $Q$ be a finite quiver. The underlying graph of $Q$ is the graph with vertex set $Q_0$ and one undirected edge between $s(a)$ and $t(a)$ for each arrow $a \in Q_1$ . The quiver $Q$ is a Dynkin quiver if its underlying graph is one of the simply laced Dynkin diagrams:

$A_n$ for $n \ge 1$ , a path with $n$ vertices;
$D_n$ for $n \ge 4$ , a path with $n - 2$ vertices and two extra vertices joined to one end vertex;
$E_6$ , $E_7$ or $E_8$ , a path with $n - 1$ vertices and one extra vertex joined to the third vertex from one end, for $n = 6, 7, 8$ .

Remarks

Example: $1 \to 2 \leftarrow 3$ is a Dynkin quiver, since its underlying graph is the path with $3$ vertices, of type $A_3$ . Non-example: the Kronecker quiver, with two arrows $1 \to 2$ , is not Dynkin, since its underlying graph has a double edge. Non-example: a quiver with a loop, or whose underlying graph is a cycle, is not Dynkin. A Dynkin quiver is connected and has no oriented cycles, and every orientation of a Dynkin diagram is a Dynkin quiver.

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Quiver concept
Dynkin quiver definition

Tag	`RX1NWE1WMD`
URI	`m2://RX1NWE1WMD`
Kind	definition
Writing	W0 stub
MSC	16G20, 16G60
Defines	Dynkin quiver
Uses	Quiver

Dynkin quiver

Definition

Remarks

Reading path

Further reading

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