---
tag: RE9YFFS3TS
kind: concept
name: Quiver
msc: [16G20, 05C20]
specializes: [0KFYGSHHF5]
sources:
  - url: https://en.wikipedia.org/wiki/Quiver_(mathematics)
same_as:
  wikidata: Q493980
  wikipedia: Quiver (mathematics)
---

## Summary

A quiver is a [[directed graph|0KFYGSHHF5]] in which several arrows between the same two vertices and loops at a vertex are allowed.
Quivers are the combinatorial input of the representation theory of algebras.
A [[representation of a quiver|TXVWFWP6VQ]] places a vector space at each vertex and a linear map along each arrow.
Every quiver gives a [[path algebra|2PYANBGXFD]], and many finite-dimensional algebras are quotients of path algebras.
In category theory, a quiver is the underlying data of a category without composition.

## Explanations

### L1 human

A quiver is a picture made of dots and arrows between them.
Two dots may be joined by several arrows, and an arrow may start and end at the same dot.
The word quiver, a case for arrows, was introduced by Pierre Gabriel.
The Kronecker quiver has two vertices and two parallel arrows from the first to the second.
Quivers are used as blueprints for systems of vector spaces and linear maps.