# underlying graph of a quiver

Let $Q=({Q}_{0},{Q}_{1},s,t)$ be a quiver, i.e. ${Q}_{0}$ is a set of vertices, ${Q}_{1}$ is a set of arrows and $s,t:{Q}_{1}\to {Q}_{0}$ are functions which take each arrow to its source and target respectively.

Definition. An underlying graph of $Q$ or graph associated with $Q$ is a graph

$$G=(V,E,\tau )$$ |

such that $V={Q}_{0}$, $E={Q}_{1}$ and $\tau :E\to {V}_{sym}^{2}$ is given by

$$\tau (\alpha )={[s(\alpha ),t(\alpha )]}_{\sim}.$$ |

In other words $G$ is a graph which is obtained from $Q$ after forgeting the orientation of arrows. The definition of a graph used here is taken from this entry (http://planetmath.org/AlternativeDefinitionOfAMultigraph).

Note, that if we know the underlying graph $G$ of a quiver $Q$, then the information we have is not enough to reconstruct $Q$ (except for a trivial case with no edges). The orientation of arrows is lost forever. In some cases it is possible to reconstruct $Q$ up to an isomorphism^{} of quivers (http://planetmath.org/MorphismsBetweenQuivers), for example graph

$$\text{xymatrix}1\text{ar}\mathrm{@}-[r]\mathrm{\&}2$$ |

uniquely (up to isomorphism) determines its quiver, but

$$\text{xymatrix}G:\mathrm{\&}1\text{ar}\mathrm{@}-[r]\mathrm{\&}2\text{ar}\mathrm{@}-[r]\mathrm{\&}3$$ |

does not uniquely determine its quiver. Indeed, there are exactly two nonisomorphic quivers with underlying graph $G$, namely:

$$\text{xymatrix}Q:\mathrm{\&}1\text{ar}[r]\mathrm{\&}2\text{ar}[r]\mathrm{\&}3{Q}^{\prime}:\mathrm{\&}1\text{ar}[r]\mathrm{\&}2\mathrm{\&}3\text{ar}[l]$$ |

Title | underlying graph of a quiver |
---|---|

Canonical name | UnderlyingGraphOfAQuiver |

Date of creation | 2013-03-22 19:16:59 |

Last modified on | 2013-03-22 19:16:59 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 14L24 |