# relations in quiver

Let $Q$ be a quiver and $k$ a field.

Definition. A relation in $Q$ is a linear combination^{} (over $k$) of paths of length at least $2$ such that all paths have the same source and target. Thus a relation is an element of the path algebra $kQ$ of the form

$$\rho =\sum _{i=1}^{m}{\lambda}_{i}\cdot {w}_{i}$$ |

such that there exist $x,y\in {Q}_{0}$ with $s({w}_{i})=x$ and $t({w}_{i})=y$ for all $i$, all ${w}_{i}$ are of length at least $2$ and not all ${\lambda}_{i}$ are zero.

If a relation $\rho $ is of the form $\rho =w$ for some path $w$, then it is called a zero relation and if $\rho ={w}_{1}-{w}_{2}$ for some paths ${w}_{1},{w}_{2}$, then $\rho $ is called a commutativity relation.

Title | relations in quiver |
---|---|

Canonical name | RelationsInQuiver |

Date of creation | 2013-03-22 19:16:45 |

Last modified on | 2013-03-22 19:16:45 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 14L24 |