# 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 RelationsInQuiver 2013-03-22 19:16:45 2013-03-22 19:16:45 joking (16130) joking (16130) 4 joking (16130) Definition msc 14L24