relations in quiver

Let Q be a quiver and k a field.

Definition. A relation in Q is a linear combinationMathworldPlanetmath (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


such that there exist x,yQ0 with s(wi)=x and t(wi)=y for all i, all wi are of length at least 2 and not all λi are zero.

If a relation ρ is of the form ρ=w for some path w, then it is called a zero relation and if ρ=w1-w2 for some paths w1,w2, then ρ 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