subquiver and image of a quiver

Let Q=(Q0,Q1,s,t) be a quiver.

Definition. A quiver Q=(Q0,Q1,s,t) is said to be a subquiver of Q, if


are such that if αQ1, then s(α),t(α)Q0. Furthermore


In this case we write QQ.

A subquiver QQ is called full if for any x,yQ0 and any αQ1 such that s(α)=x and t(α)=y we have that αQ1. In other words a subquiver is full if it ,,inherits” all arrows between points.

If Q is a subquiver of Q, then the mapping


where both i0,i1 are inclusions is a morphism of quivers. In this case i is called the inclusion morphism.

If F:QQ is any morphism of quivers Q=(Q0,Q1,s,t) and Q=(Q0,Q1,s,t), then the quadruple


where s′′,t′′ are the restrictionsPlanetmathPlanetmathPlanetmathPlanetmath of s,t to Im(F1) is called the image of F. It can be easily shown, that Im(F) is a subquiver of Q.

Title subquiver and image of a quiver
Canonical name SubquiverAndImageOfAQuiver
Date of creation 2013-03-22 19:17:19
Last modified on 2013-03-22 19:17:19
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Definition
Classification msc 14L24