translation quiver

Let Q=(Q0,Q1,s,t) be a locally finite quiver without loops. Recall that a loop is an arrow α such that s(α)=t(α). Let X,YQ0.

Definition 1. A pair (Q,τ) is said to be a translation quiver iff the following holds:

  1. 1.

    τ:XY is a bijection;

  2. 2.

    If xX and yx- is a direct predecessor of x, then the number of arrows from y to x is equal to the number of arrows from τ(x) to y.

If (Q,τ) is a translation quiver then we will say that τ(x) exists if xX and τ(x) does not exist (or it is not defined) if xX.

Definition 2. If (Q,τ) is a translation quiver, then a pair (Q,τ) is called a translationMathworldPlanetmathPlanetmath subquiver if it is a translation quiver, Q is a full subquiver ( of Q and τ(x)=τ(x) whenever x is a vertex in Q such that τ(x) exists and belongs to Q.

Example. Let Q be the following quiver:


If we put X={2,3,4,6,8}, Y={1,2,3,5,6} and


then the pair (Q,τ) is a translation quiver and


is its translation subquiver, where τ(6)=5.

Remark. It is common to write translation quivers as in example. This means that Q is ,,oriented” to the right and in rows we have vertices such that ,,jumping” two places to the left gives us τ of this vertex. Note that in the example the vertex 7 is not written in the same row as 9 because τ(7) is not 9 (indeed, τ(7) is not defined).

Title translation quiver
Canonical name TranslationQuiver
Date of creation 2013-03-22 19:17:53
Last modified on 2013-03-22 19:17:53
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Definition
Classification msc 14L24