representations of a bound quiver

Let (Q,I) be a bound quiver ( over a field k.

Let 𝕍 be a representationPlanetmathPlanetmath of Q over k composed by {f(q)}qQ1 a family of linear maps. If


is a path in Q, then we have the evaluation map


For stationary paths we define fex:VxVx by fex=0. Also, note that if ρ is a relationPlanetmathPlanetmath ( in Q, then


where all wi’s have the same source and target. Thus it makes sense to talk about evaluation in ρ, i.e.


In particular


is a linear map.

Recall that the ideal I is generated by relations (see this entry ( {ρ1,,ρn}.

Definition. A representation 𝕍 of Q over k with linear mappings {f(q)}qQ1 is said to be bound by I if


for every i=1,,n.

It can be easily checked, that this definition does not depend on the choice of (relation) generatorsPlanetmathPlanetmathPlanetmath of I.

The full subcategory of the categoryMathworldPlanetmath of all representations which is composed of all representations bound by I is denoted by REP(Q,I). It can be easily seen, that it is abelianMathworldPlanetmathPlanetmath.

Title representations of a bound quiver
Canonical name RepresentationsOfABoundQuiver
Date of creation 2013-03-22 19:16:51
Last modified on 2013-03-22 19:16:51
Owner joking (16130)
Last modified by joking (16130)
Numerical id 6
Author joking (16130)
Entry type Definition
Classification msc 14L24