morphisms of path algebras induced from morphisms of quivers

Let Q=(Q0,Q1,s,t), Q=(Q0,Q1,s,t) be quivers and let F:QQ be a morphism of quivers.

PropositionPlanetmathPlanetmath 1. If w=(α1,,αn) is a path in Q, then


is a path in Q.

Proof. Indeed, for any i=1,,n-1 we calculate


which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Proposition 2. Let w,u be paths in Q. If w is compatible ( with u then F(w) is compatible ( with F(u). The inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath implicationMathworldPlanetmath holds if and only if F0 is an injective function.

Proof. Assume that we have the following presentationsMathworldPlanetmathPlanetmath:


If t(wn)=s(u1), then


which shows the first part of the thesis.

For the second part note, that if F0 is injective, then the above equalities can be reversed to obtain that t(wn)=s(u1).

On the other hand assume that F0 is not injective, i.e. F0(a)=F0(b) for some distinct vertices a,bQ0. Then for stationary paths ea and eb we have that


so paths (F1(ea)) and (F1(eb)) are compatible (, although (ea), (eb) are not.

Definition. Let k be a field. The linear map


defined on a basis of kQ by


is said to be induced from F.

Proposition 3. The linear map F¯:kQkQ induced from F:QQ is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of algebras if and only if F0 is injective.

Proof. Indeed, we will show that F¯ preservers multiplicationPlanetmathPlanetmath of compatible paths ( If


are compatible paths ( in Q, then


which completes this part.

Now assume that w, u are paths, which are not compatible ( If F0 is injective, then by proposition 2 F(w) and F(u) are also not compatible ( and thus


On the other hand, if F0 is not injective, then there are paths w, u which are not compatible (, but F(w), F(u) are. Assume, that F¯ is a homomorphism of algebras. Then


because of the compatibility ( The contradictionMathworldPlanetmathPlanetmath shows that F¯ is not a homomorphism of algebras. This completes the proof.

Title morphisms of path algebras induced from morphisms of quivers
Canonical name MorphismsOfPathAlgebrasInducedFromMorphismsOfQuivers
Date of creation 2013-03-22 19:17:03
Last modified on 2013-03-22 19:17:03
Owner joking (16130)
Last modified by joking (16130)
Numerical id 6
Author joking (16130)
Entry type Definition
Classification msc 14L24