# morphisms between bound quivers

Let $(Q,I)$ and $(Q^{\prime},I^{\prime})$ be bound quivers (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) over the same base field $k$.

Definition. A morphism $F:Q\to Q^{\prime}$ is said to be bounded by $(I,I^{\prime})$ if the induced linear map (http://planetmath.org/MorphismsOfPathAlgebrasInducedFromMorphismsOfQuivers) $\overline{F}:kQ\to kQ^{\prime}$ is such that

 $\overline{F}(I)\subseteq I^{\prime}.$

In this case we write

 $F:(Q,I)\to(Q^{\prime},I^{\prime})$

and we say that $F$ is a morphism of bound quivers.

If $F:(Q,I)\to(Q^{\prime},I^{\prime})$ is a morphism of bound quivers, then $\overline{F}:kQ\to kQ^{\prime}$ induces a linear map

 $\overline{\overline{F}}:kQ/I\to kQ^{\prime}/I^{\prime}.$

Furthermore, if $F_{0}$ is injective, then $\overline{F}$ is a homomorphism of algebras (see this entry (http://planetmath.org/MorphismsOfPathAlgebrasInducedFromMorphismsOfQuivers) for details) and thus $\overline{\overline{F}}$ is a homormorphism of algebras.

Title morphisms between bound quivers MorphismsBetweenBoundQuivers 2013-03-22 19:17:07 2013-03-22 19:17:07 joking (16130) joking (16130) 5 joking (16130) Definition msc 14L24