# 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 $\mathrm{(}I\mathrm{,}{I}^{\mathrm{\prime}}\mathrm{)}$ if the induced linear map (http://planetmath.org/MorphismsOfPathAlgebrasInducedFromMorphismsOfQuivers) $\overline{F}:kQ\to k{Q}^{\prime}$ 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 $\overline{F}:kQ\to k{Q}^{\prime}$ induces a linear map

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

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 |
---|---|

Canonical name | MorphismsBetweenBoundQuivers |

Date of creation | 2013-03-22 19:17:07 |

Last modified on | 2013-03-22 19:17:07 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 5 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 14L24 |