# representations of a bound quiver

Let $(Q,I)$ be a bound quiver (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) over a field $k$.

Let $\mathbb{V}$ be a representation^{} of $Q$ over $k$ composed by ${\{f(q)\}}_{q\in {Q}_{1}}$ a family of linear maps. If

$$w=({\alpha}_{1},\mathrm{\dots},{\alpha}_{n})$$ |

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

$${f}_{w}=f({\alpha}_{n})\circ f({\alpha}_{n-1})\circ \mathrm{\cdots}\circ f({\alpha}_{2})\circ f({\alpha}_{1}).$$ |

For stationary paths we define ${f}_{{e}_{x}}:{V}_{x}\to {V}_{x}$ by ${f}_{{e}_{x}}=0$. Also, note that if $\rho $ is a relation^{} (http://planetmath.org/RelationsInQuiver) in $Q$, then

$$\rho =\sum _{i=1}^{m}{\lambda}_{i}\cdot {w}_{i}$$ |

where all ${w}_{i}$’s have the same source and target. Thus it makes sense to talk about evaluation in $\rho $, i.e.

$${f}_{\rho}=\sum _{i=1}^{n}{\lambda}_{i}\cdot {f}_{{w}_{i}}.$$ |

In particular

$${f}_{\rho}:{V}_{s({w}_{i})}\to {V}_{t({w}_{i}})$$ |

is a linear map.

Recall that the ideal $I$ is generated by relations (see this entry (http://planetmath.org/PropertiesOfAdmissibleIdeals)) $\{{\rho}_{1},\mathrm{\dots},{\rho}_{n}\}$.

Definition. A representation $\mathbb{V}$ of $Q$ over $k$ with linear mappings ${\{f(q)\}}_{q\in {Q}_{1}}$ is said to be bound by $I$ if

$${f}_{{\rho}_{i}}=0$$ |

for every $i=1,\mathrm{\dots},n$.

It can be easily checked, that this definition does not depend on the choice of (relation) generators^{} of $I$.

The full subcategory of the category^{} of all representations which is composed of all representations bound by $I$ is denoted by $\mathrm{REP}(Q,I)$. It can be easily seen, that it is abelian^{}.

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 |