# quiver representations and representation morphisms

Let $Q=({Q}_{0},{Q}_{1},s,t)$ be a quiver, i.e. ${Q}_{0}$ is a set of vertices, ${Q}_{1}$ is a set of arrows and $s,t:{Q}_{1}\to {Q}_{0}$ are functions such that $s$ maps each arrow to its source and $t$ maps each arrow to its target.

A representation^{} $\mathbb{V}$ of $Q$ over a field $k$ is a family of vector spaces^{} ${\{{V}_{i}\}}_{i\in {Q}_{0}}$ over $k$ together with a family of $k$-linear maps ${\{{f}_{a}:{V}_{s(a)}\to {V}_{t(a)}\}}_{a\in {Q}_{1}}$.

A morphism $F:\mathbb{V}\to \mathbb{W}$ between representations $\mathbb{V}=({V}_{i},{g}_{a})$ and $\mathbb{W}=({W}_{i},{h}_{a})$ is a family of $k$-linear maps ${\{{F}_{i}:{V}_{i}\to {W}_{i}\}}_{i\in {Q}_{0}}$ such that for each arrow $a\in {Q}_{1}$ the following relation holds:

$${F}_{t(a)}\circ {g}_{a}={h}_{a}\circ {F}_{s(a)}.$$ |

Obviously we can compose morphisms of representations and in this the case class of all representations and representation morphisms together with the standard composition is a category. This category is abelian^{}.

It can be shown that for each finite quiver $Q$ (i.e. with ${Q}_{0}$ finite) and field $k$ there exists an algebra^{} $A$ over $k$ such that the category of representations of $Q$ is equivalent to the category of modules over $A$.

A representation $\mathbb{V}$ of $Q$ is called trivial iff ${V}_{i}=0$ for each vertex $i\in {Q}_{0}$.

A representation $\mathbb{V}$ of $Q$ is called locally finite-dimensional iff $$ for each vertex $i\in {Q}_{0}$ and finite-dimensional iff $\mathbb{V}$ is locally finite-dimensional and ${V}_{i}=0$ for almost all vertices $i\in {Q}_{0}$.

Title | quiver representations and representation morphisms |
---|---|

Canonical name | QuiverRepresentationsAndRepresentationMorphisms |

Date of creation | 2013-03-22 19:16:15 |

Last modified on | 2013-03-22 19:16:15 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 5 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 14L24 |