# 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 $\mathbb{V}$ 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 $\mathrm{dim}_{k}V_{i}<\infty$ 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 QuiverRepresentationsAndRepresentationMorphisms 2013-03-22 19:16:15 2013-03-22 19:16:15 joking (16130) joking (16130) 5 joking (16130) Definition msc 14L24