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 $\mathrm{dim}_{k}V_{i}<\infty$ 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
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