[[Quiver representation theory MOC]]
# Quiver representation

A **$\mathbb{K}$-[[representation]]** of a [[quiver]] $\Gamma$ may be characterized in several different ways: #m/def/quiv 

1. A [[quiver homomorphism]] from $\Gamma$ onto a $\mathbb{K}$-[[linear quiver]]; ^R1
2. A [[functor]] from the [[free category]] $\underline{\Gamma}$ to [[Category of vector spaces]]; ^R2
3. A [[module]] over the [[Path algebra]] $\mathbb{K}[\underline \Gamma]$. ^R3

where the equivalence of [[#^R2]] and [[#^R3]] follows from [[Module over a category ring]].
Generally, it is useful to think of a quiver representation $V$ as a $\mathbb{K}[\underline \Gamma]$-representations which is also a $\Gamma E$-[[graded vector space]].

Often we are only interested in **finite-dimensional representations**, i.e. those of the form $\underline{\Gamma} \to \cat{FinVect}_{\mathbb{K}}$.
We might also consider a [[Matrix quiver representation]] $\underline \Gamma \to \Sk (\Vect_{\mathbb{K}})$.


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#state/tidy | #lang/en | #SemBr