with the subject, for instance [@Pierce88] [@Ringel1984] [@ARS] [@ASS] and [@Schiffler2014].

A *quiver* is an oriented graph. More precisely, a quiver $Q$ is given by a $4$-tuple $(Q_0, Q_1, s,t)$, where

-   $Q_0$ is a set, whose elements are called *vertices*;

-   $Q_1$ is a set, whose elements are called *arrows*;

-   $s,t:Q_1\to Q_0$ are two maps, which associate to each arrow its *source* or its *target*, respectively.

Quivers are allowed to have multiple edges, oriented cycles and even loops. Throughout these notes quivers will be assumed to *finite*, that is to say, their sets of vertices and arrows will be finite.

We will usually number the vertices of a quiver by using natural numbers, and use letters to name the arrows. We will represent quivers as oriented graphs. Here is an example: $$\xymatrix@!0{ &&& 4\\
                   1\ar[r]^b\ar@(ul,ur)^a & 2\ar@<.3ex>[r]^c & 3\ar@<.3ex>[l]^d\ar@<.3ex>[ur]^e\ar@<-.3ex>[ur]_f\ar[dr]^g \\
                   &&& 5. 
     }$$

A *path* in a quiver is a concatenation of arrows $w=a_m\cdots a_1a_0$ such that $s(a_{i+1}) = t(a_i)$ for all $i$ from $0$ to $m-1$. This means that we compose arrows from right to left. We extend the maps $s$ and $t$ to the set of all paths by putting $s(w) = s(a_0)$ and $t(w) = 