Let $K$ be a field, $Q$ a quiver, and $\mathcal{ A}$ the ideal of the path algebra $KQ$ that is generated by the arrows of $Q$. We present old and new results about the representation theories of the truncations $KQ/\mathcal{ A}^L$, $L \in \mathbb{ N}$, tracking their development as $L$ goes to infinity. The goal is to gain a better understanding of the category of those finite dimensional $KQ$-modules which arise as finitely generated modules over admissible quotients of $KQ$.
Categories: math.RT math.RA