A finite-type quiver has no self-loops

Question

I am reading through Etingof et al's notes on representation theory, and they assert in Exercise 5.4(c) on page 80 that a finite-type quiver has no self-loops. I think the way to show this is to consider the simplest case: a quiver with one vertex and one edge - the self-loop. Representations of this quiver correspond to a choice of vector space $V$ and any linear endomorphism $T: V \to V$. I should be able to simply write down an infinite family of nonisomorphic indecomposable representations, but I'm not very good at seeing which representations are indecomposable/nonisomorphic. Can someone give me a hint?

Answer 1

Here is one infinite family of pairwise non-isomorphic indecomposables. Take $V$ to be $F^n$ (your base field $F$) a nilpotent matrix with a single Jordan block for $T$: any submodule must contain the (unique up to scalars) vector in the kernel, so these are indecomposable.

Here is another, if the base field is infinite: take for $T$ the scalar matrix $T=a$ for $a \in F$, acting on a one-dimensional $V=F$. These are simple and non-isomorphic. The real issue here is the infinite dimensionality of the path algebra...