Kronecker quiver ($K_2$)

The Kronecker quiver has two vertices and two parallel arrows from the first to the second. For a representation with arrows $I,N$, an endomorphism consists of the same matrix at both vertices and must commute with $N$. If $N$ is a single Nilpotent Jordan block, this centralizer is $k[N]$, yielding an indecomposable non-brick when the block size exceeds one.