1-loop quiver and the classification of quivers

Question

Gabriel's theorem states that finite type quivers are exactly the ones whose underlying graphs are ADE type Dynkin diagrams. Furthermore, the quivers whose underlying diagrams are ADE type affine Dynkin diagrams are tame quivers. What about the 1-loop quiver? It isn't Dynkin or affine Dynkin, but I don't think it's wild type either since the indecomposables are parametrized by the dimension vector and one continuous parameter (as in a Jordan block). But people say that a quiver is either finite, tame, or wild type. So where should the 1-loop quiver belong?

Answer 1

As you already noticed the $1$-loop quiver should be considered tame as for each dimension you only need one continous parameter to describe its finite dimensional representations. Similarly this is true for the $\tilde{A}_n$-quiver with cyclic orientation.

When one says that a quiver is tame one usually restricts to the case of acyclic quivers. In this case, as you mentioned there is the generalisation of Gabriel's theorem by Donovan-Freislich and Nazarova saying that these are exactly given by the affine Dynkin diagrams. Here, it is implicitly assumed that you don't take the cyclic orientation in type $\tilde{A}_n$.

One problem with the cyclic orientation is that there is in general no tame-wild dichotomy theorem for infinite-dimensional algebras. I'm not sure whether the case of quivers with oriented cycles has already been considered.