A famous theorem of Drozd says that every finite dimensional hereditary algebra is either of tame or wild representation type. I am interested in infinite dimensional hereditary algebras. Is there a known dichotomy result for this situation? Are extra assumptions necessary?
This question appears to have been posted a long time ago, but I shall answer for the benefit of future searchers. This question (in a more general setting) has been addressed in a very recent paper by Iovanov. I have not gone through the paper in detail yet, as I have just stumbled across it myself. Although the answer appears to be that the dichotamy holds, with no assumptions on the algebra necessary.