More finite sets coming from non-commutative counting

Abstract

In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are finite. Using results by Geigle, Lenzning, Meltzer, H\"ubner for weighted projective lines we show that for any two affine acyclic quivers $Q$, $Q'$ (i.e. quivers of extended Dynkin type) there are only finitely many full triangulated subctegories in $D^b(Rep_{\mathbb K}(Q))$, which are equivalent to $D^b(Rep_{\mathbb K}(Q'))$, where ${\mathbb K}$ is an algebraically closed field. Some of the numbers counting the elements in these finite sets are explicitly determined.