More finite sets coming from non-commutative counting

TL;DR

This paper extends the class of categorical invariants, showing that for affine acyclic quivers, the set of certain triangulated subcategories is finite, with some counts explicitly determined, advancing understanding of non-commutative invariants.

Contribution

It demonstrates finiteness of specific sets of triangulated subcategories for affine acyclic quivers, expanding the scope of categorical invariants in non-commutative geometry.

Findings

Finiteness of sets of triangulated subcategories for affine acyclic quivers.

Explicit counts for some of these finite sets.

Extension of previous results to new classes of quivers.

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.