English

More finite sets coming from non-commutative counting

Category Theory 2019-07-31 v3

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 QQ, QQ' (i.e. quivers of extended Dynkin type) there are only finitely many full triangulated subctegories in Db(RepK(Q))D^b(Rep_{\mathbb K}(Q)), which are equivalent to Db(RepK(Q))D^b(Rep_{\mathbb K}(Q')), where K{\mathbb K} is an algebraically closed field. Some of the numbers counting the elements in these finite sets are explicitly determined.

Keywords

Cite

@article{arxiv.1903.00295,
  title  = {More finite sets coming from non-commutative counting},
  author = {George Dimitrov and Ludmil Katzarkov},
  journal= {arXiv preprint arXiv:1903.00295},
  year   = {2019}
}

Comments

16 pages, In v3 Corollary 5.6 does not depend on any additional conditions, because in a private communication Professor Helmut Lenzing confirmed that (21) is correct. The last section 6 and the introduction in the new version are slightly extended. The reference list is also updated

R2 v1 2026-06-23T07:55:22.401Z