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Exceptional Cycles in the Bounded Derived Categories of Quivers

Authors
  • Guo, Peng1
  • Zhang, Pu1
  • 1 Shanghai Jiao Tong University, Shanghai, 200240, P. R. China , Shanghai (China)
Type
Published Article
Journal
Acta Mathematica Sinica, English Series
Publisher
Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
Publication Date
Feb 15, 2020
Volume
36
Issue
3
Pages
207–223
Identifiers
DOI: 10.1007/s10114-020-9094-x
Source
Springer Nature
Keywords
License
Yellow

Abstract

An exceptional n-cycle in a Horn-finite triangulated category with Serre functor has been recently introduced by Broomhead, Pauksztello and Ploog. When n = 1, it is a spherical object. We explicitly determine all the exceptional cycles in the bounded derived category Db (kQ) of a finite quiver Q without oriented cycles. In particular, if Q is an Euclidean quiver, then the length type of exceptional cycles in Db (kQ) is exactly the tubular type of Q; if Q is a Dynkin quiver of type Em (m = 6, 7, 8), or Q is a wild quiver, then there are no exceptional cycles in Db (kQ); and if Q is a Dynkin quiver of type An or Dn, then the length of an exceptional cycle in Db (kQ) is either h or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{h}{2}$$\end{document}, where h is the Coxeter number of Q.

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