Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Strongness of companion bases for cluster-tilted algebras of finite type


Authors: Karin Baur and Alireza Nasr-Isfahani
Journal: Proc. Amer. Math. Soc. 146 (2018), 2409-2416
MSC (2010): Primary 16G10, 16G20, 13F60, 16S70; Secondary 05E10
DOI: https://doi.org/10.1090/proc/13977
Published electronically: February 16, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e., gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons's conjecture.


References [Enhancements On Off] (What's this?)

  • [1] I. Assem, T. Brüstle, and R. Schiffler, Cluster-tilted algebras as trivial extensions, Bull. Lond. Math. Soc. 40 (2008), no. 1, 151-162. MR 2409188, https://doi.org/10.1112/blms/bdm107
  • [2] Michael Barot, Christof Geiss, and Andrei Zelevinsky, Cluster algebras of finite type and positive symmetrizable matrices, J. London Math. Soc. (2) 73 (2006), no. 3, 545-564. MR 2241966, https://doi.org/10.1112/S0024610706022769
  • [3] Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), no. 2, 572-618. MR 2249625, https://doi.org/10.1016/j.aim.2005.06.003
  • [4] Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), no. 1, 323-332. MR 2247893, https://doi.org/10.1090/S0002-9947-06-03879-7
  • [5] Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster mutation via quiver representations, Comment. Math. Helv. 83 (2008), no. 1, 143-177. MR 2365411, https://doi.org/10.4171/CMH/121
  • [6] Philippe Caldero, Frédéric Chapoton, and Ralf Schiffler, Quivers with relations and cluster tilted algebras, Algebr. Represent. Theory 9 (2006), no. 4, 359-376. MR 2250652, https://doi.org/10.1007/s10468-006-9018-1
  • [7] P. Caldero, F. Chapoton, and R. Schiffler, Quivers with relations arising from clusters ($ A_n$ case), Trans. Amer. Math. Soc. 358 (2006), no. 3, 1347-1364. MR 2187656, https://doi.org/10.1090/S0002-9947-05-03753-0
  • [8] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529. MR 1887642, https://doi.org/10.1090/S0894-0347-01-00385-X
  • [9] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63-121. MR 2004457, https://doi.org/10.1007/s00222-003-0302-y
  • [10] Tomoki Nakanishi and Salvatore Stella, Diagrammatic description of $ c$-vectors and $ d$-vectors of cluster algebras of finite type, Electron. J. Combin. 21 (2014), no. 1, Paper 1.3, 107. MR 3177498
  • [11] Mark James Parsons, Companion bases for cluster-tilted algebras, Algebr. Represent. Theory 17 (2014), no. 3, 775-808. MR 3254769, https://doi.org/10.1007/s10468-013-9418-y
  • [12] M. J. Parsons, On indecomposable modules over cluster-tilted algebras of type $ A$, Ph.D. Thesis, University of Leicester, 2007.
  • [13] Claus Michael Ringel, Cluster-concealed algebras, Adv. Math. 226 (2011), no. 2, 1513-1537. MR 2737792, https://doi.org/10.1016/j.aim.2010.08.014
  • [14] Bin Zhu, Equivalences between cluster categories, J. Algebra 304 (2006), no. 2, 832-850. MR 2264281, https://doi.org/10.1016/j.jalgebra.2006.03.012

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16G10, 16G20, 13F60, 16S70, 05E10

Retrieve articles in all journals with MSC (2010): 16G10, 16G20, 13F60, 16S70, 05E10


Additional Information

Karin Baur
Affiliation: Institut für Mathematik und Wissenschaftliches Rechnen, Universität Graz, Heinrichstrasse 36, A-8010 Graz, Austria
Email: baurk@uni-graz.at

Alireza Nasr-Isfahani
Affiliation: Department of Mathematics, University of Isfahan, P.O. Box 81746-73441, Isfahan, Iran—and—School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
Email: nasr_a@sci.ui.ac.ir, nasr@ipm.ir

DOI: https://doi.org/10.1090/proc/13977
Keywords: Cluster-tilted algebra, companion basis, indecomposable modules, dimension vector, relation-extension algebra, root system, Euler form
Received by editor(s): March 15, 2017
Received by editor(s) in revised form: July 31, 2017, and September 10, 2017
Published electronically: February 16, 2018
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society