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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 27, Issue 2


Cluster tilting vs. weak cluster tilting in Dynkin type A infinity

Thorsten Holm
  • Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
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/ Peter Jørgensen
Published Online: 2013-03-21 | DOI: https://doi.org/10.1515/forum-2012-0093


This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category 𝖢 with the following properties.

On the one hand, the d-cluster tilting subcategories of 𝖢 have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of 𝖢 which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 ≤ ℓ ≤ d - 1, we show a weakly d-cluster tilting subcategory 𝖳 which has an indecomposable object with precisely ℓ mutations.

The category 𝖢 is the algebraic triangulated category generated by a (d + 1)-spherical object and can be thought of as a higher cluster category of Dynkin type A.

Keywords: Auslander–Reiten quiver; d-Calabi–Yau category; d-cluster tilting subcategory; Fomin–Zelevinsky mutation; functorial finiteness; left-approximating subcategory; right-approximating subcategory; spherical object; weakly d-cluster tilting subcategory

MSC: 13F60; 16G20; 16G70; 18E30

About the article

Received: 2012-07-02

Revised: 2012-12-17

Published Online: 2013-03-21

Published in Print: 2015-03-01

Funding Source: DFG

Award identifier / Grant number: HO 1880/4-1

Citation Information: Forum Mathematicum, Volume 27, Issue 2, Pages 1117–1137, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2012-0093.

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