Details
| Original language | English |
|---|---|
| Pages (from-to) | 1117-1137 |
| Number of pages | 21 |
| Journal | Forum mathematicum |
| Volume | 27 |
| Issue number | 2 |
| Publication status | Published - 1 Mar 2015 |
Abstract
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 Tℓ 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
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Forum mathematicum, Vol. 27, No. 2, 01.03.2015, p. 1117-1137.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Cluster tilting vs. weak cluster tilting in Dynkin type A infinity
AU - Holm, Thorsten
AU - Jørgensen, Peter
N1 - Funding information: This work was supported by grant number HO 1880/4-1 under the research priority programme SPP 1388 Darstellungstheorie of the Deutsche Forschungsgemeinschaft (DFG).
PY - 2015/3/1
Y1 - 2015/3/1
N2 - 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 Tℓ 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∞.
AB - 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 Tℓ 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∞.
KW - Auslander-Reiten quiver
KW - d-Calabi-Yau category
KW - d-cluster tilting subcategory
KW - Fomin-Zelevinsky mutation
KW - functorial finiteness
KW - left-approximating subcategory
KW - right-approximating subcategory
KW - spherical object
KW - weakly d-cluster tilting subcategory
UR - http://www.scopus.com/inward/record.url?scp=84925622803&partnerID=8YFLogxK
U2 - 10.1515/forum-2012-0093
DO - 10.1515/forum-2012-0093
M3 - Article
AN - SCOPUS:84925622803
VL - 27
SP - 1117
EP - 1137
JO - Forum mathematicum
JF - Forum mathematicum
SN - 0933-7741
IS - 2
ER -