Finite repetitive generalized cluster complexes and \(d\)-cluster categories. (English) Zbl 1275.05063
Summary: For any positive integer \(n\), we construct an \(n\)-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the \(n\)-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading’s generalized cluster complex as a special case when \(n=1\).
We also introduce the intermediate coverings (called generalized d-cluster categories) of \(d\)-cluster categories of hereditary algebras, and study the \(d\)-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of \(d\)-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) \(d\)-cluster tilted algebras. Moreover, we prove that the generalized \(d\)-cluster categories of hereditary algebras of finite representation type provide a category model for the \(n\)-repetitive generalized cluster complexes.
We also introduce the intermediate coverings (called generalized d-cluster categories) of \(d\)-cluster categories of hereditary algebras, and study the \(d\)-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of \(d\)-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) \(d\)-cluster tilted algebras. Moreover, we prove that the generalized \(d\)-cluster categories of hereditary algebras of finite representation type provide a category model for the \(n\)-repetitive generalized cluster complexes.
MSC:
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 05E45 | Combinatorial aspects of simplicial complexes |
| 05A15 | Exact enumeration problems, generating functions |
| 16G20 | Representations of quivers and partially ordered sets |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 17B20 | Simple, semisimple, reductive (super)algebras |
Keywords:
\(d\)-cluster categories; repetitive generalized cluster complexes; \(d\)-cluster tilted algebras; coveringsReferences:
| [1] | DOI: 10.1016/j.jpaa.2007.07.013 · Zbl 1143.16015 · doi:10.1016/j.jpaa.2007.07.013 |
| [2] | DOI: 10.1016/j.jpaa.2008.12.008 · Zbl 1183.16013 · doi:10.1016/j.jpaa.2008.12.008 |
| [3] | Baur K., Int. Math. Res. Not. 2007 (4) pp 1– |
| [4] | DOI: 10.1090/S0002-9947-08-04441-3 · Zbl 1178.16010 · doi:10.1090/S0002-9947-08-04441-3 |
| [5] | DOI: 10.1016/j.aim.2005.06.003 · Zbl 1127.16011 · doi:10.1016/j.aim.2005.06.003 |
| [6] | DOI: 10.1090/S0002-9947-06-03879-7 · Zbl 1123.16009 · doi:10.1090/S0002-9947-06-03879-7 |
| [7] | DOI: 10.1016/j.aim.2009.05.017 · Zbl 1182.16010 · doi:10.1016/j.aim.2009.05.017 |
| [8] | DOI: 10.4171/CMH/65 · Zbl 1119.16013 · doi:10.4171/CMH/65 |
| [9] | DOI: 10.1090/S0002-9947-05-03753-0 · Zbl 1137.16020 · doi:10.1090/S0002-9947-05-03753-0 |
| [10] | DOI: 10.1007/s00222-008-0111-4 · Zbl 1141.18012 · doi:10.1007/s00222-008-0111-4 |
| [11] | DOI: 10.1155/IMRN.2005.2709 · Zbl 1117.52017 · doi:10.1155/IMRN.2005.2709 |
| [12] | DOI: 10.1090/S0894-0347-01-00385-X · Zbl 1021.16017 · doi:10.1090/S0894-0347-01-00385-X |
| [13] | DOI: 10.1007/s00222-003-0302-y · Zbl 1054.17024 · doi:10.1007/s00222-003-0302-y |
| [14] | Fomin S., Math. 158 pp 977– (2003) |
| [15] | DOI: 10.1007/s00222-007-0096-4 · Zbl 1140.18007 · doi:10.1007/s00222-007-0096-4 |
| [16] | Keller B., Doc. Math. 10 pp 551– (2005) |
| [17] | DOI: 10.1016/j.aim.2006.07.013 · Zbl 1128.18007 · doi:10.1016/j.aim.2006.07.013 |
| [18] | DOI: 10.1007/s00209-007-0165-9 · Zbl 1133.18005 · doi:10.1007/s00209-007-0165-9 |
| [19] | DOI: 10.1090/S0002-9947-03-03320-8 · Zbl 1042.52007 · doi:10.1090/S0002-9947-03-03320-8 |
| [20] | DOI: 10.1016/j.jalgebra.2007.09.012 · Zbl 1155.16016 · doi:10.1016/j.jalgebra.2007.09.012 |
| [21] | DOI: 10.1016/j.jalgebra.2008.11.007 · Zbl 1177.18008 · doi:10.1016/j.jalgebra.2008.11.007 |
| [22] | DOI: 10.1016/j.jalgebra.2009.01.032 · Zbl 1196.16013 · doi:10.1016/j.jalgebra.2009.01.032 |
| [23] | DOI: 10.1016/j.jalgebra.2006.03.012 · Zbl 1106.18007 · doi:10.1016/j.jalgebra.2006.03.012 |
| [24] | DOI: 10.1016/j.jpaa.2006.06.006 · Zbl 1127.16014 · doi:10.1016/j.jpaa.2006.06.006 |
| [25] | DOI: 10.1007/s10801-007-0074-3 · Zbl 1145.16006 · doi:10.1007/s10801-007-0074-3 |
| [26] | DOI: 10.1080/00927872.2010.489082 · Zbl 1250.16015 · doi:10.1080/00927872.2010.489082 |
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