Abstract
In Igusa and Todorov(2013) we constructed topological triangulated categories \(\mathcal {C}_{c}\) as stable categories of certain topological Frobenius categories \(\mathcal {F}_{c}\). In this paper we show that these categories have a cluster structure for certain values of c including c = π. The continuous cluster categories are those \(\mathcal {C}_{c}\) which have cluster structure. We study the basic structure of these cluster categories and we show that \(\mathcal {C}_{c}\) is isomorphic to an orbit category \(\mathcal {D}_{r}/\underline F_{s}\) of the continuous derived category \(\mathcal {D}_{r}\) if c = rπ/s. In \(\mathcal {C}_{\pi }\), a cluster is equivalent to a discrete lamination of the hyperbolic plane. We give the representation theoretic interpretation of these clusters and laminations.
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The first author is supported by NSA Grant #H98230-13-1-0247. The second author is supported by NSF Grant #DMS-1103813
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Igusa, K., Todorov, G. Continuous Cluster Categories I. Algebr Represent Theor 18, 65–101 (2015). https://doi.org/10.1007/s10468-014-9481-z
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DOI: https://doi.org/10.1007/s10468-014-9481-z
Keywords
- Frobenius categories
- Triangulated categories
- Topological categories
- Laminations
- Hyperbolic plane
- Matrix factorizations