Cluster categories, introduced by Buan–Marsh–Reineke–Reiten–Todorov and later generalised by Amiot, are certain 2-Calabi–Yau triangulated categories that model the combinatorics of cluster algebras without frozen variables. When frozen variables do occur, it is natural to try to model the cluster combinatorics via a Frobenius category, with the indecomposable projective-injective objects corresponding to these special variables. Amiot–Iyama–Reiten show how Frobenius categories admitting (d-1)-cluster-tilting objects arise naturally from the data of a Noetherian bimodule d-Calabi–Yau algebra A and an idempotent e of A such that A/< e > is finite dimensional. In this work, we observe that this phenomenon still occurs under the weaker assumption that A and A^op are internally d-Calabi–Yau with respect to e; this new definition allows the d-Calabi–Yau property to fail in a way controlled by e. Under either set of assumptions, the algebra B=eAe is Iwanaga–Gorenstein, and eA is a cluster-tilting object in the Frobenius category GP(B) of Gorenstein projective B-modules. Geiß–Leclerc–Schröer define a class of cluster algebras that are, by construction, modelled by certain Frobenius subcategories Sub(Q_J) of module categories over preprojective algebras. Buan–Iyama–Reiten–Smith prove that the endomorphism algebra of a cluster-tilting object in one of these categories is a frozen Jacobian algebra. Following Keller–Reiten, we observe that such algebras are internally 3-Calabi–Yau with respect to the idempotent corresponding to the frozen vertices, thus obtaining a large class of examples of such algebras. Geiß–Leclerc–Schröer also attach, via an algebraic homogenization procedure, a second cluster algebra to each category Sub(Q_J), by adding more frozen variables. We describe how to compute the quiver of a seed in this cluster algebra via approximation theory in the category Sub(Q_J); our alternative construction has the advantage that arrows between the frozen vertices appear naturally. We write down a potential on this enlarged quiver, and conjecture that the resulting frozen Jacobian algebra A and its opposite are internally 3-Calabi–Yau. If true, the algebra may be realised as the endomorphism algebra of a cluster-tilting object in a Frobenius category GP(B) as above. We further conjecture that GP(B) is stably 2-Calabi–Yau, in which case it would provide a categorification of this second cluster algebra.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:678852 |
| Date | January 2015 |
| Creators | Pressland, Matthew |
| Contributors | King, Alastair |
| Publisher | University of Bath |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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