Title:

Frobenius categorification of cluster algebras

Cluster categories, introduced by Buan–Marsh–Reineke–Reiten–Todorov and later generalised by Amiot, are certain 2Calabi–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 projectiveinjective objects corresponding to these special variables. Amiot–Iyama–Reiten show how Frobenius categories admitting (d1)clustertilting objects arise naturally from the data of a Noetherian bimodule dCalabi–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 dCalabi–Yau with respect to e; this new definition allows the dCalabi–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 clustertilting object in the Frobenius category GP(B) of Gorenstein projective Bmodules. 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 clustertilting object in one of these categories is a frozen Jacobian algebra. Following Keller–Reiten, we observe that such algebras are internally 3Calabi–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 3Calabi–Yau. If true, the algebra may be realised as the endomorphism algebra of a clustertilting object in a Frobenius category GP(B) as above. We further conjecture that GP(B) is stably 2Calabi–Yau, in which case it would provide a categorification of this second cluster algebra.
