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- W2171153887 abstract "We construct a CalderoâChapoton map on a triangulated category with a cluster tilting subcategory which may have infinitely many indecomposable objects. The map is not necessarily defined on all objects of the triangulated category, but we show that it is a (weak) cluster map in the sense of BuanâIyamaâReitenâScott. As a corollary, it induces a surjection from the set of exceptional objects which can be reached from the cluster tilting subcategory to the set of cluster variables of an associated cluster algebra. Along the way, we study the interaction between CalabiâYau reduction, cluster structures, and the CalderoâChapoton map. We apply our results to the cluster category $mathscr {D}$ of Dynkin type $A_{infty }$ which has a rich supply of cluster tilting subcategories with infinitely many indecomposable objects. We show an example of a cluster map which cannot be extended to all of $mathscr {D}$. The case of $mathscr {D}$ also permits us to illuminate results by AssemâReutenauerâSmith on $operatorname {SL}_2$-tilings of the plane." @default.
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- W2171153887 date "2012-11-06" @default.
- W2171153887 modified "2023-09-25" @default.
- W2171153887 title "A Caldero–Chapoton map for infinite clusters" @default.
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- W2171153887 doi "https://doi.org/10.1090/s0002-9947-2012-05464-x" @default.
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