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Cluster characters for cluster categories with infinite-dimensional morphism spaces. (English) Zbl 1288.13016
Cluster algebras, invented by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)] in order to study total positivity in algebraic groups and canonical bases in quantum groups, are a class of commutative algebras endowed with a distinguished set of generators, the cluster variables. The cluster variables are grouped into finite subsets, called clusters, and are defined recursively from initial variables through mutation on the clusters. Cluster categories [A. B. Buan et al., Adv. Math. 204, No. 2, 572–618 (2006; Zbl 1127.16011)] are certain categories of representations of finite dimensional algebras which were introduced to “categorify” cluster algebras. The Caldero–Chapoton map was introduced in [P. Caldero and F. Chapoton, Comment. Math. Helv. 81, No. 3, 595–616 (2006; Zbl 1119.16013)] to formalize the connection between the cluster algebras and the cluster categories. Indeed, using the Caldero–Chapoton map, [P. Caldero and B. Keller, Ann. Sci. Éc. Norm. Supér. (4) 39, No. 6, 983–1009 (2006; Zbl 1115.18301)] established a bijection between the indecomposable rigid objects of a cluster category and the cluster variables of the corresponding cluster algebra.
Using the notion of quiver with potential [H. Derksen et al., Sel. Math., New Ser. 14, No. 1, 59–119 (2008; Zbl 1204.16008)], C. Amiot generalized the definition of cluster category [C. Amiot, Ann. Inst. Fourier 59, No. 6, 2525–2590 (2009; Zbl 1239.16011)]. In the case when the quiver with potential is Jacobi-finite, the cluster character introduced in [Y. Palu, Ann. Inst. Fourier 58, No. 6, 2221–2248 (2008; Zbl 1154.16008)] sends reachable indecomposable rigid objects of the (generalized) cluster category to cluster variables. In their works, the categories encountered are \(\text{Hom}{}\)-finite and 2-Calabi-Yau.
In the paper under review, the author studies a version of Y. Palu’s cluster characters for \(\text{Hom}{}\)-infinite cluster categories \(\mathcal{C}\), that is, cluster categories with possibly infinite-dimensional morphism spaces. This cluster character \(L \mapsto X'_L\) is not defined for all objcts \(L\) but only for those in a suitable mutation-invariant subcategory \(\mathcal{D}\). As an application, the author proves that this cluster character indeed realises in full generality a surjection from the set of indecomposable rigid reachable objects in \(\mathcal{D}\) to the set of cluster variables in the cluster algebra. It is worth mentioning that the author used this cluster character to prove several classical conjectures on cluster algebras in [Compos. Math. 147, No. 6, 1921–1954 (2011; Zbl 1244.13017)].

MSC:
13F60 Cluster algebras
18E30 Derived categories, triangulated categories (MSC2010)
16G20 Representations of quivers and partially ordered sets
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