City Research Online

Abstract

A strong connection between cluster algebras and representation theory was established by the cluster category. Cluster characters, like the original Caldero-Chapoton map, are maps from certain triangulated categories to cluster algebras and they have generated much interest. Holm and Jørgensen constructed a modified Caldero-Chapoton map from a sufficiently nice triangulated category to a commutative ring, which is a generalised frieze under some conditions. In their construction, a quotient K0sp(T)/M of a Grothendieck group of a cluster tilting subcategory T is used. In this article, we show that this quotient is the Grothendieck group of a certain extriangulated category, thereby exposing the significance of it and the relevance of extriangulated structures. We use this to define another modified Caldero-Chapoton map that recovers the one of Holm–Jørgensen. We prove our results in a higher homological context. Suppose S is a (d+2)-angulated category with subcategories X⊆T⊆S, where X is functorially finite and T is 2d-cluster tilting, satisfying some mild conditions. We show there is an isomorphism between the Grothendieck group K0(S,EX,sX) of the category S, equipped with the d-exangulated structure induced by X, and the quotient K0sp(T)/N, where N is the higher analogue of M above. When X=T the isomorphism is induced by the higher index with respect to T introduced recently by Jørgensen. Thus, in the general case, we can understand the map taking an object in S to its K0-class in K0(S,EX,sX) as a higher index with respect to the rigid subcategory X.

Publication Type:	Article
Additional Information:	This is an open access article distributed under the terms of the Creative Commons CC-BY license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Publisher Keywords:	Caldero-Chapoton map, Grothendieck group, Index, n-cluster tilting subcategory, n-exangulated category, Rigid subcategory
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology School of Science & Technology > Department of Mathematics
SWORD Depositor:	Symplectic Administrator

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