Grothendieck groups of d-exangulated categories and a modified Caldero-Chapoton map

TY - JOUR

T1 - Grothendieck groups of d-exangulated categories and a modified Caldero-Chapoton map

AU - Jørgensen, Peter

AU - Shah, Amit

PY - 2024/5

Y1 - 2024/5

N2 - 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 K 0 sp(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 K 0(S,E X,s X) of the category S, equipped with the d-exangulated structure induced by X, and the quotient K 0 sp(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 K 0-class in K 0(S,E X,s X) as a higher index with respect to the rigid subcategory X.

AB - 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 K 0 sp(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 K 0(S,E X,s X) of the category S, equipped with the d-exangulated structure induced by X, and the quotient K 0 sp(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 K 0-class in K 0(S,E X,s X) as a higher index with respect to the rigid subcategory X.

KW - Caldero-Chapoton map

KW - Grothendieck group

KW - Index

KW - Rigid subcategory

KW - n-cluster tilting subcategory

KW - n-exangulated category

UR - http://www.scopus.com/inward/record.url?scp=85181695900&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2023.107587

DO - 10.1016/j.jpaa.2023.107587

M3 - Journal article

SN - 0022-4049

VL - 228

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 5

M1 - 107587

ER -