Cluster structures in 2-Calabi–Yau triangulated categories of Dynkin type with maximal rigid objects

Abstract

In this paper, we consider two kinds of 2-Calabi–Yau triangulated categories with finitely many indecomposable objects up to isomorphisms, called An,t = D b (KA(2t+1)(n+1)−3)/τt(n+1)−1[1], where n, t ≥ 1, and Dn,t = D b (KD2t(n+1))/τ(n+1)φ n , where n, t ≥ 1, and φ is induced by an automorphism of D2t(n+1) of order 2. Except the categories An,1, they all contain non-zero maximal rigid objects which are not cluster tilting. An,1 contain cluster tilting objects. We define the cluster complex of An,t (resp. Dn,t) by using the geometric description of cluster categories of type A (resp. type D). We show that there is an isomorphism from the cluster complex of An,t (resp. Dn,t) to the cluster complex of root system of type B n . In particular, the maximal rigid objects are isomorphic to clusters. This yields a result proved recently by Buan–Palu–Reiten: Let $${R_{{A_{n,t}}}}$$ , resp. $${R_{{D_{n,t}}}}$$ , be the full subcategory of An,t, resp. Dn,t, generated by the rigid objects. Then $${R_{{A_{n,t}}}} \simeq {R_{{A_{n,1}}}}$$ and $${R_{{D_{n,t}}}} \simeq {R_{{A_{n,1}}}}$$ as additive categories, for all t ≥ 1.