Cluster algebras arising from cluster tubes

We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2−Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a maximal rigid object T in the cluster tube Γn of rank n (n > 1). For any indecomposable rigid object M in Γn, we define an analogous XM of Caldero-Chapton’s formula (or Palu’s cluster character formula) by using the geometric information of M. We show that XM , XM′ satisfy the mutation formula for cluster variables when M,M′ form an exchange pair, and that X? : M 7→ XM gives a bijection from the set of indecomposable rigid objects in Γn to the set of cluster variables of the cluster algebra of type Cn−1, which induces a bijection between the set of (basic) maximal rigid objects in Γn and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne [BMV] that the combinatorics of maximal rigid objects in the cluster tube Γn encodes the combinatorics of the cluster algebra of type Bn−1 since the combinatorics of cluster algebras of type Bn−1 and of type Cn−1 are the same by one of results of Fomin-Zelevinsky in [FZ2]. As a consequence, we give a categorification of cluster algebras of type C.