Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.