ar X iv : 0 70 9 . 08 50 v 1 [ m at h . R T ] 5 S ep 2 00 7 On the Galois coverings of a cluster - tilted algebra ∗

The cluster category was introduced in (BMRRT, 2006) and also in (CCS1, 2006) for type A, as a categorical model to understand better the cluster algebras of Fomin and Zelevinsky (FZ, 2002). It is a quotient of the bounded derived category Db(modA) of the finitely generated modules over a finite dimensional hereditary algebra A. It was then natural to consider the endomorphism algebras of tilting objects in the cluster category. Such algebras are called cluster-tilted, and have been the subject of several investigations since their introduction in (BMR1, 2007; CCS1, 2006), see, for instance (BMR2, 2006; CCS2, 2006; KR, preprint; ABS1, preprint; ABS2, preprint; BFPT, preprint). In particular, it was shown in (ABS1, preprint) that the cluster-tilted algebras are trivial extensions of tilted algebras by a certain bimodule. Now, the class of trivial extensions of tilted algebras by the minimal injective cogenerator has been extensively investigated. They play an important rôle in the classification results for self-injective algebras. In this study, one of the essential tools is the repetitive algebra, introduced by Hughes and Waschbüsch in (HW, 1983). In previous works (ABST1, 2006; ABST2, to appear), we have related the cluster category and the m-cluster category to the repetitive algebra of a hereditary algebra.