On Tilting Modules over Cluster-tilted Algebras

In this paper, we show that the tilting modules over a clustertilted algebra A lift to tilting objects in the associated cluster category CH . As a first application, we describe the induced exchange relation for tilting Amodules arising from the exchange relation for tilting object in CH . As a second application, we exhibit tilting A-modules having cluster-tilted endomorphism algebras. Cluster algebras were introduced by Fomin and Zelevinsky [FZ02] in the context of canonical basis of quantized enveloping algebras and total positivity for algebraic groups, but quickly turned out to be related to many other fields in mathematics. In the representation theory of finite dimensional algebras, the so-called cluster categories were introduced in [BMR06] (and also in [CCS06] for the An case) as a natural categorical model for the combinatorics of the corresponding cluster algebras of Fomin and Zelevinsky. The construction is as follows. Let Q be a quiver without oriented cycles. There is then, for a field k, an associated finite dimensional hereditary path algebra H = kQ. Since H has finite global dimension, its bounded derived category D(H) of the finitely generated modules has almost split triangles [Hap88]. Let τ be the corresponding translation functor. Denoting by F the composition τ[1], where [1] is the shift functor in D(H), the cluster category CH was defined as the orbit category D(H)/F , and was shown to be canonically triangulated [Kel05] and have almost split triangles [BMR06]. In this model, the exceptional objects are associated with the cluster variables of [FZ02] while the tilting objects correspond to the clusters. Remarkably, one also defines an exchange relation on the tilting objects in CH , corresponding to the exchange relation on the clusters of [FZ02]. More precisely, an almost complete tilting object T in CH has exactly two nonisomorphic indecomposable complements M and M, and these are related by triangles M g //B f //M //M[1] and M f∗ //B g∗ //M //M [1] where f, g are minimal right addT -approximations and f, g are minimal left addT -approximations (see [BMR06]). In view of the importance of tilting theory in the representation theory of finite dimensional algebras, the (opposite) endomorphism algebras of these tilting objects, called cluster-tilted, were then introduced and studied in [BMR07] (see also [CCS06]). Their module theory was shown to be to a large extent determined by the cluster categories in which they arise. Indeed, given a cluster category CH and a 2000 Mathematics Subject Classification. 16G20, 18E30.