Preprojective cluster variables of acyclic cluster algebras

It is proved that any cluster-tilted algebra defined in the cluster category C(H) has the same representation type as the initial hereditary algebra H . For any valued quiver (Γ,Ω), an injection from the subset PI(Ω) of the cluster category C(Ω) consisting of indecomposable preprojective objects, preinjective objects and the first shifts of indecomposable projective modules to the set of cluster variables of the corresponding cluster algebra AΩ is given. The images are called preprojective cluster variables. It is proved that all preprojective cluster variables other than ui have denominators udimM in their irreducible fractions of integral polynomials, where M is the corresponding preprojective module or preinjective module. In case the valued quiver (Γ,Ω) is of finite type, the denominator theorem holds with respect to any cluster. Namely, let x = (x1, · · · , xn) be a cluster of the cluster algebra AΩ, and V the cluster tilting object in C(Ω) corresponding to x, whose endomorphism algebra is denoted by Λ. Then the denominator of any cluster variable y other than xi is xdimM , where M is the indecomposable Λ−module corresponding to y. This result is a generalization of the corresponding result in [CCS2] to non simply-laced case.