Preprojective cluster variables of acyclic cluster algebras

Abstract

For any valued quiver, by using BGP-reflection functors, an injection from the set of preprojective objects in the cluster category to the set of cluster variables of the corresponding cluster algebra is given, the images are called preprojective cluster variables. It is proved that all preprojective cluster variables have denominators $u^{\underline{dim}M}$ in their irreducible fractions of integral polynomials, where $M$ is the corresponding preprojective module or preinjective module. If the quiver is of Dynkin type, we generalize the denominator theorem in [FZ2] to any seed, and also generalize the corresponding results in [CCS1] [CCS2] [CK1] to non-simply-laced case. Given a finite quiver (with trivial valuations) without oriented cycles, fixed a tilting seed $(V, B_V)$, it is proved that the existence and uniqueness of a bijection (abstractly, not in explicit form, compare [CK2]) from the set of exceptional indecomposable objects in the cluster categories to the set of cluster variables associated to $B_V$ which sends $ V_i[1]$ to $u_i$ and sends cluster tilting objects to clusters.