QUIVERS WITH POTENTIALS AND THEIR REPRESENTATIONS II: APPLICATIONS TO CLUSTER ALGEBRAS

摘要

This paper continues our study of quivers with potentials and their representations initiated in [9]. Here we develop some applications of this theory to the theory of cluster algebras. As shown in [12], the structure of cluster algebras is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F-polynomials. In the case of skew-symmetric exchange matrices (the terminology will be recalled later), we find an interpretation of g-vectors and F-polynomials in terms of representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about g-vectors and F-polynomials made in [12].