Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel

We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. In particular, we generalize and prove a conjecture of C. M. Ringel on cluster-additive functions associated to arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of positive spaces of finite type, we use tropical friezes and cluster-additive functions to explicitly express the Fock-Goncharov pairing between their tropical points and the bijections between global monomials on one and tropical points of the other.