Cluster Algebras and Polylogarithm Relations

Abstract

We seek to illuminate the connection between multiple polylogarithm relations and cluster algebras in two ways. First, we give a uniform description of the cluster modular group of affine and doubly extended cluster algebras. This will be critical for the future work of extracting polylogarithm relations from infinite type cluster algebras. Second, we introduce a differential one form, ωn, associated to each multiple polylogarithm, which can be used to compute multiple polylogarithm relations. This form satisfies a clean recurrence relation, mirroring the inductive definition of multiple polylogarithms. We are able to use this recurrence to find several families of “small” polylogarithm relations that hold in any weight. Finally for small values of n, we extract polylogarithm relations from type An and Dn cluster algebras.