Dilogarithm identities of higher degree and cluster $a$-variable periodicity

TL;DR

This paper demonstrates that higher degree dilogarithm identities, derived from cluster mutation periodicities, can be deduced from classical identities in cases where the exchange matrix is full rank, linking advanced identities to classical theory.

Contribution

It shows that Nakanishi's higher degree dilogarithm identities follow from classical identities via cluster mutation periodicities when the exchange matrix is full rank.

Findings

Higher degree dilogarithm identities are derivable from classical identities.

Cluster groupoid mutation periodicities underpin these identities.

The results apply specifically when the exchange matrix is full rank.

Abstract

We use the periodicities of cluster groupoid mutations established by Li and the second author to prove that the dilogarithm identities of higher degree obtained by Nakanishi follow from the classical dilogarithm identities associated to a periodicity of cluster mutations when the exchange matrix is full rank.