Periodicities in cluster algebras and dilogarithm identities

TL;DR

This paper explores periodic mutation sequences in cluster algebras, defining associated T- and Y-systems and dilogarithm identities, with proofs for skew-symmetric cases, advancing understanding of algebraic periodicities and identities.

Contribution

It introduces new periodicity concepts in cluster algebras, defines related T-, Y-systems, and formulates dilogarithm identities, providing proofs for skew-symmetric exchange matrices.

Findings

Defined T- and Y-systems for periodic mutations

Formulated dilogarithm identities for seed periodicities

Proved identities for skew-symmetric exchange matrices

Abstract

We consider two kinds of periodicities of mutations in cluster algebras. For any sequence of mutations under which exchange matrices are periodic, we define the associated T- and Y-systems. When the sequence is `regular', they are particularly natural generalizations of the known `classic' T- and Y-systems. Furthermore, for any sequence of mutations under which seeds are periodic, we formulate the associated dilogarithm identity. We prove the identities when exchange matrices are skew symmetric.