Periodicities in cluster algebras and dilogarithm identities

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.