# Representations of Algebras and Related Topics

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#### DOI: 10.4171/101-1/9

Periodicities in cluster algebras and dilogarithm identities

Tomoki Nakanishi[1]

(1) Nagoya University, Nagoya, Japan

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.

Keywords: Cluster algebras, T-systems, Y-systems, dilogarithm