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Functional identities for the Rogers dilogarithm associated to cluster $$Y$$-systems. (English) Zbl 1134.33303
The author proves, in the case of specific $$Y$$-systems related to clusters, that there is for each Dynkin diagram of finite type a certain functional equation for the Rogers dilogarithm. The author considers only the simply laced cases, as the other cases can be obtained from these by the usual folding procedure and do not provide any really new identities. The proof given in the paper is uniform for all simply laced root systems and it provides new proofs of previously known cases.

##### MSC:
 33B30 Higher logarithm functions 20F55 Reflection and Coxeter groups (group-theoretic aspects) 33C52 Orthogonal polynomials and functions associated with root systems 82B23 Exactly solvable models; Bethe ansatz
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