Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities
arXiv:1611.02813 · doi:10.1093/integr/xyx005
Abstract
We introduce and study a Hamiltonian formalism of mutations in cluster algebras using canonical variables, where the Hamiltonian is given by the Euler dilogarithm. The corresponding Lagrangian, restricted to a certain subspace of the phase space, coincides with the Rogers dilogarithm. As an application, we show how the dilogarithm identity associated with a period of mutations in a cluster algebra arises from the Hamiltonian/Lagrangian point of view.
ver1: 28 pages; ver2: 30 pages, Sec.5.4.3 revised + minor changes; ver3: 30 pages, Remark 6.9 replaced with Theorem 6.9 + minor changes; ver4:31 pages, Sec.5.2, 5.3 revised + minor changes; ver5: 31 pages, minor changes, final version to appear in J. Integrable Syst