Rogers dilogarithms of higher degree and generalized cluster algebras

Наканиши, Томоки

doi:10.2969/jmsj/75767576

Cited by 6 publications

(20 citation statements)

References 21 publications

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“…Denote the symplectic potential corresponding to the new triangulation by θM and introduce the Rogers dilogarithm L which for x ≥ 0 is defined by (see (1.9) of [36]):…”

Section: Tau-function As Generating Function Of Monodromy Symplectomorphismmentioning

confidence: 99%

Tau-Functions and Monodromy Symplectomorphisms

Bertola

Korotkin

2021

Commun. Math. Phys.

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We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical r-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock–Goncharov coordinates are log-canonical for the symplectic form. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the Jimbo–Miwa–Ueno tau-function as the generating function of the monodromy map. This, in particular, solves a recent conjecture by A. Its, O. Lisovyy and A. Prokhorov.

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“…Denote the symplectic potential corresponding to the new triangulation by θM and introduce the Rogers dilogarithm L which for x ≥ 0 is defined by (see (1.9) of [36]):…”

Section: Tau-function As Generating Function Of Monodromy Symplectomorphismmentioning

confidence: 99%

Tau-Functions and Monodromy Symplectomorphisms

Bertola

Korotkin

2021

Commun. Math. Phys.

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“…The following theorem was proved in [Nak11b] by a cluster algebraic method with the help of the constancy condition from [FS95]. See also [Nak16]. Theorem 6.1 (Dilogarithm identity [Nak11b, Theorems 6.4 and 6.8]).…”

Section: 42mentioning

confidence: 99%

“…Remark 3.14. The realization of quantum y-variables by the canonical variables presented here appeared in [FG09b,KN11,Nak16]. In fact, the construction of x-and y-variables in (3.7) and (3.8) is deduced from the quantum ones in [KN11,Nak16].…”

Section: Signed Mutations Let Us Introduce a Composition Of Mapsmentioning

confidence: 99%

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Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities

Gekhtman

Наканиши

Rupel

2017

Journal of Integrable Systems

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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. 1 2. Preliminaries 2.1. Mutations in cluster algebras. Let us recall two main notions in cluster algebras, namely, a seed and its mutation. See [FZ02, FZ07] for more information on cluster algebras.

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“…The function L 2,λ has been considered for the first time by Hill (compare (58) with the function denoted by D α x in [Hi]). It is a particular case of a class of functions recently introduced by Nakanishi in [Nakan4] (see also [Nakan6,§3]), the so called 'generalized (Euler) dilogarithms'. This author has proved (cf.…”

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confidence: 99%

On webs, polylogarithms and cluster algebras

Pirio

2021

Preprint

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In this text, we investigate webs which can be associated to cluster algebras from the point of view of the abelian functional equations these webs carry, focusing on the polylogarithmic ones. We introduce a general notion of webs whose rank is 'As Maximal as Possible' (AMP) and show that many webs associated either to polylogarithmic functional equations or to cluster algebras are of this type. In particular, we prove a few results and state some conjectures about cluster webs associated to (pairs of) Dynkin diagrams. Along the way, we show that many of the classical functional equations satisfied by low-order polylogarithms (such as Spence-Kummer's equation of the trilogarithm or the tetralogarithmic one of Kummer) are of cluster type.

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Rogers dilogarithms of higher degree and generalized cluster algebras

Cited by 6 publications

References 21 publications

Tau-Functions and Monodromy Symplectomorphisms

Tau-Functions and Monodromy Symplectomorphisms

Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities

On webs, polylogarithms and cluster algebras

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