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Poisson reduction of twisted Heisenberg double and finite-dimensional integrable systems. (English) Zbl 0952.37023

From the text: The author shows that the Poisson structure on the reduced space coincides with the one of the moduli space of flat connections on a Riemann surface of genus one. The recent developments in the theory of integrable many-body systems are mainly related with the discovery of dynamical \(r\)-matrices, i.e. \(r\)-matrices depending on phase variables. One natural way to understand the origin of dynamical \(r\)-matrices is to consider the reduction procedure. In this approach he starts with an initial phase space \({\mathcal P}\) supplied with a symplectic action of some symmetry group. Considering a relatively simple invariant Hamiltonian and factorizing the corresponding dynamics by the symmetry group he gets a smaller phase space \({\mathcal P}_{\text{red}}\) with a nontrivial dynamic. Then the \(L\)-operator coming in the Lax representation \(dL/dt= [M, L]\) appears as a specific coordinate on \({\mathcal P}_{\text{red}}\) while the dynamical \(r\)-matrix describes the Poisson (Dirac) bracket on the reduced space.
In this note the author clarifies the Poisson reduction procedure for the twisted Heisenberg double (HD). The twisted HD is a lattice analog of the affine HD. The importance of the affine HD becomes clear due to its relation with integrable many-body systems of Calogero type. As was shown earlier the integrable Calogero-Moser and Ruijsenaars-Schneider (rational and trigonometric in the latter case) hierarchies are obtained by the reduction from the hierarchy of phase spaces.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81R12 Groups and algebras in quantum theory and relations with integrable systems
82B23 Exactly solvable models; Bethe ansatz
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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