On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system

Abstract

We suggest a Hamiltonian formulation for the spin Ruijsenaars-Schneider system in the trigonometric case. Within this interpretation, the phase space is obtained by a quasi-Hamiltonian reduction performed on (the cotangent bundle to) a representation space of a framed Jordan quiver. For arbitrary quivers, analogous varieties were introduced by Crawley-Boevey and Shaw, and their interpretation as quasi-Hamiltonian quotients was given by Van den Bergh. Using Van den Bergh's formalism, we construct commuting Hamiltonian functions on the phase space and identify one of the flows with the spin Ruijsenaars-Schneider system. We then calculate all the Poisson brackets between local coordinates, thus answering an old question of Arutyunov and Frolov. We also construct a complete set of commuting Hamiltonians and integrate all the flows explicitly.

arXiv comment30 pages. v2: improved exposition in Section 3.1, added Remarks 5.6 and 5.7. v3: References added, accepted version

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Cited by in corpus (17)

• Superintegrable Systems on Moduli Spaces of Flat Connections
• Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers
• Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities
• Elliptic generalization of integrable q-deformed anisotropic Haldane-Shastry long-range spin chain
• Lax equations for relativistic $\mathrm{GL}(NM,\mathbb{C})$ Gaudin models on elliptic curve
• Relativistic interacting integrable elliptic tops
• Integrable System of Generalized Relativistic Interacting Tops
• Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems
• Hamiltonian structures for integrable nonabelian difference equations
• Superintegrability of Calogero-Moser systems associated with the cyclic quiver
• Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
• Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $U(n)$
• Poisson reductions of master integrable systems on doubles of compact Lie groups
• Wreath Macdonald polynomials as eigenstates
• Integrable systems on multiplicative quiver varieties from cyclic quivers
• Poisson-Lie analogues of spin Sutherland models revisited
• Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n{)}^{\ast }$-valued spins from Poisson reduction