Poisson-Lie analogues of spin Sutherland models

Abstract

We present generalizations of the well-known trigonometric spin Sutherland models, which were derived by Hamiltonian reduction of `free motion' on cotangent bundles of compact simple Lie groups based on the conjugation action. Our models result by reducing the corresponding Heisenberg doubles with the aid of a Poisson-Lie analogue of the conjugation action. We describe the reduced symplectic structure and show that the `reduced main Hamiltonians' reproduce the spin Sutherland model by keeping only their leading terms. The solutions of the equations of motion emerge from geodesics on the compact Lie group via the standard projection method and possess many first integrals. Similar hyperbolic spin Ruijsenaars--Schneider type models were obtained previously by L.-C. Li using a different method, based on coboundary dynamical Poisson groupoids, but their relation with spin Sutherland models was not discussed.

27 pages, v3: final version, some clarifying remarks are added

References in corpus (10)

• Physics and Mathematics of Calogero particles
• From Spinning Conformal Blocks to Matrix Calogero-Sutherland Models
• Torus Actions and Integrable Systems
• Symmetries and Integrability
• Quantum Lax pairs via Dunkl and Cherednik operators
• Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction
• Quasi-compact Higgs bundles and Calogero-Sutherland systems with two types spins
• Twisted spin Sutherland models from quantum Hamiltonian reduction
• Coboundary dynamical Poisson groupoids and integrable systems
• The non-Abelian momentum map for Poisson-Lie symmetries on the chiral WZNW phase space

Cited by in corpus (14)

• 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
• Integrable System of Generalized Relativistic Interacting Tops
• Relativistic interacting integrable elliptic tops
• Reduction of a bi-Hamiltonian hierarchy on $T^{\ast }\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models
• Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
• Poisson reductions of master integrable systems on doubles of compact Lie groups
• Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $U(n)$
• Integrable systems on multiplicative quiver varieties from cyclic quivers
• Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n{)}^{\ast }$-valued spins from Poisson reduction
• On the bi-Hamiltonian structure of the trigonometric spin Ruijsenaars--Sutherland hierarchy
• Poisson-Lie analogues of spin Sutherland models revisited