Bi-Hamiltonian structure of spin Sutherland models: the holomorphic case

Abstract

We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of GL(n,C), which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard GL(n,C) Poisson--Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

Expanded to 20 pages, contains a simplified formula of the second reduced Poisson bracket, more detailed derivations, and added references

References in corpus (2)

• Three lectures on classical integrable systems and gauge field theories
• Real Forms of Holomorphic Hamiltonian Systems

Cited by in corpus (4)

• Universal Spinning Casimir Equations and Their Solutions
• 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
• A note on quadratic Poisson brackets on $\mathrm{gl}(n,\mathbb{R})$ related to Toda lattices