Bi-Hamiltonian Structure of Spin Sutherland Models: The Holomorphic Case

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 syinplectic 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.