Abstract

We review our results on bi-Hamiltonian structures of trigonometric spin Sutherland models builton collective spin variables. Our basic observation was that the cotangent bundle T'*U(n) and itsholomorphic analogue T'*GL(n, C), as well as T"*GL(n, C)k, carry a natural quadratic Poissonbracket, which is compatible with the canonical linear one. The quadratic bracket arises by changeof variables and analytic continuation from an associated Heisenberg double. Then, the reductionsof 'T'*U(n) and T"*GL(n, C) by the conjugation actions of the corresponding groups lead to thereal and holomorphic spin Sutherland models, respectively, equipped with a bi-Hamiltonianstructure. The reduction of T'*GL(n, C)k by the group U(n)xU(n) gives a generalizedSutherland model coupled to two u(n)*-valued spins. We also show that a bi-Hamiltonian structureon the associative algebra gl(n, IR) that appeared in the context of Toda models can be interpretedas the quotient of compatible Poisson brackets on T"*GL(n,R). Before our work, all thesereductions were studied using the canonical Poisson structures of the cotangent bundles, withoutrealizing the bi-Hamiltonian aspect.