Reduction of a bi-Hamiltonian hierarchy on $T^{\ast }\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models

Abstract

We first exhibit two compatible Poisson structures on the cotangent bundle of the unitary group $\mathrm{U}(n)$ in such a way that the invariant functions of the $\mathfrak{u}(n{)}^{\ast }$-valued momenta generate a bi-Hamiltonian hierarchy. One of the Poisson structures is the canonical one and the other one arises from embedding the Heisenberg double of the Poisson-Lie group $\mathrm{U}(n)$ into $T^{\ast }\mathrm{U}(n)$, and subsequently extending the embedded Poisson structure to the full cotangent bundle. We then apply Poisson reduction to the bi-Hamiltonian hierarchy on $T^{\ast }\mathrm{U}(n)$ using the conjugation action of $\mathrm{U}(n)$, for which the ring of invariant functions is closed under both Poisson brackets. We demonstrate that the reduced hierarchy belongs to the overlap of well-known trigonometric spin Sutherland and spin Ruijsenaars--Schneider type integrable many-body models, which receive a bi-Hamiltonian interpretation via our treatment.

20 pages, added proofs of Proposition 2.1 and Proposition 2.2 in v2

References in corpus (5)

• A simple way of making a Hamiltonian system into a bi-Hamiltonian one
• Poisson-Lie analogues of spin Sutherland models
• Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction
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Cited by in corpus (5)

• Bi-Hamiltonian structure of spin Sutherland models: the holomorphic case
• Poisson reductions of master integrable systems on doubles of compact Lie groups
• 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