Related papers: Bi-Hamiltonian structure of Sutherland models coup…
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 $\mathfrak{u}(n)^*$-valued momenta generate a bi-Hamiltonian hierarchy. One of…
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…
We report on the the trigonometric spin Ruijsenaars--Sutherland hierarchy derived recently by Poisson reduction of a bi-Hamiltonian hierarchy associated with free geodesic motion on the Lie group U(n). In particular, we give a direct proof…
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…
A deformation of the classical trigonometric BC(n) Sutherland system is derived via Hamiltonian reduction of the Heisenberg double of SU(2n). We apply a natural Poisson-Lie analogue of the Kazhdan-Kostant-Sternberg type reduction of the…
We derive a Hamiltonian structure for the $N$-particle hyperbolic spin Ruijsenaars-Schneider model by means of Poisson reduction of a suitable initial phase space. This phase space is realised as the direct product of the Heisenberg double…
We consider two families of commuting Hamiltonians on the cotangent bundle of the group GL(n,C), and show that upon an appropriate single symplectic reduction they descend to the spectral invariants of the hyperbolic Sutherland and of the…
We investigate the finite dimensional dynamical system derived by Braden and Hone in 1996 from the solitons of $A_{n-1}$ affine Toda field theory. This system of evolution equations for an $n\times n$ Hermitian matrix $L$ and a real…
Motivated by the group of Galilean transformations and the subgroup of Galilean transformations which fix time zero, we introduce the notion of a $b$-Lie group as a pair $(G,H)$ where $G$ is a Lie group and $H$ is a codimension-one Lie…
We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of…
This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian…
Some generalizations of spin Sutherland models descend from `master integrable systems' living on Heisenberg doubles of compact semisimple Lie groups. The master systems represent Poisson--Lie counterparts of the systems of free motion…
We consider three 'classical doubles' of any semisimple, connected and simply connected compact Lie group $G$: the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson double. On each double we identify a pair of…
On a cotangent bundle $T\sp*G$ of a Lie group $G$ one can describe the standard Liouville form $\theta$ and the symplectic form $d \theta$ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of $G$ on…
We study Hamiltonian reductions of the free geodesic motion on a non-compact simple Lie group using as reduction group the direct product of a maximal compact subgroup and the fixed point subgroup of an arbitrary involution commuting with…
Let $\Sigma $ be a compact connected and oriented surface with nonempty boundary and let $G$ be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. The moduli space of flat principal $G$-bundles over $\Sigma$ which are…
Given a Lie-Poisson completely integrable bi-Hamiltonian system on $\mathbb{R}^n$, we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson…
We demonstrate the common bihamiltonian nature of several integrable systems. The first one is an elliptic rotator that is an integrable Euler-Arnold top on the complex group GL(N) for any $N$, whose inertia ellipsiod is related to a choice…
We provide a complete classification of all the ways the Pais-Uhlenbeck osicllator might be embedded in two dimensional space. We discuss the Bi-Hamiltonian structures of this model, and examine how alternative Hamiltonian structures might…
The BC(n) Sutherland Hamiltonian with coupling constants parametrized by three arbitrary integers is derived by reductions of the Laplace operator of the group U(N). The reductions are obtained by applying the Laplace operator on spaces of…