香蕉视频

Title: Spin Ruijsenaars-Schneider-Sutherland models with a bi-Hamiltonian structure.

Abstract:聽We report our recent study of 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 $n imes n$ Hermitian matrix $L$ and a real diagonal matrix $q$ with distinct eigenvalues was interpreted as a special case of the spin Ruijsenaars--Schneider models due to Krichever and Zabrodin. A decade later, L.-C. Li re-derived the model from a general framework built on coboundary dynamical Poisson groupoids. This led to a Hamiltonian description of the gauge invariant content of the model, where the gauge transformations act as conjugations of $L$ by diagonal unitary matrices. We shall explain that the same dynamics can be interpreted also as a special case of the spin Sutherland systems obtained by reducing the free geodesic motion on symmetric spaces, studied jointly with Pusztai in 2006; the relevant symmetric space being $mathrm{GL}(n,mathbb{C})/ mathrm{U}(n)$. This construction provides an alternative Hamiltonian interpretation of the Braden--Hone dynamics. It will be demonstrated that two Poisson brackets are compatible and yielda bi-Hamiltonian description of the standard commuting flows of the model. The talk is mainly based on the preprint arXiv:1901.03558. If time permits, we shall also sketch generalizations of these results.