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We introduce a bi-Hamiltonian hierarchy on the cotangent bundle of the real Lie group ${mathrm GL}(n,{mathbb{C}})$, and study its Poisson reduction with respect to the action of the product group ${{mathrm U}(n)} times {{mathrm U}(n)}$ arising from left- and right-multiplications. One of the pertinent Poisson structures is the canonical one, while the other is suitably transferred from the real Heisenberg double of ${mathrm GL}(n,{mathbb{C}})$. When taking the quotient of $T^*{mathrm GL}(n,{mathbb{C}})$ we focus on the dense open subset of ${mathrm GL}(n,{mathbb{C}})$ whose elements have pairwise distinct singular values. We develop a convenient description of the Poisson algebras of the ${{mathrm U}(n)} times {{mathrm U}(n)}$ invariant functions, and show that one of the Hamiltonians of the reduced bi-Hamiltonian hierarchy yields a hyperbolic Sutherland model coupled to two ${mathfrak u}(n)^*$-valued spins. Thus we obtain a new bi-Hamiltonian interpretation of this model, which represents a special case of Sutherland models coupled to two spins obtained earlier from reductions of cotangent bundles of reductive Lie groups equipped with their canonical Poisson structure. Upon setting one of the spins to zero, we recover the bi-Hamiltonian structure of the standard hyperbolic spin Sutherland model that was derived recently by a different method.
The Oriented Associativity equation plays a fundamental role in the theory of Integrable Systems. In this paper we prove that the equation, besides being Hamiltonian with respect to a first-order Hamiltonian operator, has a third-order non-local homo
We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove that these
We solve perturbatively the quantum elliptic Calogero-Sutherland model in the regime in which the quotient between the real and imaginary semiperiods of the Weierstrass ${cal P}$ function is small
There exists a large class of quantum many-body systems of Calogero-Sutherland type where all particles can have different masses and coupling constants and which nevertheless are such that one can construct a complete (in a certain sense) set of exa
We provide a list of explicit eigenfunctions of the trigonometric Calogero-Sutherland Hamiltonian associated to the root system of the exceptional Lie algebra E8. The quantum numbers of these solutions correspond to the first and second order weights of the Lie algebra.