It is shown that the factorization relation on simple Lie groups with standard Poisson Lie structure restricted to Coxeter symplectic leaves gives an integrable dynamical system. This system can be regarded as a discretization of the Toda flow. In case of SL n the integrals of the factorization dynamics are integrals of the relativistic Toda system. A substantial part of the paper is devoted to the study of symplectic leaves in simple complex Lie groups, its Borel subgroups and their doubles.
In this paper, we investigate flows on discrete curves in C 2 , CP 1 , and C. A novel interpretation of the one-dimensional Toda lattice hierarchy and reductions thereof as flows on discrete curves are given.
We extend the action for evolution equations of KdV and MKdV type which was derived in NC] to the case of not periodic, but only equivariant phase space variables, introduced in FV2]. The di erence of these variables may be interpreted as reduced phase space variables via a Marsden-Weinstein reduction where the monodromies play the role of the momentum map. As an example we obtain the doubly discrete sine-Gordon equation and the Hirota equation and the corresponding symplectic structures.
of the original Volterra algebra. Since the Hamiltonian of the Volterra model is the lattice rather then being an algebra on the edges of the lattice, as in the case C-Algebra. This subalgebra can be depictured as being an algebra on the faces of on a subalgebra of the original Volterra algebra, which we would like to call the The fundamental L-operator of the Volterra model is, as found in [V], dependent ln the algebraic Bethe ansatz for this L-operator was given.Volterra-L-operator by expressing it in terms of the sine-(,`¤ordon H and fields. The content of this paper is mainly based on the work of Faddeev and Volkov
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