Geodesic
Hamiltonian structures for integrable models of field theory
Teoretičeskaâ i matematičeskaâ fizika, Tome 56 (1983) no. 3, pp. 323-343

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It is shown that for classical continuous integrable field theory models the Poisson brackets, defined in $r$-matrix form, admit a simple geometrical interpretation in terms of current algebra. In such an interpretation, the phase spaces of the models are integral manifolds of a standard symplectic structure on the current algebra. For discrete integrable systems, integral manifolds are constructed for discrete $r$-matrix brackets for rational r matrices associated with the classical Lie algebras. It is shown that in the discrete ease there is a multiplicative operation of averaging that makes it possible to obtain trigonometric and elliptic $L$ operators from rational operators. This averaging is explicitly performed for the single-pole $L$ operator associated with the algebra $\mathfrak{sl}(2)$. 
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     author = {N. Yu. Reshetikhin and L. D. Faddeev},
     title = {Hamiltonian structures for integrable models of field theory},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {323--343},
     publisher = {mathdoc},
     volume = {56},
     number = {3},
     year = {1983},
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     url = {http://geodesic.mathdoc.fr/item/TMF_1983_56_3_a0/}
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N. Yu. Reshetikhin; L. D. Faddeev. Hamiltonian structures for integrable models of field theory. Teoretičeskaâ i matematičeskaâ fizika, Tome 56 (1983) no. 3, pp. 323-343. http://geodesic.mathdoc.fr/item/TMF_1983_56_3_a0/