Hamiltonian structures for integrable models of field theory

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 $\mathfrak{sl}(2)$.