Quantum current lie algebra as the universal algebraic structure of the symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics

Abstract

A new and extremely important property of the algebraic structure of symmetries of nonlinear infinite-dimensional integrable Hamiltonian dynamical systems is described. It is that their invariance groups are isomorphic to a unique universal Banach Lie group of currents G = l /circled dot/Diff (T/sup n/) on an n-dimensional torus T/sup n/. Applications of this phenomenon to the problem of constructing general criteria of integrability of nonlinear dynamical systems of theoretical and mathematical physics are considered.