DYNAMICAL-SYSTEMS ON QUANTUM TORI LIE-ALGEBRAS

We use quantum tori Lie algebras (QTLA), which are a one-parameter family of sub-algebras of gl(infinity), to describe local and non-local versions of the Toda systems. It turns out that the central charge of QTLA is responsible for the non-locality. There are two regimes in the local systems - conformal for irrational values of the parameter and non-conformal and integrable for its rational values. We also consider infinite-dimensional analogs of rigid tops. Some of these systems give rise to ''quantized'' (magneto-)hydrodynamic equations of an ideal fluid on a torus. We also consider infinite dimensional versions of the integrable Euler and Clebsch cases.