Elliptic hydrodynamics and quadratic algebras of vector fields on a torus

We construct a quadratic Poisson algebra of Hamiltonian functions on a two-dimensional torus compatible with the canonical Poisson structure. This algebra is an infinite-dimensional generalization of the classical Sklyanin–Feigin–Odesskii algebras. It yields an integrable modification of the two-dimensional hydrodynamics of an ideal fluid on the torus. The Hamiltonian of the standard two-dimensional hydrodynamics is defined by the Laplace operator and thus depends on the metric. We replace the Laplace operator with a pseudodifferential elliptic operator depending on the complex structure. The new Hamiltonian becomes a member of a commutative bi-Hamiltonian hierarchy. In conclusion, we construct a Lie bialgebroid of vector fields on the torus.