Bihamiltonian Structures and Quadratic Algebras in Hydrodynamics and on Non-Commutative Torus

Abstract

We demonstrate the common bihamiltonian nature of several integrable systems. The first one is an elliptic rotator that is an integrable Euler-Arnold top on the complex group GL(N) for any $N$, whose inertia ellipsiod is related to a choice of an elliptic curve. Its bihamiltonian structure is provided by the compatible linear and quadratic Poisson brackets, both of which are governed by the Belavin-Drinfeld classical elliptic $r$-matrix. We also generalize this bihamiltonian construction of integrable Euler-Arnold tops to several infinite-dimensional groups, appearing as certain large $N$ limits of GL(N). These are the group of a non-commutative torus (NCT) and the group of symplectomorphisms $SDiff(T^2)$ of the two-dimensional torus. The elliptic rotator on symplectomorphisms gives an elliptic version of an ideal 2D hydrodynamics, which turns out to be an integrable system. In particular, we define the quadratic Poisson algebra on the space of Hamiltonians on $T^2$ depending on two irrational numbers. In conclusion, we quantize the infinite-dimensional quadratic Poisson algebra in a fashion similar to the corresponding finite-dimensional case.