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, C) 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 infinitedimensional groups, appearing as certain large N limits of GL(N, C). These are the group of a non-commutative torus (NCT) and the group of symplectomorphisms SDif f (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.

Compatible Poisson brackets, quadratic Poisson algebras and classical r-matrices

“…those with the simple poles in some fixed set of points) and for general meromorphic T (λ) that define the structure of the infinite dimensional quadratic algebra. This result generalize a set of recent results [12]- [14] about the compatibility of the linear and quadratic rmatrix brackets for the cases of the Belavins classical elliptic r-matrix [15] and monodromy matrices T (λ) possessing simple poles. In the present paper we show that d(λ) = detT (λ) is a generating function of the Casimir functions of the brackets (1) for any classical r-matrix r(λ − µ) taking the values in sl(n) ⊗ sl(n).…”

“…Using the fact that T (λ) is meromorphic functions this can always be done using the decomposition in Laurent power series. In the special cases the other decompositions may be also used [12]. In the present paper we will not do these decompositions explicitly leaving all the expressions in the "convoluted" form of the generating functions.…”

“…Although there exist a lot of examples of such the compatible brackets in the literature (see for example [4], [12]- [14]), there seems to be no general theory of the compatibility of such the brackets.…”

Classification of isomonodromy problems on elliptic curves

“…Это эллиптическая система Калоджеро и эллиптиче-ский волчок [61]. В изомонодромном случае обе модели становятся неавтоном-ными [42], [43], [110]. При этом вследствие соответствия Пенлеве-Калоджеро пары Лакса остаются теми же, что и в автономной интегрируемой механике.…”

Quadratic algebras related to elliptic curves

“…This algebra was derived in [8] for n = 1 and is the classical Sklyanin algebra for n = 1 and N = 2 [3].…”