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(T2) 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 T2 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.
Similar content being viewed by others
References
Ziglin, S.: Non-integrability of the problem of the motion of four point vortices. Dokl. Akad. Nauk SSSR 250(6), 1296–1300 (1980)
Reyman, A.G., Semenov-Tyan-Shanskii, M.A.: Lie algebras and Lax equations with spectral parameter on elliptic curve. Notes of the LOMI Seminar 150, 104–118 (1986)
Levin, A., Olshanetsky, M., Zotov, A.: Hitchin Systems: symplectic Hecke correspondence and two-dimensional version. Commun. Math. Phys. 236(1), 93–133 (2003)
Belavin, A., Drinfeld, V.: Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funct. Anal Appl. 16(3), 1–29 (1982)
Feigin, B., Odesski, A.: Sklyanin’s elliptic algebras. Funct. Anal. Appl. 23(3), 207–214 (1989)
Arnold, V.I.: The Hamiltonian nature of the Euler equation in the dynamics of rigid body and of an ideal fluid. Russ. Math. Surv. 24(3), 225–226 (1969)
Arnold, V.I., Khesin, B.A.: Topological methods in Hydrodynamics. Applied Mathematical Science 125, New York: Springer-Verlag, 1998
Manakov, S.: A remark on the integration of the Eulerian equations of the dynamics of an n-dimensional rigid body. Funct. Anal. Appl. 10(4), 93–94 (1976)
Ward, R.S.: Infinite-dimensional gauge groups and special nonlinear gravitons. J. Geom. Phys. 8(1–4), 317–325 (1992)
Khesin, B., Ovsienko, V.: The super Korteweg-de Vries equation as an Euler equation. Funct. Anal. Appl. 21(4), 81–82 (1987)
Segal, G.: The geometry of the KdV equation. In: Topological methods in quantum field theory (Trieste, 1990). Internat. J. Modern Phys. A6(16), 2859–2869 (1991)
Khesin, B., Misiolek, G.: Euler equations on homogeneous spaces and virasoro orbits. Adv. Math. 176, 116–144 (2003)
Belavin, A.: Discrete groups and integrability of quantum systems. Funct. Anal. Appl. 14, 18–26 (1980)
Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Funct. Anal. Appl. 16(4), 27–34 (1982)
Olshanetsky, M.: Integrable tops and non-commutative torus. In: Proc. of Int. Workshop Supersymmetries and Quantum Symmetries, Sept. 2001, Karpacz, Poland, http://arxiv.org/abs/nlin.SI/0203003, 2002
Etingof, P.I., Frenkel, I.B.: Central extensions of current groups in two dimensions. Commun. Math. Phys. 165(3), 429–444 (1994)
Semenov-Tyan-Shansky, M.: What a classical r-matrix is. Funct. Anal. Appl. 17(4), 17–33 (1983)
Braden, H.W., Dolgushev, V.A., Olshanetsky, M.A., Zotov, A.V.: Classical R-Matrices and the Feigin-Odesskii Algebra via Hamiltonian and Poisson Reductions. J. Phys. A36, 6979–7000 (2003)
Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, 1156–1162 (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Takhtajan
Rights and permissions
About this article
Cite this article
Khesin, B., Levin, A. & Olshanetsky, M. Bihamiltonian Structures and Quadratic Algebras in Hydrodynamics and on Non-Commutative Torus. Commun. Math. Phys. 250, 581–612 (2004). https://doi.org/10.1007/s00220-004-1150-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-004-1150-3