Abstract
We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators (with scalar or matrix coefficients) on the line and on the circle. This defines a Poisson-Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, GL n -KdV (or GL n -Adler-Gelfand-Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this “universal” Poisson-Lie group. Moreover, the reduced (=SL n ) versions of these manifolds (orW n -algebras in physical terminology) can be viewed as certain subspaces of the quotient of this Poisson-Lie group by the dressing action of the group of functions on the circle (or as a result of a Poisson reduction). Finally we define an infinite set of commuting functions on the Poisson-Lie group that give the standard families of Hamiltonians when restricted to the submanifolds mentioned above. The Poisson structure and Hamiltonians on the whole group interpolate between the Poisson structures and Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical meaning ofW ∞ as a limit of Poisson algebrasW ε as ε→0.
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References
Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equation. Invent. Math 50, no. 3, 219–248 (1979)
Arnold, V.I.: Mathematical methods of classical mechanics Berlin Heidelberg, New York, Springer, 2nd ed. (1989)
Bakas, I. Khesin, B. Kiritsis, E.: The logarithm of the derivative operator and higher spin algebras ofW ∞ type. Commun. Math. Phys.151, 233–243 (1993)
Date, E. Jimbo, M. Kashiwara, M. Miwa, T.: Transformation group for soliton equations. Publ. RIMS18, 1077–1110 (1982)
Dickey, L.A.: Solition equations and Hamiltonian systems. Advanced Series in Math. Physics, Vol. 12, Singapore, World Scientific, (1991)
Drinfeld, V.G.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations. Doklady Akademii Nauk SSSR268, no. 2, 285–287 (1983) (Russian)
Drinfeld, V.G. Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type. Current problems in mathematics (Moscow), Itogi Nauki i Tekhniki, Vol.24, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., 1984, pp. 81–180 (Russian) English transl. in J. Sov. Math. Vol.30, 1975–2036 (1985)
Enriques, B. Khoroshkin, S. Radul, A. Rosly, A. Rubstov, V.: Poisson-Lie aspects of classical W-algebras. Preprint, Ecole Polytechnique, (1993)
Feigin, B.L.: Lie algebras gl(λ) and cohomology of a Lie algebra of differential operators. Russ. Math. Surv.43, no. 2, 169–170 (1988)
Feigin, B.L.: Private communication
Figueroa, J.M. Mas, J. Ramos, E.: A one-parameter family of Hamiltonian structures for the KP hierarchy and a continuous deformation of the nonlinearW KP algebra. Preprint, hepth/9207092, July (1992)
Gelfand, I.M. Dickey, L.A.: A family of Hamiltonian structures associated with nonlinear integrable differential equations. Preprint, IPM AN SSSR, Moscow, (1978)
Gervais, J.-L.: Infinite family of polynomial functions of the Virasoro generators. Phys. Lett. B160, 277–278 (1985)
Kac, V.G.: Infinite-dimensional Lie algebras. Cambridge: Cambridge University Press, 1990, 3rd ed.
Kac, V.G. Peterson, D.H.: Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc. Nat. Ad. Sci. USA78, 3308–3312 (1981)
Khesin, B. Lyubashenko, V. Roger, C.: Extensions and contractions of the Lie algebra ofq-pseudodifferential operators. Preprint hep-th 9403189, March (1994)
Khesin, B. Malikov, F.: Universal Drinfeld-Sokolov reduction and matrices of complex size. Preprint hep-th 9405116, May (1994), to appear in Comm. Math. Phys.
Khesin, B.A. Zakharevich, I.S.: Poisson Lie group of pseudodifferential symbols and fractional KP-KdV hierarchies. C. R. Acad. Sci.316, 621–626 (1993)
Khesin, B.A. Zakharevich, I.S.: The Gelfand-Dickey structure and an extension of the Lie algebra of pseudodifferential symbols. In preparation
Khovanova, T.G.: Gelfand-Dickey Lie algebras and Virasoro algebra. Funct. Anal. Appl20, no. 4, 89–90 (1986)
Kravchenko, O.S. Khesin, B.A.: Central extension of the algebra of pseudodifferential symbols. Funct. Anal. Appl.25, no. 2, 83–85 (1991)
Kupershmidt, B.A. Wilson, G.: Modifying Lax equations and the second Hamiltonian structure. Invent. Math.62, no. 3, 403–436 (1981)
Lebedev, D.R. Manin, Yu.I.: Conservation laws and Lax representations of Benney's long wave equations. Phys. Lett. A74, no. 3-4, 154–156 (1979)
Lebedev, D.R. Manin, Yu.I.: The Gelfand-Dikii Hamiltonian operator and the coadjoint representation of the Volterra group. Akademiya Nauk SSSR. Funktionalnyi Analiz i ego Prilozheniya13, no. 4, 40–46 (1979) (Russian)
Lebedev, D.R. Manin, Yu.I.: Benney's long wave equations. II. The Lax representations and conservation laws. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V.A.: Steklova Akademii Nauk SSSR (LOMI)96, 169–178 (1980) (Russian); Boundary value problems of mathematical physics and related questions in the theory of functions, 12.
Li, W.L.: 2-cocyles on the algebra of differential operators. J. of Algebra122, 64–80 (1989)
Lu, J.-H. Weinstein, A.: Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Diff. Geo.31, no. 2, 501–526 (1990)
Ovsienko, V.Yu. Khesin, B.A.: Korteveg-de Vries superequation as an Euler equation. Funct. Anal. Appl.21, no. 4, 329–331 (1987)
Pope, C.N. Romans, L.J. Shen, X.:W ∞ and the Racah-Wigner algebra. Nuc. Phys.339B, 191–221 (1990)
Radul, A.O.: Central extension of Lie algebra of differential operators on a circle and w algebras. JETP Letters50, no. 8, 371–373 (1989)
Radul, A.O.: The Lie algebras of differential operators, their central extensions and W-algebras. Funct. Anal. Appl.25, no. 1, 33–49 (1991)
Roger, C.: Extensions centrales d'algébres et de groups de Lie de dimension infinie, algébre de Virasoro et génŕalization. Preprint, Institut de Mathématique, Université de Liège, Belgique, (1993)
Segal, G. Wilson, G.: Loop groups and equations of KdV type. Institut des Hautes Etudes Scientifiques. Publications Mathematiques no. 61, pp. 5–65 (1985)
Semenov-Tyan-Shanskii, M.A.: What a classicalr-matrix is Akademiya Nauk SSSR. Funktsionalnyi Analiz i ego Prilozheniya17, no. 4, 17–33 (1983) (Russian) English transl. in Funct. Anal. Appl. Vol.17, 259–272 (1983)
Semenov-Tyan-Shanskii, M.A.: Dressing transformations and Poisson group actions. Kyoto University. Research Institute for Mathematical Sciences. Publications21, no. 6, 1237–1260 (1985)
Semenov-Tyan-Shanskii, M.A.: Group-theoretical methods in the theory of integrable systems. Ph.D. thesis, LOMI, Leningrad, (1985)
Wodzicki, M.: Cyclic homology of differential operators. Duke Math.54, no. 2, 641–647 (1987)
Wodzicki, M.: Noncommutative residue. I. Fundamentals.K-theory, arithmetic and geometry, Lecture Notes in Math., Vol.1289, Moscow, 1984–1986, Berlin, Heidelberg, New York, Springer, pp 320–399 (1987)
Zakharevich, I.: The Second Gelfand-Dickey structure as a bracket on a Poisson-Lie Grassmannian. Adv. in Sov. Math.16, part 2, 179–208 (1993)
Zakharevich, I.: A short report on quantization of differential operators. In preparation
Dzhumadildaev, A.S.: Derivations and central extensions of the Lie algebra of formal pseudodifferential operators. St. Peter. Math. J. Vol.6, 140–158 (1994)
Kontsevich, M., Vishik, S.: Determinants of elliptic pseudo-differential operators. Preprint MPI/94-30 (1994), to appear in GAFA
Kontsevich, M., Vishik, S.: Geometry of determinants of elliptic operators. Preprint MPI/94-57 (1994)
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Khesin, B., Zakharevich, I. Poisson-Lie group of pseudodifferential symbols. Commun.Math. Phys. 171, 475–530 (1995). https://doi.org/10.1007/BF02104676
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DOI: https://doi.org/10.1007/BF02104676