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On classification of poisson vertex algebras

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Abstract

We describe a conjectural classification of Poisson vertex algebras of CFT type and of Poisson vertex algebras in one differential variable (= scalar Hamiltonian operators).

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References

  1. А. М. Асташов, Нормальные формы гамцльmонвых оnераmоров в mеорццnоля, ДАН CCCP 270 (1983), 1033–1037. Engl. transl.: A. M. Astashov, Normal forms of Hamiltonian operators in field theory, Sov. Math., Dokl. 27 (1983), 685–689.

  2. A. M. Astashov, A. M. Vinogradov, On the structure of Hamiltonian operators in field theory, J. Geom. Phys. 2 (1986), 263–287.

    Article  MathSciNet  Google Scholar 

  3. A. Beilinson, V. G. Drinfeld, Chiral Algebras, AMS Colloquium Publications, Vol 51, Amer. Math. Soc., Providence, RI, 2004.

  4. A. Barakat, A. De Sole, V. G. Kac, Poisson vertex algebras in the theory of Hamiltonian equations, Jpn. J. Math. 4 (2009), 141–252.

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Cooke, Classification results and Darboux' theorem for low order Hamiltonian operators, J. Math. Phys. 32 (1991), 109–119.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. Cooke, Compatibility conditions for Hamiltonian pairs, J. Math. Phys. 32 (1991), 3071–3076.

    Article  MATH  MathSciNet  Google Scholar 

  7. I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, New York, 1993.

    Google Scholar 

  8. A. De Sole, V. G. Kac, Finite vs. affine W-algebras, Jpn. J. Math. 1 (2006), 137–261.

    Article  MATH  MathSciNet  Google Scholar 

  9. И. М. Гельфанд, И. Дорфман, Гамцльmонвы оnераmоры ц связанные с нцмц алгебрацкце сmрукmуры, Функц. анал. и его прилож. 13 (1979), no. 4, 13–30. Engl. transl.: I. M. Gelfand, I. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl. 13 (1980), 248–262.

  10. Л. А. Тахтаджяд, Л. Д. Фаддеев, Гомцльmонов nодход в mеорцц солцmоное, Наука, М., 1986. Engl. transl.: L. D. Faddeev, L. A. Takhtadzhyan, The Hamiltonian Methods in the Theory of Solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987.

  11. V. G. Kac, Vertex Algebras for Beginners, University Lecture Series, Vol. 10, Amer. Math. Soc., Providence, RI, 1996. Second edition, 1998. Russian transl.: В. Г. Кац, Верmексные алгебры для начцнающцх, МЩНМО, M., 2005.

  12. F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156–1162.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. V. Mikhailov, A. V. Shabat, V. V. Sokolov The symmetry approach to classification of integrable equations, in: V. E. Zakharov, ed., What is Integrabilty?, Springer Series in Non-Linear Dynamics, Springer-Verlag, New York, 1991, pp, 115–184.

  14. О. И. Мохов, Гамцльmоновы дцфференццальные оnераnоры ц конmакmная геомеmрця, Функц. анал. и его прллож. 21 (1987), no. 3, 53–60. Engl. transl.: O. I. Mokhov, Hamiltonian differential operators and contact geometry, Funct. Anal. Appl. 21 (1987), 217–223.

  15. P. J. Olver, Darboux' theorem for Hamiltonian operators, J. Differential Equations 71 (1988), 10–33.

    Article  MATH  MathSciNet  Google Scholar 

  16. А. М. Виноградов, О гамцльmоновых сmрукmурах в mеорцц nоля, ДАН СССР 241 (1978), 18–21. Engl. transl.: A. M. Vinogradov, On Hamiltonian structures in field theory, Sov. Math., Dokl. 19 (1978), 790–794.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “La Sapienza”, 00185, Roma, Italy

    Alberto De Sole & Minoru Wakimoto

  2. Department of Mathematics, MIT, Cambridge, MA, 02139, USA

    Victor G. Kac & Minoru Wakimoto

Authors
  1. Alberto De Sole
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  2. Victor G. Kac
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  3. Minoru Wakimoto
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Correspondence to Alberto De Sole.

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Dedicated to Vladimir Morozov on the 100th anniversary of his birth

Supported in part by the Department of Mathematics, MIT, Cambridge, MA 02139, USA, and by PRIN and AST grants

Supported in part by NSF grants.

Supported in part by the Department of Mathematics, MIT, Cambridge, MA 02139, USA.

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De Sole, A., Kac, V.G. & Wakimoto, M. On classification of poisson vertex algebras. Transformation Groups 15, 883–907 (2010). https://doi.org/10.1007/s00031-010-9110-9

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