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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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.
A. M. Astashov, A. M. Vinogradov, On the structure of Hamiltonian operators in field theory, J. Geom. Phys. 2 (1986), 263–287.
A. Beilinson, V. G. Drinfeld, Chiral Algebras, AMS Colloquium Publications, Vol 51, Amer. Math. Soc., Providence, RI, 2004.
A. Barakat, A. De Sole, V. G. Kac, Poisson vertex algebras in the theory of Hamiltonian equations, Jpn. J. Math. 4 (2009), 141–252.
D. Cooke, Classification results and Darboux' theorem for low order Hamiltonian operators, J. Math. Phys. 32 (1991), 109–119.
D. Cooke, Compatibility conditions for Hamiltonian pairs, J. Math. Phys. 32 (1991), 3071–3076.
I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, New York, 1993.
A. De Sole, V. G. Kac, Finite vs. affine W-algebras, Jpn. J. Math. 1 (2006), 137–261.
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.
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.
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.
F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156–1162.
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.
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.
P. J. Olver, Darboux' theorem for Hamiltonian operators, J. Differential Equations 71 (1988), 10–33.
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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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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DOI: https://doi.org/10.1007/s00031-010-9110-9