Abstract
We prove that all classical affine W-algebras 𝒲(𝔤; f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G2, one in F4, and five in E8.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
19 June 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00031-021-09664-x
References
M. Adler, On a trace functional for formal pseudodifferential operators and the symplectic structure of the Korteweg-de Vries equation, Invent. Math. 50 (1979), 219–248.
A. Barakat, A. De Sole, V. G. Kac, Poisson vertex algebras in the theory of Hamiltonian equations, Japan. J. Math. 4 (2009), n. 2, 141–252.
D. Collingwood, W. McGovern, Nilpotent Orbits in Semisimple Lie Algebra, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993.
J. Dadok, V. G. Kac, Polar representations, J. Algebra 92 (1985), no. 2, 504–524.
A. De Sole, M. Jibladze, V. G Kac, D. Valeri, Integrable triples in semisimple Lie algebras, arXiv: 2012.12913 (2020).
A. De Sole, V. G. Kac, Finite vs. affine 𝒲-algebras, Japan. J. Math. 1 (2006), no.1, 137–261.
A. De Sole, V. G. Kac, D. Valeri, Classical 𝒲-algebras and generalized Drinfeld–Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras, Comm. Math.Phys. 323 (2013), n. 2, 663–711.
A. De Sole, V. G. Kac, D. Valeri, Classical 𝒲-algebras and generalized Drinfeld–Sokolov hierarchies for minimal and short nilpotents, Comm. Math. Phys. 331 (2014), n. 2, 623–676. Erratum in Commun. Math. Phys. 333 (2015), n. 3, 1617–1619.
V. Drinfeld, V. Sokolov, Lie algebras and equations of KdV type, Soviet J. Math. 30 (1985), 1975–2036.
A. G. Elashvili, V. G. Kac, E. B. Vinberg, Cyclic elements in semisimple Lie algebras, Transform. Groups 18 (2013), 97–130.
L. Fehér, L. O’Raifeartaigh, P. Ruelle, I. Tsutsui, A. Wipf, On Hamiltonian reductions of the Wess–Zumino–Novikov–Witten theories, Phys. Rep. 222 (1992), no.1, 1–64.
C. Fernández-Pousa, M. Gallas, L. Miramontes, J. Sánchez Guillén, 𝒲-algebras from soliton equations and Heisenberg subalgebras, Ann. Physics 243 (1995), no. 2, 372–419.
C. S. Gardner, J. M. Greene, Kruskal, R. M. Miura, Method for solving the Korteweg–deVries equation, Phys. Rev. Lett. 19, no. 19, 1095–1097.
И. M. Гельфанд, Л. А. Дикиӥ , Дробные степени операторов и гамиль-тоновы системы, Функц. анализ и его прил. 10 (1976), bьш. 4, 13–29. Engl. transl.: I. M. Gelfand, L. A. Dickey, Fractional powers of operators and Hamiltonian systems, Funct. Anal. Appl. 10 (1976), 259–273.
I. M. Gelfand, L. A. Dickey, Family of Hamiltonian structures connected with integrable non-linear equations, preprint, IPM, Moscow (in Russian), 1978. English version in: Collected Papers of I. M. Gelfand, Vol. 1, Springer-Verlag, Berlin, 1987, pp. 625–646.
P. D. Lax, Integrals of non-linear equations of evolution and solitary waves, Comm. Pure and Appl. Math., 21 (1968), no. 5, 467–490.
F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), no. 5, 1156–1162.
B. E. Захаров, Л. Д. Фаддеев, Уравнение Кортевега–де Фриза–вполне интегрируема гамиль~тонова система, Функц. анализ и его прил. 5 (1971), bьш. 4, 18–27. Engl. transl.: V. E. Zakharov, L. D. Faddeev, Korteweg–de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl. 5 (1971), 280–287.
Author information
Authors and Affiliations
Corresponding author
Additional information
Funding Information
Open access funding provided by Università degli Studi di Roma La Sapienza within the CRUI-CARE Agreement.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
To the memory of Ernest Borisovich Vinberg
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
SOLE, A.D., JIBLADZE, M., KAC, V.G. et al. INTEGRABILITY OF CLASSICAL AFFINE W-ALGEBRAS. Transformation Groups 26, 479–500 (2021). https://doi.org/10.1007/s00031-021-09645-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-021-09645-0