Abstract
In our recent paper “The variational Poisson cohomology” (2011) we computed the dimension of the variational Poisson cohomology \({{\mathcal{H}^\bullet_K(\mathcal{V})}}\) for any quasiconstant coefficient ℓ × ℓ matrix differential operator K of order N with invertible leading coefficient, provided that \({{\mathcal{V}}}\) is a normal algebra of differential functions over a linearly closed differential field. In the present paper we show that, for K skewadjoint, the \({{\mathbb{Z}}}\) -graded Lie superalgebra \({{\mathcal{H}^\bullet_K(\mathcal{V})}}\) is isomorphic to the finite dimensional Lie superalgebra \({{\widetilde{H}(N\ell,S)}}\) . We also prove that the subalgebra of “essential” variational Poisson cohomology, consisting of classes vanishing on the Casimirs of K, is zero. This vanishing result has applications to the theory of bi-Hamiltonian structures and their deformations. At the end of the paper we consider also the translation invariant case.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barakat A., De Sole A., Kac V.G.: Poisson vertex algebras in the theory of Hamiltonian equations. Japan. J. Math. 4, 141–252 (2009)
Degiovanni L., Magri F., Sciacca V.: On deformation of Poisson manifolds of hydrodynamic type. Commun. Math. Phys. 253(1), 1–24 (2005)
De Sole A., Kac V.G.: Finite vs. affine W-algebras. Japan. J. Math. 1, 137–261 (2006)
De Sole A., Kac V.G.: Lie conformal algebra cohomology and the variational complex. Commun. Math. Phys. 292, 667–719 (2009)
De Sole, A., Kac, V.G.: The variational Poisson cohomology. http://arXiv.org/abs/1106.0082v1 [math-ph], 2011
De Sole A., Kac V.G., Wakimoto M.: On classification of Poisson vertex algebras. Transform. Groups 15(4), 883–907 (2010)
Dorfman I.Ya.: Dirac structures and integrability of nonlinear evolution equations. Nonlinear Sci. Theory Appl. New York, John Wiley & Sons (1993)
Getzler E.: A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111(3), 535–560 (2002)
Guillemin V.W., Sternberg S.: An algebraic model of transitive differential geometry. Bull. Amer. Math. Soc. 70, 16–47 (1964)
Kac V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)
Krasilshchik I.S.: Schouten brackets and canonical algebras. Lecture Notes in Math. 1334, New York (1988)
Kupershmidt, B.A.: Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms. In: Geometric methods in Mathematical Physics, Lecture Notes in Math. 775, New York: Springer Verlag, 1980, pp. 162–218
Lichnerowicz A.: Les varietes de Poisson et leur algebres de Lie associees. J. Diff. Geom. 12, 253–300 (1977)
Liu S.-Q., Zhang Y.: Jacobi structures of evolutionary partial differential equations. Adv. Math. 227, 73–130 (2011)
Liu, S.-Q., Zhang, Y.: Private communication, Beijing, June 2011
Magri F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, 129–134 (1978)
Olver, P.J.: BiHamiltonian systems, in Ordinary and partial differential equations. Pitman Research Notes in Mathematics Series 157, New York: Longman Scientific and Technical, 1987, pp. 176–193
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Supported by a PRIN grant and Fondi Ateneo, from the University of Rome.
Supported in part by an NSF grant.
Rights and permissions
About this article
Cite this article
De Sole, A., Kac, V.G. Essential Variational Poisson Cohomology. Commun. Math. Phys. 313, 837–864 (2012). https://doi.org/10.1007/s00220-012-1461-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-012-1461-8