Abstract
We compute the formal Poisson cohomology of quadratic Poisson structures. We first recall that, generically, quadratic Poisson structures are diagonalizable. Then we compute the formal cohomology of diagonal Poisson structures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnol'd, V. I.: Geometrical Methods in the Theory of Di¡erential Equations, 2nd edn, Grundlehren Math. Wiss., Springer, Berlin, 1988.
Dufour, J. P.: Quadratisation de structures de Poisson à partie quadratique diagonale, Séminaire Gaston Darboux, 1992-1993 (Montpellier).
Dufour, J. P. and Haraki, A.: Rotationnel et structures de Poisson quadratiques, C.R. Acad. Sci. Paris 312(I) (1991), 137–140.
Dufour J. P. and Wade, A.: Formes normales de structures de Poisson ayant un 1-jet nul en un point, J. Geom. Phys. 26 (1998), 79–96.
El Galiou, M. and Tihami, Q.: Star-Product of a quadratic Poisson structure, Tokyo J. Math. 19(2) (1996), 475–498.
Fernandes, R. J. and Vanhaecke, P.: Hyperelliptic Prym varieties and integrable systems, Comm. Math. Phys. 221 (2001), 169–196.
Ginzburg, V. L. and Weinstein, A.: Lie-Poisson structures on some Poisson Lie groups, J. Amer. Math. Soc. 5 (1992), 445–453.
Haraki, A.: Quadratisation de certaines structures de Poisson, J. London Math. Soc. (2) 56 (1997), 384–394.
Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom. 12 (1977), 253–300.
Liu, Z. J. and Xu, P.: On quadratic Poisson structures, Lett. Math. Phys. 26 (1992), 33–42.
Manchon, D., Masmoudi, M. and Roux, A.: On quantization of quadratic Poisson structures, Comm. Math. Phys. 225 (2002), 121–130.
Monnier, Ph.: Poisson cohomology in dimension two, Preprint math.DG/0005261, to appear in Israel J. Math.
Nakanishi, N.: Poisson cohomology of plane quadratic Poisson structures, Publ. Res. Inst. Math. Sci. 33 (1997), 73–89.
Omori, H., Maeda, Y. and Yoshioka, A.: Deformation quantization of Poisson algebras, Contemp. Math. 179 (1994), 213–240.
Roger, C. and Vanhaecke, P.: Poisson cohomology of the affine plane, Preprint, to appear in J. Algebra.
Sklyanin, E. K.: Some algebraic structures connected with the Yang-Baxter equation, Funct. Anal. Appl. 16 (1982), 263–270.
Vaisman, I.: Remarks on the Lichnerowicz-Poisson cohomology, Ann. Inst. Fourier 40 (1990), 951–963.
Vaisman, I.: Lectures on the Geometry of Poisson Manifold, Progr. Math. 118, Birkhäuser, Basel, 1984.
Wade, A.: Normalisation de structures de Poisson, Thesis, Montpellier 1996.
Weinstein, A.: The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379–394.
Xu, P.: Poisson cohomology of regular Poisson manifolds, Ann. Inst. Fourier 42 (1992), 967–988.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Monnier, P. Formal Poisson Cohomology of Quadratic Poisson Structures. Letters in Mathematical Physics 59, 253–267 (2002). https://doi.org/10.1023/A:1015513632414
Issue Date:
DOI: https://doi.org/10.1023/A:1015513632414