Communications in Mathematical Physics

, Volume 180, Issue 1, pp 99–108 | Cite as

On tangential star products for the coadjoint Poisson structure

Article

Abstract

We derive necessary conditions on a Lie algebra from the existence of a star product on a neighbourhood of the origin in the dual of the Lie algebra for the coadjoint Poisson structure which is both differential and tangential to all the coadjoint orbits. In particular we show that when the Lie algebra is semisimple there are no differential and tangential star products on any neighbourhood of the origin in the dual of its Lie algebra.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Lett. Math. Phys.1, 521–530 (1977)CrossRefGoogle Scholar
  2. 2.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys.111, 61–110 (1978)CrossRefGoogle Scholar
  3. 3.
    Cahen, M., Gutt, S., Rawnsley, J.: Non-linearisability of the Iwasawa Poisson Lie structure. Lett. Math. Phys.24, 79–83 (1992)CrossRefGoogle Scholar
  4. 4.
    Conn, J.F.: Normal forms for smooth Poisson structures. Ann. Math.121, 565–593 (1985)Google Scholar
  5. 5.
    De Wilde, M., Lecomte, P.B.: Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys.7, 487–496 (1983)CrossRefGoogle Scholar
  6. 6.
    Fedosov, B.V.: A simple geometrical construction of deformation quantization. J. Diff. Geom.40, 213–238 (1994)Google Scholar
  7. 7.
    Lu, J.H., Ratiu, T.: On the nonlinear convexity theorem of Kostant. J. Am. Math. Soc.4, 349–363 (1991)Google Scholar
  8. 8.
    Lu, J.H., Weinstein, A.: Poisson-Lie groups, dressing transformations and Bruhat decompositions. J. Diff. Geom.31, 501–526 (1990)Google Scholar
  9. 9.
    Ginzburg, V.L., Weinstein, A.: Lie-Poisson structure on some Poisson Lie groups. J. Am. Math. Soc.5, 445–453 (1992)Google Scholar
  10. 10.
    Masmoudi, M.: Tangential formal deformations of the Poisson bracket and tangential star products on a regular Poisson manifold. J. Geom. Phys.9, 155–171 (1992)CrossRefGoogle Scholar
  11. 11.
    Omori, H., Maeda, Y., Yoshioka, A.: Weyl manifolds and deformation quantization. Adv. Math.85, 224–255 (1991)CrossRefGoogle Scholar
  12. 12.
    Omori, H., Maeda, Y., Yoshioka, A.: Deformation quantization of Poisson algebras. Contemp. Math.179, 213–240 (1994)Google Scholar
  13. 13.
    Vey, J.: Déformation du crochet de Poisson sur une variété symplectique. Comment. Math. Helvet.50, 421–454 (1975)Google Scholar
  14. 14.
    Weinstein, A.: The local structure of Poisson manifolds. J. Diff. Geom.18, 523–557 (1983)Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité Libre de BruxellesBruxellesBelgium
  2. 2.Département de MathématiquesUniversité de MetzMetz cedexFrance
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations