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Classifications of star products and deformations of Poisson brackets

Philippe Bonneau

Banach Center Publications (2000)

  • Volume: 51, Issue: 1, page 25-29
  • ISSN: 0137-6934

Abstract

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On the algebra of functions on a symplectic manifold we consider the pointwise product and the Poisson bracket; after a brief review of the classifications of the deformations of these structures, we give explicit formulas relating a star product to its classifying formal Poisson bivector.

How to cite

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Bonneau, Philippe. "Classifications of star products and deformations of Poisson brackets." Banach Center Publications 51.1 (2000): 25-29. <http://eudml.org/doc/209038>.

@article{Bonneau2000,
abstract = {On the algebra of functions on a symplectic manifold we consider the pointwise product and the Poisson bracket; after a brief review of the classifications of the deformations of these structures, we give explicit formulas relating a star product to its classifying formal Poisson bivector.},
author = {Bonneau, Philippe},
journal = {Banach Center Publications},
keywords = {algebra of functions; symplectic manifold; Poisson bracket; deformations; star product; classifying formal Poisson bivector},
language = {eng},
number = {1},
pages = {25-29},
title = {Classifications of star products and deformations of Poisson brackets},
url = {http://eudml.org/doc/209038},
volume = {51},
year = {2000},
}

TY - JOUR
AU - Bonneau, Philippe
TI - Classifications of star products and deformations of Poisson brackets
JO - Banach Center Publications
PY - 2000
VL - 51
IS - 1
SP - 25
EP - 29
AB - On the algebra of functions on a symplectic manifold we consider the pointwise product and the Poisson bracket; after a brief review of the classifications of the deformations of these structures, we give explicit formulas relating a star product to its classifying formal Poisson bivector.
LA - eng
KW - algebra of functions; symplectic manifold; Poisson bracket; deformations; star product; classifying formal Poisson bivector
UR - http://eudml.org/doc/209038
ER -

References

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  2. [B] P. Bonneau, Fedosov star products and one-differentiable deformations, Lett. Math. Phys. 45 (1997), 363-376. Zbl1006.53075
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  4. [F1] B. V. Fedosov, A simple geometrical construction of deformation quantization, J. Differential Geom. 40 (1994), 213-238. Zbl0812.53034
  5. [F2] B.V. Fedosov, Deformation Quantization and Index Theory, Akademie Verlag, Berlin, 1996. Zbl0867.58061
  6. [FLS] M. Flato, A. Lichnerowicz and D. Sternheimer, Déformations 1-différentiables des algèbres de Lie attachées à une variété symplectique ou de contact, Comp. Math. 31 (1975), 47-82. Zbl0317.53039
  7. [Ge] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1964), 59-103. Zbl0123.03101
  8. [Gu] S. Gutt, Equivalence of deformations and associated *-products, Lett. Math. Phys. 3 (1979), 297-309. Zbl0424.53031
  9. [K] M. Kontsevich, Deformation quantization of Poisson manifolds I, q-alg/9709040. Zbl1058.53065
  10. [Le] P.B.A. Lecomte, Applications of the cohomology of graded Lie algebras to formal deformations of Lie algebras, Lett. Math. Phys. 13 (1987), 157-166. Zbl0628.17009
  11. [Li] A. Lichnerowicz, Existence and equivalence of twisted products on a symplectic manifold, Lett. Math. Phys. 3 (1979), 495-502. 
  12. [M] J. Moyal, Quantum mechanics as a statistical theory, Proc. Camb. Phil. Soc. 45 (1949), 99-124. Zbl0031.33601
  13. [NT] R. Nest and B. Tsygan, Algebraic index theorem, Comm. Math. Phys. 172 (1995), 223-262. Zbl0887.58050

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