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An inductive explicit construction of $*$-products on some Poisson manifolds

Author: Santos Asin Lares
Journal: Proc. Amer. Math. Soc. 128 (2000), 1853-1857
MSC (1991): Primary 58F06, 53Z05; Secondary 81Q99.
Published electronically: September 30, 1999
MathSciNet review: 1654089
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Abstract: We extend the Cahen Gutt coboundary construction on cotangent bundles of $n$-dimensional parallelisable manifolds to manifolds which admit $n$ global vector fields defining a parallelisation on a dense open set. This result is used to give an inductive explicit construction of $*$-products on certain Poisson manifolds.

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  • 1. S. Asin. On tangential properties of the Gutt $*$-product. Journal of Geometry and Physics, 24:164-172, 1988. MR 98i:58108
  • 2. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer. Deformation theory and quantization I and II. Ann. Physics, 111(61-151), 1978. MR 58:14737a
  • 3. M. Cahen and S. Gutt. Regular $*$-representations of Lie algebras. Lett. in Math. Physics, 6:395-404, 1982.
  • 4. M. Cahen, S. Gutt, and J.H. Rawnsley. On tangential star products for the coadjoint Poisson structure. Comm. Math. Physics, 180:99-108, 1996. MR 97e:58101
  • 5. B. Fedosov. A simple geometrical construction of deformation quantization. J. Diff Geom., 40:213-238, 1994. MR 95h:58062
  • 6. M. Gerstenhaber. On the deformation of rings and algebras. Ann. of Math., 79:59-103, 1964. MR 30:2034
  • 7. J. Grabowski. Hochschild cohomology and quantization of Poisson structures. Rend. Circ. Matem. di Palermo, 2(Suppl. no 37):87-91, 1994. MR 96f:58068
  • 8. S. Gutt. An explicit $*$-product on the cotangent bundle of a Lie group. Lett. Math. Phys., 7:249-258, 1983. MR 85g:58037
  • 9. M. Kontsevich. Deformation quantization of Poisson manifolds I. q-alg 9709040, 1997.
  • 10. J.H. Lu and A. Weinstein. Poisson-Lie groups, dressing transformations and Bruhat decompositions. J. Diff Geom., 31:501-526, 1990. MR 91c:22012
  • 11. H. Omori, Y. Maeda, and A. Yoshioka. Weyl manifolds and deformation quantization. Advances in Math., 85:224-255, 1991. MR 92d:58071
  • 12. J. Vey. Deformation du crochet de Poisson sur une varieté symplectique. Commentarii Math. Helvet., 50:421-454, 1975. MR 54:8765
  • 13. A. Weinstein. Tangential deformation quantization and polarized symplectic groupoids. In D. Sternheimer et al., editor, Deformation theory and symplectic geometry, volume 20 of Mathematical Physics Studies, pages 301-327. Kluwer, 1997. CMP 98:04
  • 14. M. De Wilde and P. Lecomte. Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. in Math. Physics, 7:487-496, 1983. MR 85j:17021

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Additional Information

Santos Asin Lares
Affiliation: Mathematical Institute, University of Warwick, CV4-7AL, United Kingdom

Keywords: $*$-product, deformation quantization
Received by editor(s): July 21, 1998
Published electronically: September 30, 1999
Additional Notes: The author was supported by a grant from the University of Warwick.
Communicated by: Peter Li
Article copyright: © Copyright 2000 American Mathematical Society

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