publication . Article . Preprint . 2010

star products with separation of variables admitting a smooth extension

Karabegov, Alexander;
Open Access
  • Published: 26 Dec 2010 Journal: Letters in Mathematical Physics, volume 101, pages 125-142 (issn: 0377-9017, eissn: 1573-0530, Copyright policy)
  • Publisher: Springer Science and Business Media LLC
Abstract
Given a complex manifold $M$ with an open dense subset $\Omega$ endowed with a pseudo-Kaehler form $\omega$ which cannot be smoothly extended to a larger open subset, we consider various examples where the corresponding Kaehler-Poisson structure and a star product with separation of variables on $(\Omega, \omega)$ admit smooth extensions to $M$. We suggest a simple criterion of the existence of a smooth extension of a star product and apply it to these examples.
Subjects
arXiv: Mathematics::Differential Geometry
free text keywords: Mathematical Physics, Statistical and Nonlinear Physics, A* search algorithm, law.invention, law, Mathematics, Complex manifold, Smooth structure, Discrete mathematics, Separation of variables, Topology, Dense set, Mathematics - Quantum Algebra, 53D55, 53D17, 53B35

[1] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Physics 111 (1978), no. 1, 61 - 110. [OpenAIRE]

[2] Bordemann, M. and Waldmann, S.: A Fedosov star product of the Wick type for K¨ahler manifolds. Lett. Math. Phys. 41 (3) (1997), 243 - 253.

[3] Engliˇs, M.: Weighted Bergman kernels and quantization, Commun. Math. Phys. 227 (2002), 211-241.

[4] Helgason, S.: Some Results on Invariant Differential Operators on Symmetric Spaces. Amer. J. Math. 114 (1992), 789-811.

[5] Karabegov, A.: Deformation quantizations with separation of variables on a K¨ahler manifold. Commun. Math. Phys. 180 (1996), 745-755. [OpenAIRE]

[6] Karabegov, A.: A covariant Poisson deformation quantization with separation of variables up to the third order. Lett. Math. Phys. 61 (2002), 255 - 261. [OpenAIRE]

[7] Karabegov, A.: Formal symplectic groupoid of a deformation quantization. Commun. Math. Phys. 258 (2005), 223-256. [OpenAIRE]

[8] Karabegov, A.: Deformation quantization of a K¨ahler-Poisson structure vanishing on a Levi nondegenerate hypersurface. Contemporary Math. 450 (2008), 163 - 171.

[9] Kontsevich, M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66 (2003), 157 - 216. [OpenAIRE]

[10] Leichtnam, E., Tang, X., and Weinstein, A.: Poisson geometry and deformation quantization near a strictly pseudoconvex boundary. Journal of the European Mathematical Society, 9, (2007), 681-704. (Alexander Karabegov) Department of Mathematics, Abilene Christian University, ACU Box 28012, Abilene, TX 79699-8012 E-mail address: axk02d@acu.edu

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