Karabegov, Alexander V. Deformation quantizations with separation of variables on a Kähler manifold. (English) Zbl 0866.58037 Commun. Math. Phys. 180, No. 3, 745-755 (1996). A deformation quantization on a symplectic manifold \(M\) is an associative algebra structure on the space \(C^\infty(M)[[v]]\) of formal power series such that the algebra multiplication * is a deformation of the ordinary multiplication of functions on \(M\) and the *-commutator is a deformation of the Poisson bracket. In this paper the author considers deformation quantizations on Kähler manifolds that satisfy the following separation of variables property. For each open set \(U\subseteq M\) the *-multiplication from the left by a holomorphic function and from the right by an antiholomorphic function coincides with the ordinary multiplication. He shows that these quantizations are in 1-1 correspondence with the formal deformations of the original Kähler metric. Reviewer: V.Perlick (Berlin) Cited in 10 ReviewsCited in 53 Documents MSC: 53D50 Geometric quantization 81S10 Geometry and quantization, symplectic methods 53B35 Local differential geometry of Hermitian and Kählerian structures Keywords:deformation quantization; Kähler manifolds; separation of variables PDF BibTeX XML Cite \textit{A. V. Karabegov}, Commun. Math. Phys. 180, No. 3, 745--755 (1996; Zbl 0866.58037) Full Text: DOI arXiv OpenURL References: [1] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys.111, 1–151 (1978) · Zbl 0377.53025 [2] Berezin, F.A.: Quantization. Math. USSR Izv.8, 1109–1165 (1974) · Zbl 0312.53049 [3] Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds, II. Trans. Am. Math. Soc.337, 73–98 (1993) · Zbl 0788.53062 [4] Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds, IV. Lett. Math. Phys.34, 159–168 (1995) · Zbl 0831.58026 [5] Karabegov, A.V.: On deformation quantization on a Kähler manifold associated to Berezin’s quantization. To appear in Funct. Anal. Appl. · Zbl 0869.53045 [6] Moreno, C.: *-products on some Kähler manifolds. Lett. Math. Phys.11, 361–372 (1986) · Zbl 0618.53049 [7] Moreno, C.: Invariant star products and representations of compact semisimple Lie groups. Lett. Math. Phys.12, 217–229 (1986) · Zbl 0681.53036 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.