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We study noncommutative deformation of manifolds by constructing star products. We start from a noncommutative d and discuss more genaral noncommutative manifolds. In general, star products can not be described in concrete expressions without some exceptions. In this article we introduce new examples of noncommutative manifolds with explicit star products. Karabegov's deformation quantization of PN and HN with separation of variables gives explicit calulable star products represented by gamma functions. Using the results of star products between inhomogeneous coordinates, we find creation and anihilation operators and obtain the Fock representation of the noncommutative PN and HN.

NoncommutativeCPNandCHNand their physics

NoncommutativeCP^NandCH^Nand their physics