Quantization of K\"ahler manifolds admitting $H$-projective mappings

Abstract

We discuss the quantization of mechanical systems for which the Hamiltonian vector fields of observables form the deformation of $n$-dimensional oscilator algebra. Because of this fact these systems can be considered as "deformations" of the harmonic oscillator. The set of abovementioned mechanical systems are realized at the classical level in the form of K\"ahler manifolds of constant holomorphic curvature. Such mechanical systems are quantized later with the help of the geometric quantization approach. We also discuss the quantization of more general K\"ahler manifolds (not necessarily of constant holomorphic curvature) admitting $H$-projective mappings.