Abstract

Symplectic geometry is an important branch of modern differential geometry. The purpose of the work is a description four-dimensional isotropically-faithful homogeneous spaces with an invariant non-degenerate almost symplectic structure over the field C. In the work the basic concepts are defined: almost symplectic structure, generalized module, virtual pair, isotropic representation, isotropically-faithful pair, virtual structure. The algorithm for classifying isotropically-faithful homogeneous spaces with an invariant non-degenerate almost symplectic structure is presented. Using this algorithm, we explicitly describe four-dimensional isotropically-faithful almost symplectic homogeneous spaces in the complex case. The algorithms described in the work can be computerized and used to solve similar problems in large dimensions. The results obtained in this paper can be applied in various areas of mathematics and physics, in particular, the symplectic manifold allows us to introduce Hamiltonian mechanics in a natural geometric way and provides a visual interpretation of many of its properties, the apparat of symplectic geometry is transferred from geometric optics and classical mechanics to quantum mechanics.