FOUR-DIMENSIONAL HOMOGENEOUS SPACES WITH ALMOST SYMPLECTIC STRUCTURE. THE REAL CASE
UDC 514.144
Key words: Lie algebra, homogeneous space, real form, isotropic representation, almost symplectic structure.
For citation: Mozhey N. P. Four-dimensional homogeneous spaces with almost symplectic structure. The real case. Proceedings of BSTU, issue 3, Physics and Mathematics. Informatics, 2021, no. 2 (248), pp. 15–21 (In Russian). DOI: https://doi.org/10.52065/2520-6141-2021-248-2-15-21.
Abstract
The purpose of the work is a description of four-dimensional isotropically-faithful homogeneous spaces with an invariant non-degenerate almost symplectic structure over the field of real numbers. It defines the basic concepts: almost symplectic structure, isotropic representation, isotropically-faithful pair, complexification of Lie algebra, anti-involution, real form. The algorithm for classifying isotropically-faithful homogeneous spaces with an invariant non-degenerate almost symplectic structure is presented. Finding real forms of both subalgebras of linear Lie algebras and isotropically-faithful pairs is described, and an explicit description of four-dimensional isotropically-faithful almost symplectic homogeneous spaces in the real case is given. The features of the methods presented in the work are the application of a purely algebraic approach to the description of homogeneous spaces and structures on them. The results obtained in the paper can be used in works on differential geometry, differential equations, topology, as well as in other areas of mathematics and physics. The algorithms given in the work can be computerized and used for the decision of similar problems in large dimensions.
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