Homogeneous spaces and quasigroups

The paper under review is a survey on quasigroup methods in the theory of homogeneous spaces essentially based on the works of the author and his students. The main tool here is the so-called R. Baer-L. V. Sabinin construction establishing the equivalence of categories of left loops and left homogeneous transversalled spaces, which is displayed in the smooth case. Reductive, symmetric and generalized symmetric spaces are treated as smooth loops with identities. The unifying concept of transsymmetric space is also treated (as a smooth correct left F-quasigroup). par In this framework a new concept of hyporeductive homogeneous space is described in terms of an appropriate hyporeductive algebra. The quasigroup analysis of homogeneous spaces leads to new classes of such spaces as almost symmetric and antisymmetric spaces and generates new linear algebras that are close to Malʹtsev and Sagle algebras. par The bibliography (29 titles) contains some titles related to physical applications.