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Homogeneous spaces and quasigroups. (English. Russian original) Zbl 0874.53037
Russ. Math. 40, No. 7, 74-81 (1996); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1996, No. 7(410), 77-84 (1996).
The paper is a review dedicated to loops and homogeneous spaces. Let \(G/H\) be a homogeneous space, \(Q\) be a submanifold of the Lie group \(G\) through the unit of \(G\) transversal to the cosets \(gH\). Then we can define a binary operation on \(Q\) by projection on \(Q\) along the leaves \(gH\). Conversely, one can reconstruct a homogeneous space \(G/H\) from the given loop \(Q\). The connection between \(Q\) and \(G/H\) gives a possibility to describe the homogeneous space properties in terms of quasigroup and loop theory.

MSC:
53C30 Differential geometry of homogeneous manifolds
20N05 Loops, quasigroups
22A22 Topological groupoids (including differentiable and Lie groupoids)
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