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On the theory of \(\varphi\)-spaces. (Russian) Zbl 0809.22004
Let \(G/H\) be a left homogeneous \(\varphi\)-space [see N. A. Stepanov, Izv. Vyssh. Uchebn. Zaved., Mat. 1967, 88-95 (1967; Zbl 0149.276)] and \(Q\) be its quasi-reductant. Such a space is called regular if \(Q = \{g \varphi (g^{-1})\}\), \(g \in G\). It is proved that a regular \(\varphi\)-space is reductive, and \(Q\) is its reductant. A homogeneous \(\varphi\)-space is called minimal if the subgroup \(\widetilde {Q} = \langle Q \rangle_ G\) generated by the reductant \(Q\) coincides with \(G\). Let a homogeneous \(\varphi\)-space be endowed with a non-associative multiplication: \((xH) \circ (yH) = x \varphi (x^{-1}) y \cdot H\). A number of claims are proved describing properties of homogeneous regular \(\varphi\)-spaces in terms of this multiplication.
MSC:
22A22 Topological groupoids (including differentiable and Lie groupoids)
20N05 Loops, quasigroups
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
22A30 Other topological algebraic systems and their representations
53A60 Differential geometry of webs
22E05 Local Lie groups
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