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Coverings of topological loops. (English. Russian original) Zbl 1181.22009
J. Math. Sci., New York 137, No. 5, 5098-5116 (2006); translation from Sovrem. Mat. Prilozh. 2004, No. 22 (2004).
The authors discuss the question: For a given connected Lie group $$\widetilde{G}$$ having a continuous, sharply transitive section $$\widetilde{\sigma }: \widetilde{G}/ \widetilde{H} \to \widetilde{G}$$ with respect to a connected subgroup $$\widetilde{H}$$, in which groups $$G$$ having $$\widetilde{G}$$ as a covering group, does there exist a sharply transitive section $$\sigma : G/ H \to G$$ such that the loop $$\widetilde{L}$$ corresponding to the section $$\widetilde{\sigma }$$ is a covering of the loop $$L$$ corresponding to $$\sigma$$.
The authors show: If $$\widetilde{G}$$ is the simply connected $$3$$-dimensional, non-Abelian, nilpotent Lie group, then for any section $$\widetilde{\sigma }: \widetilde{G}/ \widetilde{H} \to \widetilde{G}$$ corresponding to the topological loop $$\widetilde{L}$$ and any group $$G$$ covered by $$\widetilde{G}$$, there exists a section $$\sigma : G/ H \to G$$ in $$G$$ such that the corresponding loop $$L$$ is covered by the loop $$\widetilde{L}$$. But there are higher-dimensional nilpotent Lie groups $$G$$ of nilpotency class $$2$$ whose universal coverings contain sharply transitive sections corresponding to loops $$\widetilde{L}$$ such that $$G$$ cannot be the group topologically generated by the left translations of topological loops $$L$$ having $$\widetilde{L}$$ as the universal covering. The group $$\widetilde{G}$$, which is the direct product of $$\mathbb R$$ and the $$2$$-dimensional affine group, has differentiable sections $$\widetilde{\sigma }: \widetilde{G}/ \widetilde{H} \to \widetilde{G}$$ such that the exponential image $$\exp (T_1 \widetilde{\sigma } (\widetilde{G}/ \widetilde{H}))$$ of the tangent space $$T_1 \widetilde{\sigma } (\widetilde{G}/ \widetilde{H})$$ at $$1 \in \widetilde{G}$$ is contained in the image $$\widetilde{\sigma } (\widetilde{G}/ \widetilde{H})$$ of the section $$\widetilde{\sigma }$$ for which there exists no loop having a group $$G$$ covered by $$\widetilde{G}$$ as the group topologically generated by left translations. Moreover, the authors prove in two different ways that no proper covering of the group $$PSL_2(\mathbb R)$$ can occur as the group topologically generated by the left translations of a $$2$$-dimensional topological loop.

##### MSC:
 22A22 Topological groupoids (including differentiable and Lie groupoids)
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##### References:
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