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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.

22A22 Topological groupoids (including differentiable and Lie groupoids)
Full Text: DOI
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