Figula, Ágota; Strambach, Karl Loops on spheres having a compact-free inner mapping group. (English) Zbl 1175.22002 Monatsh. Math. 156, No. 2, 123-140 (2009). Let \((L,*)\) be an almost topological proper loop (i.e., \(L\) is a locally compact space and the multiplication \(*:L\times L \to L\) is continuous) homeomorphic to a sphere or to a real projective space. Assume that the group \(G\) topologically generated by the left translations of \(L\) is a Lie group and the stabilizer \(H\) of the identity of \(L\) in \(G\) is a compact-free subgroup of \(G\). Under these assumptions, using results of H. Salzman, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen, M. Stroppel [Compact projective planes, de Gruyter, Berlin, New York (1995; Zbl 0851.51003)], J. F. Adams [Ann. Math. (2) 72, 20–104 (1960; Zbl 0096.17404)], V. V. Gorbatsevich and A. L. Onishchik [Lie groups and Lie algebras I. Foundations of Lie theory. Lie transformation groups. Encycl. Math. Sci. 20, 95–229 (1993); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 20, 103–240 (1988; Zbl 0781.22004)], T. Asoh [Osaka J. Math. 24, 271–298 (1987; Zbl 0706.57021)] and P. T. Nagy and the second author [Loops in group theory and Lie theory. de Gruyter Expositions in Mathematics 35. (Berlin 2002; Zbl 1050.22001)], the authors prove that \(L\) is homeomorphic to the \(1\)-sphere and \(G\) is a finite covering of the group \(PSL_2(\mathbb{R})\). Moreover, using the theory of Fourier series, they classify the \(1\)-dimensional connected \(\mathcal{C}^1\)-differentiable loops \(L\) such that the group topologically generated by the left translations of \(L\) is isomorphic to the group \(SL_2(\mathbb{R})\) (implying that \(L\) is compact). Reviewer: Mircea Craioveanu (Timişoara) Cited in 1 ReviewCited in 3 Documents MSC: 22A30 Other topological algebraic systems and their representations 57S20 Noncompact Lie groups of transformations 22E99 Lie groups 20N05 Loops, quasigroups 22F30 Homogeneous spaces Keywords:locally compact loop; differentiable loop; multiplications on spheres; sharply transitive section Citations:Zbl 0096.17404; Zbl 0781.22004; Zbl 0706.57021; Zbl 1050.22001; Zbl 0851.51003 PDFBibTeX XMLCite \textit{Á. Figula} and \textit{K. Strambach}, Monatsh. Math. 156, No. 2, 123--140 (2009; Zbl 1175.22002) Full Text: DOI arXiv References: [6] Kamke E (1951) Differentialgleichungen Lösungsmethoden und Lösungen. Mathematik und Ihre Anwendungen in Physik und Technik. Band 181. Leipzig: Akademische Verlagsgesellschaft · Zbl 0026.31801 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.