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Loops on spheres having a compact-free inner mapping group. (English) Zbl 1175.22002

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).

MSC:

22A30 Other topological algebraic systems and their representations
57S20 Noncompact Lie groups of transformations
22E99 Lie groups
20N05 Loops, quasigroups
22F30 Homogeneous spaces
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[6] Kamke E (1951) Differentialgleichungen Lösungsmethoden und Lösungen. Mathematik und Ihre Anwendungen in Physik und Technik. Band 181. Leipzig: Akademische Verlagsgesellschaft · Zbl 0026.31801
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