Loops as sections in compact Lie groups

Abstract

We prove that there does not exist any connected topological proper loop homeomorphic to a quasi-simple Lie group and having a compact Lie group as the group topologically generated by its left translations. Moreover, any connected topological loop homeomorphic to the 7-sphere and having a compact Lie group as the group of its left translations is classical. We give a particular simple general construction for proper loops such that the compact group of their left translations is direct product of at least 3 factors.

Our Friend and Mathematician Karl Strambach

“…Moreover, they showed that any topological loop L homeomorphic to a sphere or to a real projective space and having a compact-free Lie subgroup as the stabilizer of the identity of L in the Lie group topologically generated by all left translations is homeomorphic to the circle. Applying the investigation of H. Scheerer, who has clarified for which compact connected Lie groups G and for which closed subgroups H the natural projection G → G/H has a continuous section σ, in [18] K. Strambach and Á. Figula proved that there does not exist any connected topological proper loop homeomorphic to a quasisimple Lie group and having a compact Lie group as the group topologically generated by its left translations. Similarly, any connected topological loop L homeomorphic to the 7-sphere and having a compact Lie group as the group topologically generated by its left translations is either the Moufang loop O of octonions of norm 1 or the factor loop O/Z, where Z is the centre of O.…”