Loops as sections in compact Lie groups (Q525193)

Consider a coset projection \(p: G \to G/H\) from a compact connected Lie group \(G\). \textit{H. Scheerer} [Math. Ann. 206, 149--155 (1973; Zbl 0247.22010)] has given necessary and sufficient conditions for the existence of a continuous section \(\sigma : G/H \to G\) for \(p\). Roughly he showed that the only nontrivial building block for such structures comes from the action of \(SO_8\mathbb R\) on the 7-sphere, the section being obtained from octonion multiplication. If no such block occurs, then \(G/H\) is homeomorphic to a compact Lie group. The present article addresses the question, raised by Scheerer, in which cases a section \(\sigma\) defines a loop strucure on \(L = G/H\) via \(x\cdot y : = p(\sigma(x)\cdot \sigma(y))\), or in other words, in which cases \(\sigma(L)\) is sharply transitive on \(L\). So assume that \(L = G/H\) is a loop. If \(L\) is homeomorphic to \(\mathbb S_7\) or to \(P_7\mathbb R\), then it is shown that \(L\) is the Moufang loop of octonions of norm one or its quotient modulo \(-1\). If \(L\) is not a group and is homeomorphic to a compact semisimple Lie group, then also \(G\) is semisimple with at least two simple factors, and \(\dim G \geq 14\). If \(\dim G = 14\), then \(G\) is locally isomorphic to \(Spin_3\times SU_3 \times Spin_3\). A general construction for loops coming from groups \(G\) with three factors is presented.