Loops as sections in compact Lie groups

TL;DR

This paper investigates the structure of certain topological loops related to compact Lie groups, proving non-existence results and providing constructions for specific cases, thereby advancing understanding of loop-Lie group relationships.

Contribution

It establishes non-existence of certain loops homeomorphic to quasi-simple Lie groups with compact translation groups and offers a construction method for loops with product group translations.

Findings

No connected proper loops homeomorphic to quasi-simple Lie groups with compact translation groups exist.

Connected loops homeomorphic to the 7-sphere with compact translation groups are classical.

A simple construction for proper loops with translation groups as products of at least three factors.

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.