Lie 2-groups from loop group extensions

Understand

We give a very simple construction of the string 2-group as a strict Fr\'echet Lie 2-group.

• The corresponding crossed module is defined using the conjugation action of the loop group on its central extension, which drastically simplifies several constructions previously given in the literature.
• More generally, we construct strict 2-group extensions for a Lie group from a central extension of its based loop group, under the assumption that this central extension is disjoint commutative.
• We show in particular that this condition is automatic in the case that the Lie group is semisimple and simply connected.