Regularity and completeness of half-Lie groups

Half-Lie groups exist only in infinite dimensions: They are smooth manifolds and topological groups such that right translations are smooth, but left translations are merely required to be continuous. The main examples are groups of H^{s} or C^{k} diffeomorphisms and semidirect products of a Lie group with kernel an infinite-dimensional representation space. Here, we investigate mainly Banach half-Lie groups, the groups of their C^{k} -elements, extensions, and right-invariant strong Riemannian metrics on them: surprisingly, the full Hopf–Rinow theorem holds, which is not the case in general even for Hilbert manifolds.