  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 wrong in general even for Hilbert
manifolds.
