Regularity and completeness of half-Lie groups

Abstract

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.