Regularity and completeness of half-Lie groups
Martin Bauer, Philipp Harms, Peter W. Michor
TL;DR
This work develops a comprehensive theory of half-Lie groups—infinite-dimensional manifolds with one-sided differentiability between the group and manifold structures. It proves that for Banach half-Lie groups carrying a right-invariant local addition, the $C^k$-elements $G^k$ are regular Banach half-Lie groups and the inverse limit $G^ ty$ is a regular Fréchet Lie group, extending prior semidirect-product results. It extends extension theory to half-Lie groups via extension data $(oldsymbol{ ho},f)$, including a detailed treatment of split and central extensions, with explicit constructions and differentiability results; it then applies these ideas to automorphism and gauge groups of principal bundles and to fiber-preserving diffeomorphisms. In the Riemannian direction, the paper proves Hopf–Rinow-type completeness for strong right-invariant metrics and a no-loss-no-gain property for geodesic evolution, with applications to Sobolev diffeomorphism groups and curvature formulas adapted to the one-sided setting. Overall, the framework provides new completeness, regularity, and geometric insights for infinite-dimensional diffeomorphism/extension groups and their structured variants.
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.