Half-Lie groups

TL;DR

This paper investigates a class of topological groups with Banach manifold structures that are not fully smooth, showing they share many properties with Banach-Lie groups, including solutions to differential equations and group regularity.

Contribution

It introduces and analyzes half-Lie groups, a new class of topological groups with Banach manifold structures but non-smooth multiplication, extending Lie theoretic properties to this setting.

Findings

Solutions to initial value problems exist for regulated functions.

One-parameter groups satisfy the Trotter formula.

Subgroups of smooth orbit elements form Fréchet-Lie groups.

Abstract

In this paper we study the Lie theoretic properties of a class of topological groups which carry a Banach manifold structure but whose multiplication is not smooth. If $G$ and $N$ are Banach-Lie groups and $\pi : G \to \mathrm{Aut}(N)$ is a homomorphism defining a continuous action of $G$ on $N$, then $H := N \rtimes_\pi G$ is a Banach manifold with a topological group structure for which the left multiplication maps are smooth, but the right multiplication maps need not to be. We show that these groups share surprisingly many properties with Banach-Lie groups: (a) for every regulated function $\xi : [0,1] \to T_1H$ the initial value problem $\dot\gamma(t) = \gamma(t)\xi(t)$, $\gamma(0)= 1_H$, has a solution and the corresponding evolution map from curves in $T_1H$ to curves in $H$ is continuous; (b) every $C^1$-curve $\gamma$ with $\gamma(0) = 1$ and $\gamma'(0) = x$ satisfies $\lim_{n \to \infty} \gamma(t/n)^n = \exp(tx)$; (c) the Trotter formula holds for $C^1$ one-parameter groups in $H$; (d) the subgroup $N^\infty$ of elements with smooth $G$-orbit maps in $N$ carries a natural Fr\'echet-Lie group structure for which the $G$-action is smooth; (e) the resulting Fr\'echet-Lie group $H^\infty := N^\infty \rtimes G$ is also regular in the sense of (a).