Half-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 $H:=N{⋊}_{\pi }G$ 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 $\xi :[0,1]\to T_{1}H$ 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 $T_{1}H$ to curves in $H$ is continuous; (b) every $C^1$-curve $\gamma$ with $\gamma (0)=1$ and ${\gamma }^{\prime }(0)=x$ 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 Fréchet-Lie group structure for which the $G$-action is smooth; (e) the resulting Fréchet-Lie group $H^{\infty }:=N^{\infty }⋊G$ is also regular in the sense of (a).

28 pages; minor improvements, title changed to be consistent with previously defined terminology

References in corpus (1)

• An exotic zoo of diffeomorphism groups on ${\mathbb{R}}^n$

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