Weighted diffeomorphism groups of Banach spaces and weighted mapping groups
Volume 484 / 2012
Abstract
In this work, we construct and study certain classes of infinite-dimensional Lie groups that are modelled on weighted function spaces. In particular, we construct a Lie group $\mathrm{Diff}_{\mathcal W}(X)$ of diffeomorphisms, for each Banach space $X$ and each set $\mathcal W$ of weights on $X$ containing the constant weights. We also construct certain types of “weighted mapping groups”. These are Lie groups modelled on weighted function spaces of the form $\mathcal C_{\mathcal W}^k(U,{\mathbf L}(G))$, where $G$ is a given (finite- or infinite-dimensional) Lie group. Both the weighted diffeomorphism groups and the weighted mapping groups are shown to be regular Lie groups in Milnor's sense.
We also discuss semidirect products of such groups. Moreover, we study the integrability of Lie algebras of vector fields of the form $\mathcal C_{\mathcal W}^\infty(X,X)\rtimes \mathbf L(G)$, where $X$ is a Banach space and $G$ a Lie group acting smoothly on $X$.
