Weighted diffeomorphism groups of Banach spaces and weighted mapping groups

TL;DR

This paper develops and analyzes infinite-dimensional Lie groups modeled on weighted function spaces, including weighted diffeomorphism and mapping groups, demonstrating their regularity and exploring their algebraic structures.

Contribution

It introduces new classes of weighted diffeomorphism and mapping groups as regular Lie groups, extending the theory of infinite-dimensional Lie groups on Banach spaces.

Findings

Weighted diffeomorphism groups are regular Lie groups.

Construction of weighted mapping groups as regular Lie groups.

Analysis of semidirect products and integrability of related Lie algebras.

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 of weighted diffeomorphisms on a Banach space. Further, we also construct certain types of weighted mapping groups. 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 the former groups. Moreover, we study the integrability of Lie algebras of certain vector fields.