Infinite-Dimensional Lie Groups

TL;DR

This book provides a comprehensive introduction to infinite-dimensional Lie groups, covering foundational calculus, manifold theory, and various classes of these groups with detailed structural and topological analysis.

Contribution

It develops the theory of infinite-dimensional Lie groups using Bastiani's smooth maps, including new insights into their structure, subgroups, quotients, and specific classes like diffeomorphism groups.

Findings

Analysis of regular Lie groups and exponential maps

Detailed study of subgroups and quotient structures

Examination of specific classes like diffeomorphism groups

Abstract

This is a preliminary version of a book on infinite-dimensional Lie groups. It covers the basics of calculus and manifolds in the context of locally convex spaces, based on Bastiani's notion of a smooth map. Starting from this concept, we develop the basics of smooth manifolds and define Lie groups as manifolds with smooth group operations. We discuss in particular several classes of Lie groups, such as regular ones, or those with an exponential function that is a local diffeomorphism. The local theory, subgroups and quotients are explored in some detail. Classes of Lie groups that are discussed in detail include: unit groups of continuous inverse algebras, groups of smooth maps, direct limit groups and groups of diffeomorphism. We also included chapters on the topology of infinite-dimensional Lie group and on various selected topics.