Towards a Lie theory for locally convex groups

TL;DR

This survey reviews the current state of Lie theory for locally convex groups, focusing on integrability issues, regularity, and specific classes like direct limit and diffeomorphism groups.

Contribution

It synthesizes recent advances in understanding integrability and regularity in locally convex Lie groups, highlighting their applications to various specific group classes.

Findings

Regularity and local exponentiality facilitate solving fundamental Lie theory problems.

Integrability of Lie algebras and subalgebras varies across different classes of locally convex groups.

Specific classes like direct limit groups and diffeomorphism groups exhibit unique integrability properties.

Abstract

In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfying answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.