Direct limits of infinite-dimensional Lie groups

TL;DR

This survey explores the structure and properties of infinite-dimensional Lie groups formed as unions of ascending sequences of Lie groups, focusing on their direct limit characteristics, regularity, homotopy, subgroups, and construction methods.

Contribution

It compiles general results and clarifies the theory behind infinite-dimensional Lie groups expressed as ascending unions, highlighting new insights into their structure and properties.

Findings

Analysis of direct limit properties of Lie groups

Results on regularity in Milnor's sense

Descriptions of homotopy groups and subgroup structures

Abstract

Many infinite-dimensional Lie groups of interest can be expressed as a union of an ascending sequence of (finite- or infinite-dimensional) Lie groups. In this survey article, we compile general results concerning such ascending unions, describe the main classes of examples, and explain what the general theory tells us about these. In particular, we discuss: (1) Direct limit properties of ascending unions of Lie groups in the relevant categories; (2) Regularity in Milnor's sense; (3) Homotopy groups of direct limit groups and of Lie groups containing a dense union of Lie groups; (4) Subgroups of direct limit groups; (5) Constructions of Lie group structures on ascending unions of Lie groups.