Simplified Proofs for the Pro-Lie Group Theorem and the One-Parameter Subgroup Lifting Lemma

TL;DR

This paper provides simplified, more elementary proofs for key theorems in the theory of pro-Lie groups, improving understanding and correctness in the field.

Contribution

It offers shorter, clearer proofs for the Pro-Lie Group Theorem and the One-Parameter Subgroup Lifting Lemma, enhancing the theoretical framework.

Findings

Shorter proof of the One-Parameter Subgroup Lifting Lemma

Corrected proof of the Pro-Lie Group Theorem

Clarified the relationship between pro-Lie groups and projective limits of Lie groups

Abstract

This note is devoted to the theory of projective limits of finite-dimensional Lie groups, as developed in the recent monograph ``The Lie Theory of Connected Pro-Lie Groups'' by K.H. Hofmann and S.A. Morris. We replace the original, highly non-trivial proof of the One-Parameter Subgroup Lifting Lemma given in the monograph by a shorter and more elementary argument. Furthermore, we shorten (and correct) the proof of the so-called Pro-Lie Group Theorem, which asserts that pro-Lie groups and projective limits of Lie groups coincide.