Regularity of Lie Groups

TL;DR

This paper investigates the regularity properties of Milnor's infinite-dimensional Lie groups, establishing conditions for the continuity and differentiability of the evolution map in various topological settings, and exploring implications for Lie group integrability.

Contribution

It provides necessary and sufficient conditions for regularity of Milnor's Lie groups in $C^0$ and $C^k$ topologies, including new criteria for Mackey completeness and $ ext{k}$-confined curves.

Findings

The evolution map is $C^0$-continuous iff the group is locally $ extmu$-convex.

If the evolution map is defined on all smooth curves, then the group is Mackey complete.

Under local $ extmu$-convexity, $C^k$-curves are integrable iff the group is Mackey complete and $ ext{k}$-confined.

Abstract

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the $C^0$-topological context, and provide necessary and sufficient regularity conditions for the (standard) $C^k$-topological setting. We prove that the evolution map is $C^0$-continuous on its domain $\textit{iff}\hspace{1pt}$ the Lie group $G$ is locally $\mu$-convex. We furthermore show that if the evolution map is defined on all smooth curves, then $G$ is Mackey complete. Under the assumption that $G$ is locally $\mu$-convex, we show that each $C^k$-curve for $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$ is integrable (contained in the domain of the evolution map) $\textit{iff}\hspace{1pt}$ $G$ is Mackey complete and $\mathrm{k}$-confined. The latter condition states that each $C^k$-curve in the Lie algebra $\mathfrak{g}$ of $G$ can be uniformly approximated by a special type of sequence that consists of piecewise integrable curves. A similar result is proven for the case $k\equiv 0$; and, we provide several mild conditions that ensure that $G$ is $\mathrm{k}$-confined for each $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$. We finally discuss the differentiation of parameter-dependent integrals in the (standard) $C^k$-topological context. In particular, we show that if the evolution map is defined and continuous on $C^k([0,1],\mathfrak{g})$ for $k\in \mathbb{N}\sqcup\{\infty\}$, then it is smooth thereon $\textit{iff}\hspace{1pt}$ it is differentiable at zero $\textit{iff}\hspace{1pt}$ $\mathfrak{g}$ is $\hspace{0.2pt}$ Mackey$\hspace{1pt}/ \hspace{1pt}$integral$\hspace{1pt}$ complete for $k\in \mathbb{N}_{\geq 1}\sqcup\{\infty\}\hspace{1pt}/\hspace{1pt}k\equiv 0$. This result is obtained by calculating the directional derivatives explicitly, recovering the standard formulas that hold, e.g., in the Banach case.