Regularity of Lie Groups

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 $\text{iff}$ 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 $k\in {\mathbb{N}}_{\ge 1}\bigsqcup \{\mathrm{lip},\infty \}$ is integrable (contained in the domain of the evolution map) $\text{iff}$ $G$ 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 $k\in \mathbb{N}\bigsqcup \{\mathrm{lip},\infty \}$. 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 $k\in \mathbb{N}\bigsqcup \{\infty \}$, then it is smooth thereon $\text{iff}$ it is differentiable at zero $\text{iff}$ $\mathfrak{g}$ is Mackey$/$integral complete for $k\in {\mathbb{N}}_{\ge 1}\bigsqcup \{\infty \}/k\equiv 0$. This result is obtained by calculating the directional derivatives explicitly, recovering the standard formulas that hold, e.g., in the Banach case.

72 pages. Version as published in Communications in Analysis and Geometry