Regularity properties of infinite-dimensional Lie groups, and semiregularity

TL;DR

This paper investigates the regularity properties of infinite-dimensional Lie groups, establishing conditions under which these groups are semiregular or regular, and applies these results to groups of diffeomorphisms.

Contribution

It introduces new criteria for regularity of infinite-dimensional Lie groups and proves regularity for important classes like diffeomorphism groups.

Findings

C^k-semiregularity implies C^m-regularity under certain conditions

Continuity at zero of evol leads to smooth evol in C^0-semiregular groups

Diff(M) of smooth diffeomorphisms is C^1-regular

Abstract

Let G be a Lie group modelled on a locally convex space, with Lie algebra g, and k be a non-negative integer or infinity. We say that G is C^k-semiregular if each C^k-curve c in g admits a left evolution Evol(c) in G. If, moreover, the map taking c to evol(c):=Evol(c)(1) is smooth, then G is called C^k-regular. For G a C^k-semiregular Lie group and m an order of differentiability, we show that evol is C^m if and only if Evol is C^m. If evol is continuous at 0, then evol is continuous. If G is a C^0-semiregular Lie group, then continuity of evol implies its smoothness (so that G will be C^0-regular), if smooth homomorphisms from G to C^0-regular Lie groups separate points on G and g is (e.g.) sequentially complete. Further criteria for regularity properties are provided, and used to prove regularity for several important classes of Lie groups. Notably, we find that the Lie group Diff(M) of smooth diffeomorphisms of a paracompact finite-dimensional smooth manifold M (which need not be sigma-compact) is C^1-regular. We also provide tools which enable to show that the Lie group of analytic diffeomorphisms of a compact real analytic manifold is C^1-regular.