Regularity of Infinite-Dimensional Lie Groups by Metric Space Methods

Abstract

Regularity of infinite dimensional Lie groups was defined by Hideki Omori et al. and John Milnor. Up to now the only known sufficient conditions for regularity are analytic in nature and they are included in the definition of strong \textit{ILB}-Lie groups, since there are no existence theorems for ordinary differential equations on non-normable locally convex spaces. We prove that regularity can be characterized by the existence of a family of so called Lipschitz-metrics in all interesting cases of infinite dimensional Lie groups. On Lipschitz-metrizable groups all product integrals converge to the solutions of the respective equations if some weak conditions satisfied by all known Lie groups are given. Lipschitz-metrizable groups provide a framework to solve differential equations on infinite dimensional Lie groups. Furthermore Lipschitz-metrics are the non-commutative generalization of the concept of seminorms on a Fréchet-space viewed as abelian Lie group.