Regularity of infinite-dimensional Lie groups by metric space methods (Q5946890)

An infinite-dimensional Lie group \(G\) with Lie algebra \(\mathfrak g\) (in the convenient setting of \textit{A. Kriegl} and \textit{P. Michor} [The convenient setting of global analysis, Math. Surv. Monogr. 53, Providence, RI, AMS (1997; Zbl 0889.58001)]) is called \textit{regular} if there is a smooth evolution map \({\text{Evol}}^{r}:C^\infty({\mathbb R},{\mathfrak g})\rightarrow C^\infty({\mathbb R},G)\) such that \({\text{Evol}}^{r}(X)(0)=e\) and \(\delta^r(\text{Evol}^{r}(X))(t)=X(t)\), where \(\delta^r\) is the right logarithmic derivative. In particular, the evolution with respect to a constant curve is a smooth one-parameter subgroup. In other words, the concept of regularity means that one can solve all non-autonomous Cauchy problems on the Lie group \(G\). The author shows in the paper that the regularity can be characterized by the existence of a family of Lipschitz metrics on \(G\), where Lipschitz metric is an appropriate non-commutative generalization of the concept of seminorm on a Fréchet space viewed as an abelian group.