Complexifications of infinite-dimensional manifolds and new constructions of infinite-dimensional Lie groups

Abstract

Let M be a real analytic manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of the latter (as a complex manifold) is uniquely determined. If M is regular and the complexified modeling space of M is normal, then a regular complexification exists for some neighborhood of K. For each (real or complex) analytic regular manifold M modeled on a metrizable locally convex space and Banach-Lie group H, this allows the group Germ(K,H) of germs of H-valued analytic maps around K in M to be turned into an analytic Lie group which is regular in Milnor's sense. A special case is a regular real analytic Lie group structure on the group of real analytic H-valued maps on a compact real analytic manifold M (which, previously, had only been treated in the convenient setting of analysis). Combining our results concerning groups of germs with an idea by Neeb and Wagemann, one can also obtain a regular Lie group structure on the group of all real analytic H-valued mappings on the real line.