An example of an infinite Lie group

Abstract:

We study the complex l.c.s. X of germs of holomorphic mappings around the origin of ${C^n}$, with values in ${C^n}$, vanishing at the origin. We show that X is isomorphic to $M(n,C) \times {H_2}$, where $M(n,C)$ is the set of complex matrices $n \times n$ and ${H_2}$ is the vector topological subspace of X of germs with vanishing jacobian matrix at the origin. We study the subset $\Omega$ of invertible germs of X. We show that $\Omega$ is open, connected and that ${\pi _1}(\Omega ) = {\mathbf {Z}}$. We define in $\Omega$ a topological and a Lie group structure. We determine its infinitesimal transformation, the differential equation of its law of composition and a fundamental bound of its right side. This work is a part of a larger research on infinite Lie groups, which started with a summary of results in [P$_{1}$]. In a subsequent paper we shall study the covering group of $\Omega$.