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Proceedings of the American Mathematical Society

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An example of an infinite Lie group

Author: Domingos Pisanelli
Journal: Proc. Amer. Math. Soc. 62 (1977), 156-160
MSC: Primary 58H05; Secondary 32M05, 22E65
MathSciNet review: 0436234
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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 $.

References [Enhancements On Off] (What's this?)

  • [P$ _{1}$] Domingos Pisanelli, Théorèmes d’Ovcyannicov, Frobenius, d’inversion et groupes de Lie locaux dans une échelle d’espaces de Banach, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A943–A946 (French). MR 343323
  • [P$ _{2}$] Domingos Pizanelli, Applications analytiques en dimension infinie, Bull. Sci. Math. (2) 96 (1972), 181–191 (French). MR 331061
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Keywords: Locally convex space, germ of holomorphic mapping, inductive limit topology, Silva space, direct topological sum, topological group, Lie group, local Lie group, LF-analyticity, Lie algebra
Article copyright: © Copyright 1977 American Mathematical Society