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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An example of an infinite Lie group
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by Domingos Pisanelli PDF
Proc. Amer. Math. Soc. 62 (1977), 156-160 Request permission


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 $\Omega$.
  • 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
  • Domingos Pizanelli, Applications analytiques en dimension infinie, Bull. Sci. Math. (2) 96 (1972), 181–191 (French). MR 331061
  • J. Sebastião e Silva, Sui fondamenti della teoria dei funzionali analitici, Portugal. Math. 12 (1953), 1–47 (Italian). MR 52677
  • José Sebastião e Silva, Su certe classi di spazi localmente convessi importanti per le applicazioni, Rend. Mat. e Appl. (5) 14 (1955), 388–410 (Italian). MR 70046
  • C. Chevalley, Theory of Lie groups, Princeton Univ. Press, Princeton, N.J., 1946. MR 7, 412.
  • Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 62 (1977), 156-160
  • MSC: Primary 58H05; Secondary 32M05, 22E65
  • DOI:
  • MathSciNet review: 0436234