Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T04:01:20.214Z Has data issue: false hasContentIssue false
  • English
  • Français

An Open Mapping Theorem For Pro-Lie Groups

Part of: Topological linear spaces and related structures Lie groups Topological and differentiable algebraic systems

Published online by Cambridge University Press:  09 April 2009

Karl H. Hofmann  and
Sidney A. Morris
Karl H. Hofmann
Affiliation:
Fachbereich MathematikDarmstadt University of TechnologySchlossgartenstr. 7 D-64289 [email protected]
Sidney A. Morris
Affiliation:
School of Information Technology and Mathematical Sciences University of BallaratP.O. Box 663 Ballarat Victoria [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A pro-Lie group is a projective limit of finite dimensional Lie groups. It is proved that a surjective continuous group homomorphism between connected pro-Lie groups is open. In fact this remains true for almost connected pro-Lie groups where a topological group is called almost connected if the factor group modulo the identity component is compact. As consequences we get a Closed Graph Theorem and the validity of the Second Isomorphism Theorem for pro-Lie groups in the almost connected context.

MSC classification

Secondary: 22A05: Structure of general topological groups 22E65: Infinite-dimensional Lie groups and their Lie algebras 46A30: Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 1 , August 2007 , pp. 55 - 78
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Bourbaki, N., Groupes et algebres de Lie, chapters 2–3 (Hermann, Paris, 1972).Google Scholar
[2], Bourbaki, N., Topologie generate, chapters 5–10 (Hermann, Paris, 1974).Google Scholar
[3]Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis. Vol. I Structure of topological groups. Integration theory, group representations (Academic Press, Inc., Publishers, New York, 1963).Google Scholar
[4]Hofmann, K. H., ‘On a category of topological groups suitable for a structure theory of locally compact groups’, Topology Proceedings 26 (2001–2002), 651–665.Google Scholar
[5]Hofmann, K. H. and Morris, S. A., The Structure of Compact Groups (De Gruyter Berlin, 1998).Google Scholar
[6], Hofmann, K. H. and Morris, S. A., ‘Projective limits of finite dimensional Lie groups’, Proc. London Math. Soc. 87 (2003), 647–676.CrossRefGoogle Scholar
[7], Hofmann, K. H. and Morris, S. A., “The structure of abelian pro-Lie groups’, Math. Z. 248 (2004), 867–891.Google Scholar
[8], Hofmann, K. H. and Morris, S. A., ‘SophusLie's third fundamental theorem and the adjoint functor theorem’, J. Group Theory 8 (2005), 115–133.CrossRefGoogle Scholar
[9], Hofmann, K. H. and Morris, S. A., The Lie Theory of Connected Pro-Lie Groups- the Structure of Pro-Lie Algebra, Pro-Lie Groups and Locally Compact Groups (EMS Publishing House, Zurich, 2007).Google Scholar
[10]Hofmann, K. H., Morris, S. A. and Poguntke, D., ‘The exponential function of locally connected compact abelian groups’, Forum Math. 16 (2003), 1–16.Google Scholar
[11]Jones, F. Burton, ‘Connected and disconnected plane sets and the functional equation f(x + y) = f(x) + f(y)’, Bull. Amer. Math. Soc. 48 (1942), 115–120.Google Scholar
[12]Koshi, Sh. and Takesaki, M., ‘An open mapping theorem on homogeneous spaces’, J. Aust. Math. Soc., Sen A. 53 (1992), 51–54.CrossRefGoogle Scholar
[13]Montgomery, D. and Zippin, L., Topological Transformation Groups (Interscience Publishers, New York, 1955).Google Scholar
[14]Roelcke, W. and Dierolf, S., Uniform Structures on Topological Groups and their Quotients (McGraw-Hill, New York, 1981).Google Scholar
[15]Yamabe, H., ‘Generalization of a theorem of Gleason’, Ann. of Math. 58 (1953), 351–365.Google Scholar
[16], Yamabe, H., ‘On the conjecture of Iwasawa and Gleason’, Ann. of Math. 58 (1953), 48–54.Google Scholar