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Pro-Lie groups which are infinite-dimensional Lie groups

Published online by Cambridge University Press:  01 March 2009

K. H. HOFMANN  and
K.-H. NEEB
K. H. HOFMANN
Affiliation:
Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany. e-mail: hofmann@mathematik.tu-darmstadt.de, neeb@mathematik.tu-darmstadt.de
K.-H. NEEB
Affiliation:
Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany. e-mail: hofmann@mathematik.tu-darmstadt.de, neeb@mathematik.tu-darmstadt.de
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Abstract

A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 351 - 378
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[Bou89]Bourbaki, N.Lie Groups and Lie Algebras (Chapters 1–3) (Springer–Verlag, 1989).Google Scholar
[Dix57]Dixmier, J.L'application exponentielle dans les groupes de Lie résolubles. Bull. Soc. Math. Fr. 85 (1957), 113–121.CrossRefGoogle Scholar
[GI06]Glöckner, H.Implicit functions from topological vector spaces to Banach spaces. Israel J. Math. 155 (2006), 205–252CrossRefGoogle Scholar
[GN06]Glöckner, H. and Neeb, K.-H.Infinite-Dimensional Lie Groups, book in preparation.Google Scholar
[Hel78]Helgason, S.Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, 1978).Google Scholar
[HHL89]Hilgert, J., Hofmann, K. H. and Lawson, J. D.Lie Groups, Convex Cones and Semigroups (Clarendon Press, 1989).Google Scholar
[Ho65]Hochschild, G.The Structure of Lie Groups (Holden Day, 1965).Google Scholar
[HoMs661]Hofmann, K. H. and Mostert, P. S.Elements of Compact Semigroups (Charles E. Merrill Books, 1966).Google Scholar
[HoMo98]Hofmann, K. H. and Morris, S. A.The Structure of Compact Groups. Studies in Math. (de Gruyter, 1998), 2nd ed. (2006).CrossRefGoogle Scholar
[HoMo07]Hofmann, K. H. and Morris, S. A.The Lie Theory of Connected Pro-Lie Groups–a Structure Theory for Pro-Lie Algebras, Pro-Lie Groups and Connected Locally Compact Groups (EMS Publishing House, 2007).CrossRefGoogle Scholar
[Iwa49]Iwasawa, K.On some types of topological groups. Ann. of Math. 50 (1949), 507–558.CrossRefGoogle Scholar
[KM97]Kriegl, A. and Michor, P.The convenient setting of global analysis. Math. Surveys and Monographs 53 (Amer. Math. Soc., 1997).CrossRefGoogle Scholar
[Kur59]Kuranishi, M.On the local theory of continuous infinite pseudo groups I. Nagoya Math. J. 15 (1959), 225–260.CrossRefGoogle Scholar
[Lew39]Lewis, D.Formal power series transformations. Duke Math. J. 5 (1939), 794–805.CrossRefGoogle Scholar
[Mil84]Milnor, J. Remarks on infinite-dimensional Lie groups. pp. 1007–1057. In: DeWitt, B. and Stora, R. (eds), Relativité, Groupes et Topologie II (Les Houches, 1983), (North Holland, 1984).Google Scholar
[Ne99]Neeb, K.-H. Holomorphy and convexity in Lie theory. Expositions in Mathematics 28 (de Gruyter Verlag, 1999).Google Scholar
[Ne06]Neeb, K.-H.Towards a Lie theory of locally convex groups. Jap. J. Math. 1 (2006), 291–468.CrossRefGoogle Scholar
[Omo80]Omori, H.A method of classifying expansive singularities. J. Diff. Geom. 15 (1980), 493–512.Google Scholar
[Rob97]Robart, T.Sur l'intégrabilité des sous-algèbres de Lie en dimension infinie. Canad. J. Math. 49:4 (1997), 820–839.CrossRefGoogle Scholar
[Sai57]Saito, M.Sur certains groupes de Lie resolubles. Sci. Pap. Coll. Gen. Educ. Univ. Tokyo 7 (1957), 1–11.Google Scholar
[Sze74]Szente, J.On the topological characterization of transitive Lie group actions. Acta Sci. Math. (Szeged) 36 (1974), 323–344.Google Scholar
[St61]Sternberg, S.Infinite Lie groups and the formal aspects of dynamical systems. J. Math. Mech. 10 (1961), 451–474.Google Scholar