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Fundamentals of direct limit Lie theory

Published online by Cambridge University Press:  04 December 2007

Helge Glöckner
Helge Glöckner
Affiliation:
Department of Mathematics, AG 5, Darmstadt University of Technology, Schlossgartenstrasse 7, 64289 Darmstadt, Germanygloeckner@mathematik.tu-darmstadt.de
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Abstract

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We show that every countable direct system of finite-dimensional real or complex Lie groups has a direct limit in the category of Lie groups modelled on locally convex spaces. This enables us to push all basic constructions of finite-dimensional Lie theory to the case of direct limit groups. In particular, we obtain an analogue of Lie's third theorem: every countable-dimensional locally finite real or complex Lie algebra arises as the Lie algebra of some regular Lie group (a suitable direct limit group).

Keywords

direct limitinductive limitinfinite-dimensional Lie grouplocally finite Lie algebraenlargeabilityintegration of Lie algebrasregular Lie groupuniversal complexificationconvenient differential calculushomogeneous spaceextension of chartsprincipal bundle22E65 (primary)26E1526E2026E3046T05 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 141 , Issue 6 , November 2005 , pp. 1551 - 1577
Copyright
Foundation Compositio Mathematica 2005