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Proceedings of the American Mathematical Society

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

 

 

An extension of a theorem of Nagano on transitive Lie algebras


Author: Héctor J. Sussmann
Journal: Proc. Amer. Math. Soc. 45 (1974), 349-356
MSC: Primary 58A15; Secondary 57D25
DOI: https://doi.org/10.1090/S0002-9939-1974-0356116-6
MathSciNet review: 0356116
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Abstract: Let $ M$ be a real analytic manifold, and let $ L$ be a transitive Lie algebra of real analytic vector fields on $ M$. A concept of completeness is introduced for such Lie algebras. Roughly speaking, $ L$ is said to be complete if the integral trajectories of vector fields in $ L$ are defined ``as far as $ L$ permits". Examples of situations where this assumption is satisfied: (i) $ L$ = a transitive Lie algebra all of whose elements are complete vector fields, and (ii) $ L$ = the set $ V(M)$ of all real analytic vector fields on $ M$. Our main result is: if $ M,M'$ are connected manifolds, then every Lie algebra isomorphism $ F:L \to L'$ between complete transitive Lie algebras of real analytic vector fields on $ M,M'$ which carries the isotropy subalgebra $ {L_m}$ of a point $ m$ of $ M$ to the isotropy subalgebra $ {L_{m'}}$ of $ m'\in M'$ is induced by a (unique) real analytic diffeomorphism $ f:M \to M'$ such that $ f(m) = m'$, provided that one of the following two conditions is satisfied: (l) $ M$ and $ M'$ are simply connected, or (2) the Lie algebras $ L$ and $ L'$ separate points. Nagano had proved this result for the case $ L = V(M),L' = V(M'),M$ and $ M'$ compact.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0356116-6
Article copyright: © Copyright 1974 American Mathematical Society