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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
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.

References [Enhancements On Off] (What's this?)

  • [1] T. Nagano, Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan 18 (1966), 398-404. MR 33 #8005. MR 0199865 (33:8005)
  • [2] K. Shiga, Some aspects of real analytic manifolds and differentiable manifolds, J. Math. Soc. Japan 16 (1964), 128-142; ibid. 17 (1965), 216-217. MR 30 #2517; 31 #6243. MR 0172298 (30:2517)

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