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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Transitivity of families of invariant vector fields on the semidirect products of Lie groups


Authors: B. Bonnard, V. Jurdjevic, I. Kupka and G. Sallet
Journal: Trans. Amer. Math. Soc. 271 (1982), 525-535
MSC: Primary 49E15; Secondary 22E15, 58F40
MathSciNet review: 654849
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Abstract: In this paper we give necessary and sufficient conditions for a family of right (or left) invariant vector fields on a Lie group $ G$ to be transitive. The concept of transitivity is essentially that of controllability in the literature on control systems. We consider families of right (resp. left) invariant vector fields on a Lie group $ G$ which is a semidirect product of a compact group $ K$ and a vector space $ V$ on which $ K$ acts linearly. If $ \mathcal{F}$ is a family of right-invariant vector fields, then the values of the elements of $ \mathcal{F}$ at the identity define a subset $ \Gamma $ of $ L(G)$ the Lie algebra of $ G$. We say that $ \mathcal{F}$ is transitive on $ G$ if the semigroup generated by $ { \cup _{X \in \Gamma }}\{ \exp (tX):t \geqslant 0\} $ is equal to $ G$. Our main result is that $ \mathcal{F}$ is transitive if and only if $ \operatorname{Lie} (\Gamma )$, the Lie algebra generated by $ \Gamma $, is equal to $ L(G)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0654849-4
PII: S 0002-9947(1982)0654849-4
Article copyright: © Copyright 1982 American Mathematical Society