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On Lie Algebras of Vector Fields

1977; American Mathematical Society; Volume: 226; Linguagem: Inglês

10.2307/1997943

1088-6850

Akira Koriyama, Yoshiaki Maeda, Hideki Omori,

Advanced Topics in Algebra

This paper has two purposes.The first is a generalization of the theorem of Pursell-Shanks [10].Our generalization goes by assuming the existence of a nontrivial core of a Lie algebra.However, it seems to be a necessary condition for the theorems of Pursell-Shanks type.The second is the classification of cores under the assumption that the core itself is infinitesimally transitive at every point.As naturally expected, we have the nonelliptic, primitive infinite-dimensional Lie algebras.0. Introduction.Let M and TV be connected C00 manifolds and ï(M) (resp.3c(N)) the Lie algebra of all C°° vector fields with compact support on M (resp.N).A well-known theorem of Pursell-Shanks [10] may be stated as follows:Theorem.There exists a Lie algebra isomorphism $ of£(M) onto X(N) if and only if there exists a C00 diffeomorphism <p of M onto N such that $ = dtp.The above result still holds for Lie algebras of all infinitesimal automorphisms of several geometric structures on M and N. Indeed, Omori [9, §X] proved the corresponding result in case of volume structures, symplectic structures, contact structures and fibering structures with compact fibers, and Koriyama [4] showed this is still true for submanifolds regarding a submanifold M' as a geometric structure on M. Recently, Amemiya [2] gave a generalization of the theorem of Pursell-Shanks in the case that the Lie algebra 8 is a module over C00 functions on M and there is no point on which every m E g vanishes.Our first purpose of this paper is to give a unified proof for the above results.Now, we consider a Lie algebra e (resp.'g) of C00 vector fields on a connected, complete C°° riemannian manifold M (resp.TV) with the following conditions:(C,l) Every u £ e is complete (integrable).(C,2)Ad(exp u)q ■» 8 f°r every u E g. (C,3) g is a locally closed LAS.(C,4) a has nontrivial core.