李叠加定理的局部和整体性质
Local and Global Aspects of Lie's Superposition Theorem
摘要：
本文给出了一个常微分方程在经典意义下存在解的叠加律的整体条件。这就完成了著名的李叠加定理。我们引入严格的预传递李群作用和Lie-Vessiot系统的概念。证明了预传递的李群作用是传递的。证明了一个常微分方程存在叠加律当且仅当它是预传递的Lie-Vessiot系统。这意味着它的包络代数被预传递李群作用的基本域所张成。讨论了叠加律与微分伽罗瓦论的关系，并回顾了Lie的经典结果。
英文摘要：
In this paper we give the global conditions for an ordinary differential equation to admit a superposition law of solutions in the classical sense. This completes the well-known Lie superposition theorem. We introduce rigorous notions of pretransitive Lie group action and Lie-Vessiot systems. We proof that pretransitive Lie group actions are transitive. We proof that an ordinary differential equation admit a superposition law if and only if it is a pretransitive Lie-Vessiot system. It means that its enveloping algebra is spanned by fundamental fields of a pretransitive Lie group action. We discuss the relationship of superposition laws with differential Galois theory and review the classical result of Lie.
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