可积Pfronian系统的群不变性
Group invariance of integrable Pfaffian systems
摘要：
设$\mathcal{S}$是一个可积的Pfronian系统.如果它在有限维实李代数$g$的横截自由无穷小作用下不变，从而在李群$G$的局部作用下不变，我们证明了$\mathcal{S}$的垂直变分上同调等于$g$的李代数上同调，其值在最大维的水平上同调空间中.这一结果，除了给一个有效的算法计算的变分上同调不变的Pfweian系统，提供了一种方法来检测障碍的存在，有限或无穷小的行动离开一个给定的系统不变。
英文摘要：
Let $\mathcal{S}$ be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra $g$ and consequently invariant under the local action of a Lie group $G$, we show that the vertical variational cohomology of $\mathcal{S}$ is equal to the Lie algebra cohomology of $g$ with values in the space of the horizontal cohomology in maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of finite or infinitesimal actions leaving a given system invariant.
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