论文标题
$ \ mathfrak {sl} _2（\ mathbb {c}）$的泊松线性化问题。第一部分：Poisson的共同体学
The Poisson linearization problem for $\mathfrak{sl}_2(\mathbb{C})$. Part I: Poisson cohomology
论文作者
论文摘要
这是两篇论文中的第一篇，其中我们证明了CONN的线性化定理的LIE代数$ \ Mathfrak {SL} _2（\ Mathbb {C}）\ Simeq \ Mathfrak {so}（3,1）$。也就是说，我们表明的任何泊松结构的线性近似为零的结构与与$ \ mathfrak {sl} _2 _2（\ Mathbb {c}）$相关的泊松结构是同构的。在第一部分中，我们计算了与$ \ mathfrak {sl} _2（\ Mathbb {C}）$相关的Poisson共同体，并且我们为原始位于原点平坦的多站场的Poisson copplect构建了有限的同型操作员。在第二部分中，我们将获得线性化结果，该结果适用于更一般的Lie代数类别。为了证明证明，我们将开发一种NASH-MOSER方法，用于在某个时刻平坦的功能。
This is the first of two papers, in which we prove a version of Conn's linearization theorem for the Lie algebra $\mathfrak{sl}_2(\mathbb{C})\simeq \mathfrak{so}(3,1)$. Namely, we show that any Poisson structure whose linear approximation at a zero is isomorphic to the Poisson structure associated to $\mathfrak{sl}_2(\mathbb{C})$ is linearizable. In this first part, we calculate the Poisson cohomology associated to $\mathfrak{sl}_2(\mathbb{C})$, and we construct bounded homotopy operators for the Poisson complex of multivector fields that are flat at the origin. In the second part, we will obtain the linearization result, which works for a more general class of Lie algebras. For the proof, we will develop a Nash-Moser method for functions that are flat at a point.
