用Kontsevich图演算求Poisson双向量的无穷小变形
Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus
摘要：
设$P$是有限维仿射实流形上的Poisson结构。$P$是否可以变形以保持泊松分布？语言的Kontsevich图提供了一个通用的方法-关于所有仿射泊松流形-找到一类解决方案，这个变形问题。为此，需要几种类型的图。在本文中，我们概述的算法来生成这些图。编码变形的图形按内部顶点$k$的数量进行分类;对于$k \leqslant 4$，我们提出了变形问题的所有解决方案。对于$k \geqslant 5$，首先再现五角轮图片建议在$k=6$的Kontsevich和Willwacher，我们构建的七边轮上循环，产生一个新的独特的解决方案，没有$2$ -循环和蝌蚪在$k=8$。
英文摘要：
Let $P$ be a Poisson structure on a finite-dimensional affine real manifold. Can $P$ be deformed in such a way that it stays Poisson? The language of Kontsevich graphs provides a universal approach -- with respect to all affine Poisson manifolds -- to finding a class of solutions to this deformation problem. For that reasoning, several types of graphs are needed. In this paper we outline the algorithms to generate those graphs. The graphs that encode deformations are classified by the number of internal vertices $k$; for $k \leqslant 4$ we present all solutions of the deformation problem. For $k \geqslant 5$, first reproducing the pentagon-wheel picture suggested at $k=6$ by Kontsevich and Willwacher, we construct the heptagon-wheel cocycle that yields a new unique solution without $2$-loops and tadpoles at $k=8$.
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