关于非对易形变的局部结构
On the local structure of noncommutative deformations
摘要：
设$(M,π,\mathcal{D})$是一个Poisson流形，它具有平坦的无挠逆变联络.证明了若$\mathcal{D}$是$\mathcal{F}$ -联络，则存在张量$\mathbf{T}$使得$\mathcal{D}\mathbf{T}$是E.霍金斯在他关于非对易变形的著作中。在这种情况下，我们计算$\mathbf{T}$和亚曲率张量，并表明，如果$\mathbf{T}=0$，那么，在任何正则点附近，$π$和$\mathcal{D}$的定义在一个自然的方式由一个李代数行动和经典的杨-巴克斯特方程的解决方案。此外，当$\mathcal{D}$是与$π$和黎曼度量相关联的逆变Levi-Civita联络时，李代数作用保持度量。
英文摘要：
Let $(M,π,\mathcal{D})$ be a Poisson manifold endowed with a flat, torsion-free contravariant connection. We show that if $\mathcal{D}$ is an $\mathcal{F}$-connection then there exists a tensor $\mathbf{T}$ such that $\mathcal{D}\mathbf{T}$ is the metacurvature tensor introduced by E. Hawkins in his work on noncommutative deformations. We compute $\mathbf{T}$ and the metacurvature tensor in this case, and show that if $\mathbf{T}=0$ then, near any regular point, $π$ and $\mathcal{D}$ are defined in a natural way by a Lie algebra action and a solution of the classical Yang-Baxter equation. Moreover, when $\mathcal{D}$ is the contravariant Levi-Civita connection associated to $π$ and a Riemannian metric, the Lie algebra action preserves the metric.
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