On the local structure of noncommutative deformations

Abstract

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.