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In §5.3 of Kontsevich's Formality Conjecture he writes: This (...) gives a remarkable vector field on the space of bi-vector fields on $\mathbf{R}^d$. The evolution with respect to the time $t$ is …

I'm no expert on operads but it seems that a "bracket system" can be formalized as an operad; see e.g. the Poisson operad here, the "product-complete" condition could be related to having a Hopf ope …

With every finite-dimensional Lie algebra $\mathfrak{g}$ one can associate a linear Poisson bracket on $\mathfrak{g}^\ast$. With some more restrictions on $\mathfrak{g}$ and some extra ingredients, th …

Lie ideals in Lie algebras also define Poisson ideals of the associated Lie-Poisson structure. Consider the Lie-Poisson structure associated to the Lie algebra of upper-triangular $3\times 3$ matrice …

In Deformation quantization of Poisson manifolds, Kontsevich gives the quantization formula $$f \star g = \sum_{n=0}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma,\alpha}(f,g).$$ He gives …

The answer is that the black underlined terms do not cancel. Instead, they contribute an extra term which gives precisely the Jacobi identity (times $2/3$). (The reason I missed this is that I prev …