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You might want to have a look at §2.2 of An $L_\infty$ algebra structure on polyvector fields by Boris Shoikhet, where Boris constructs an exotic $L_\infty$-structure on poly-vector fields on a (possi …

One possible answer is in Toën-Vezzozi paper From HAG to DAG, who were themselves inspired by Ciocan-Fontanine and Kapranov (Derived Quot schemes and Derived Hilbert schemes). This approach works well …

The answer to your second question is "no", I think. Let's assume that sufficiently nice means that it is a smooth algebraic variety over a field of characteristic $0$. Then, as written in Severin Bar …

Fedosov's work seems to be also available with details in a book Deformation quantization and index theory. Are the two references overlapping ? Yes, indeed. The book contains strictly more than …

In addition to the references pointed out by Stefan, I would like to add Déformation, quantification, et théorie de Lie, by Catteno, Keller, and Torossian (Part I and Part III are actually in Engl …

If, by deformation, you mean formal one-parameter deformation (like, say, in deformation quantization), then it is already known that the Koszul duality between the symmetric and the exterior algebra …

This is probably not really an answer to this question, but there are two different context I know where deformation quantization produces something not exactly associative, but associative in a large …

Welcome to mathoverflow! There is actually a whole paper (in French) about choices of signs for Kontsevich formality: https://arxiv.org/pdf/math/0003003.pdf For instance, they define the Hochschild …

The construction is functorial with respect to algebraic morphisms of Lie algebroids (as opposed to geometric ones): see for instance my paper with Van de Bergh https://arxiv.org/pdf/0708.2725.pdf (it …

I think you should have a look at the various papers of Louis Boutet de Monvel. But there is actually a construction of star-products on a symplectic manifold which makes use of the index theorem, due …

My question is: Does the vanishing of the Poisson bracket plays an important role in finding and proving Duflo's isomorphism theorem? Or it is just an literally first step? Let $A_0$ be a …

This is explained in Section 4.5.2 of "deformation quantization of poisson manifolds" by Kontsevich (http://arxiv.org/abs/q-alg/9709040). The way you wrote the homotopy between two Maurer-Cartan ele …

A $G_\infty$-morphism $\phi$ is determined by structure maps $\phi^{k_1,\dots,k_n}$, $n\geq1$, $k_1,\dots,k_n\geq1$. The $L_\infty$-part of $\phi$ is the $L_\infty$-morphism $\ell$ with structure ma …

Here is a sketch of topological description of a Tamarkin-Tsygan precalculus. Consider the compactified configuration spaces $C_n$ and $D_{1,n}$ of $n$ points on $\mathbb{R}^2$ and $\mathbb{R}^2-\{( …

Deformation of relations Answer to question 2 is the following: a deformation of an algebra $A_0$ parametrized by a pointed affine scheme $*\to X=Spec(B\to k)$ is the data of a $B$-algebra $A$ such t …