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Let's try this: assume that the complex vector bundle you start with has even fiber dimension such that the corresponding Clifford algebra bundle is a bundle of complex matrix algebras. Suppose furthe …

In addition to the answer of Bertram Arnold, let me point out that there is a very explicit formula for "fermionic" Weyl-Moyal product. Let us assume that your vector space $V$ (or module) is defined …

Not a complete answer but some observations. First it might be necessary not to take formal power series but formal Laurent series in order to get a reasonable behaviour of the Hochschild cohomology. …

a lot of questions, let me try on some of them :) The bad news is that in most of the interesting situations the higher order terms of the star product, the $B_i$ will not vanish. Heuristically this …

Deformation theory is of course a very very wide field and one can take many different points of view on it. Working in deformation quantization, i.e. formal associative deformations of algebras and t …

Unfortunately, there is no real textbook on DQ around. One has Fedosov's book on his construction of star products including a detailed exposition of his index theorem. There is a chapter on DQ in t …

Yes, it's just putting signs correctly. Martin Bordemann has a preprint from the 90s where he adapted Fedosov's construction in the graded setting. If you are only interested in the flat situation thi …

to 1) A $\mathbb{k}[[\hbar]]$-linear map between $A[[\hbar]]$ and $B[[\hbar]]$ is necessarily of the form $T = T_0 + \hbar T_1 + \cdots$ with $T_r\colon A \longrightarrow B$ being $\mathbb{k}$-linear …

OK, let me give a try on this question. There are several problems hidden underneath which one has to address. First, for physical reasons a formal deformation is not sufficient. $\hbar$ is a constan …

Concerning your first question I have a couple of suggestions: first coisotropic is in some sense the best we can have in a truely Poisson situation: there is nothing like lagrangian (unfortunately). …

Well, this is probably not the deep insight you are looking for, but if you consider the polyvector fields on a manifold, you can first define the Lie derivative $L_X$ of a polyvector field $X$ on dif …

Background: in deformation quantization one wants to construct formal associative deformations (the star products $\star$) of the algebra of smooth complex-valued functions on a Poisson manifold $M$ i …

In this generality, I would say that this is not possible: the same cohomology can be responsible for controlling quite different deformations problems. Just an example: in formal deformation quantiz …