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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. …

Let me just try a few more comments on this and on the answer of @Qiaochu Yuan: One reasons for confusion might be the actual definition of the Picard group. If I understand you correct then you take …

One should definitely take a look at the work of Bordemann, Meinrenken, and Schlichenmaier: they provide a Berezin-Toeplitz inspired deformation quantization for all compact quantizable (i.e. the Kähl …

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 …

In deformation quantization there is a full classification available: let us first focus on the symplectic case which is easier. If $(M, \omega)$ is a symplectic manifold (like the $\mathbb{R}^2$ in y …

This will not be an honest answer but just a long remark: it reminds me a bit on the relation between Weyl quantization and standard ordered quantization. Here one has the polynomials in $q$ and $p$ w …

Hi Igor, there is a quite elementary way to see that star products restrict to open subsets: it's essentially part of the definition of a star product. Here, I will focus on the case of smooth (sympl …

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). …

Oh, well, there is maybe some way out (I learned from Martin Bordmann at some point) It does not avoid all compuptations but works slightly more transparent than just brute force... The main point is …

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 …

Working in "quantum mathematics" myself, I should tend to defend this teminology a bit ;) The term is clearly motivated by the usage in physics and, nowadays, is typically used in situations where you …

This is not at all an innocent question as there are really many notions of "quantizing stuff" around. A systematic comparison is probably not available (yet) for several reasons. Let me just try to i …