On the algebra of functions on a symplectic manifold we consider the pointwise product and the Poisson bracket; after a brief review of the classifications of the deformations of these structures, we give explicit formulas relating a star product to its classifying formal Poisson bivector.

A cluster ensemble is a pair $(𝒳,𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $\Gamma$. The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p:𝒜\to 𝒳$. The space $𝒜$ is equipped with a closed $2$-form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role...

On a flat manifold $M={ℝ}^d$, M. Kontsevich’s formality quasi-isomorphism is compatible with cup-products on tangent cohomology spaces, in the sense that for any formal Poisson $2$-tensor $\hslash \gamma$ the derivative at $\hslash \gamma$ of the quasi-isomorphism induces an isomorphism of graded commutative algebras from Poisson cohomology space to Hochschild cohomology space relative to the deformed multiplication built from $\hslash \gamma$ via the quasi-isomorphism. We give here a detailed proof of this result, with signs and orientations precised....

We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-riemannian manifold $(M,g)$. In other words, we establish a canonical isomorphism between the spaces of polynomials on $T^{*}M$ and of differential operators on tensor densities over $M$, both viewed as modules over the Lie algebra $\mathrm{o}(p+1,q+1)$ where $p+q=dim\left(M\right)$. This quantization exists for generic values of the weights of the tensor densities and we compute the critical values of the weights yielding...