The standard Berezin and Berezin-Toeplitz quantizations on a Kähler manifold are based on operator symbols and on Toeplitz operators, respectively, on weighted L2-spaces of holomorphic functions (weighted Bergman spaces). In both cases, the construction basically uses only the fact that these spaces have a reproducing kernel. We explore the possibilities of using other function spaces with reproducing kernels instead, such as L2-spaces of harmonic functions, Sobolev spaces, Sobolev spaces of holomorphic...

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...

El trabajo que presentamos constituye una revisión de varios procedimientos de cuantización basados en un espacio de fases clásico M. Estos métodos consideran a la mecánica cuántica como una "deformación" de la mecánica clásica por medio de la "transformación" del álgebra conmutativa C∞(M) en una nueva álgebra no conmutativa C∞(M)ħ. Todas estas ideas conducen de modo natural a los grupos cuánticos como deformación (o cuantización en un sentido amplio) de los grupos de Poisson-Lie, lo cual también...

It is well known that one can often construct a star-product by expanding the product of two Toeplitz operators asymptotically into a series of other Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin-Toeplitz quantization. We show that one can obtain in a similar way in fact any star-product which is equivalent to the Berezin-Toeplitz star-product, by using instead of Toeplitz operators other suitable mappings from compactly supported smooth functions...

First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section  we generalize the Maurer-Cartan equation to strongly homotopy Lie algebras and prove the homotopy invariance of the moduli space of solutions of this equation. In the last...

We show that a graded commutative algebra A with any square zero odd differential operator is a natural generalization of a Batalin-Vilkovisky algebra. While such an operator of order 2 defines a Gerstenhaber (Lie) algebra structure on A, an operator of an order higher than 2 (Koszul-Akman definition) leads to the structure of a strongly homotopy Lie algebra ($L_{\infty }$-algebra) on A. This allows us to give a definition of a Batalin-Vilkovisky algebra up to homotopy. We also make a conjecture which is a...

Let $X$ be a (generalized) flag manifold of a complex semisimple Lie group $G$. We investigate the problem of constructing a graded star product on $ℛ=R\left(T^{☆}X\right)$ which corresponds to a $G$-equivariant quantization of symbols into twisted differential operators acting on half-forms on $X$. We construct, when $ℛ$ is generated by the momentum functions ${\mu }^x$ for $G$, a preferred choice of $☆$ where ${\mu }^x☆\phi$ has the form ${\mu }^x\phi +\frac{1}{2}\{{\mu }^x,\phi \}t+{\Lambda }^x\left(\phi \right)t^2$. Here ${\Lambda }^x$ are operators on $ℛ$. In the known examples, ${\Lambda }^x$ ($x\ne 0$) is not a differential operator, and so the star product ${\mu }^x☆\phi$...