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Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the $\hslash \to 0$ limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. Moreover, we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the $\hslash \to 0$ case) on the fiber C*-algebras.

Hilbert bimodules are morphisms between C*-algebraic models of quantum systems, while symplectic dual pairs are morphisms between Poisson geometric models of classical systems. Both of these morphisms preserve representation-theoretic structures of the relevant types of models. Previously, it has been shown that one can functorially associate certain symplectic dual pairs to Hilbert bimodules through strict deformation quantization. We show that, in the inverse direction, strict deformation quantization also allows one to functorially take the classical limit of a Hilbert bimodule to reconstruct a symplectic dual pair.

Using methods of computer algebra, especially, Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet–Loday formula deforming the Poisson operad into the associative operad.

In this paper we investigate the Hochschild cohomology groups H²(A) and H³(A) for an arbitrary polynomial algebra A. We also show that the corresponding cohomology groups which are built from differential operators inject in H²(A) and H³(A) and we give an application to deformation theory.

Let $k$ be an arbitrary field of characteristic $0$. We prove that the group of automorphisms of a free Poisson field $P(x,y)$ in two variables $x,y$ over $k$ is isomorphic to the Cremona group ${\mathrm{\text{Cr}}}_2(k)$. We also prove that the universal enveloping algebra $P{(x_1,\dots ,x_n)}^e$ of a free Poisson field $P(x_1,\dots ,x_n)$ is a free ideal ring and give a characterization of the Poisson dependence of two elements of $P(x_1,\dots ,x_n)$ via universal derivatives.

We study the deformation of Courant pairs with a commutative algebra base. We consider the deformation cohomology bi-complex and describe a universal infinitesimal deformation. In a sequel, we formulate an extension of a given deformation of a Courant pair to another with extended base, which leads to describe the obstruction in extending a given deformation. We also discuss the construction of versal deformation of Courant pairs. As an application, we compute universal infinitesimal deformation of Poisson algebra structures on the three-dimensional complex Heisenberg Lie algebra. We compare the second deformation cohomology spaces of these Poisson algebra structures by considering them in the category of Leibniz pairs and Courant pairs, respectively.

Let K be a field of characteristic zero and $X_n=\{x_1,\dots ,x_n\}$ be a finite set of variables. Consider the free metabelian Poisson algebra $P_n$ of rank n generated by $X_n$ over K. An element in $P_n$ is called symmetric if it is preserved under any change of variables, i.e. under the action of each permutation in $S_n$. In this study, we determine the algebra $P_n^{S_n}$ of symmetric polynomials of $P_n$.

We construct a method to obtain the algebraic classification of Poisson algebras defined on a commutative associative algebra, and we apply it to obtain the classification of the $3$-dimensional Poisson algebras. In addition, we study the geometric classification, the graph of degenerations and the closures of the orbits of the variety of $3$-dimensional Poisson algebras. Finally, we also study the algebraic classification of the Poisson algebras defined on a commutative associative null-filiform or filiform algebra and, to enrich this classification, we study the degenerations between these particular Poisson algebras.