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Global constructions of quantization deformation and obstructions are discussed for an arbitrary complex analytic space in terms of adapted (analytic) Hochschild cohomology. For K3-surfaces an explicit global construction of a Poisson…
The Donald-Flanigan conjecture asserts that any group algebra of a finite group has a separable deformation. We apply an inductive method to deform group algebras from deformations of normal subgroup algebras, establishing an infinite…
This paper does not contain any new results, it is just an attempt to present, in a systematic way, one construction which establishes an interesting relationship between some ideas and notions well-known in the theory of integrable systems…
We present an elementary approach in characterizing Lie polynomials in the generators $A,B$ of an algebra with a defining relation that is in the form of a deformed or twisted commutation relation $AB=\sigma(BA)$ where the deformation or…
Starting with a Lie algebroid ${\cal A}$ over a space $M$ we lift its action to the canonical transformations on the affine bundle ${\cal R}$ over the cotangent bundle $T^*M$. Such lifts are classified by the first cohomology $H^1({\cal…
There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for any two functions f and g. We notice that this non-commutative differential algebra has a geometrical…
We introduce a formalism for derived moduli functors on differential graded associative algebras, which leads to non-commutative enhancements of derived moduli stacks and naturally gives rise to structures such as Hall algebras. Descent…
Let $A = \Bbbk Q / I$ be the path algebra of any finite quiver $Q$ modulo any two-sided ideal $I$ of relations and let $R$ be any reduction system satisfying the diamond condition for $I$. We introduce an intrinsic notion of deformation of…
A new class of Poisson algebras, the class of {\em generalized Weyl Poisson algebras}, is introduced. It can be seen as Poisson algebra analogue of generalized Weyl algebras or as giving a Poisson structure to (certain) generalized Weyl…
Recantly, William Crawley-Boevey proposed the definition of a Poisson structure on a noncommutative algebra $A$ based on the Kontsevich principle. His idea was to find the {\it weakest} possible structure on $A$ that induces standard…
We develop deformation theory of algebras over quadratic operads where the parameter space is a commutative local algebra. We also give a construction of a distinguised deformation of an algebra over a quadratic operad with a complete local…
We briefly summarize our systematic construction procedure of q-deforming maps for Lie group covariant Weyl or Clifford algebras.
A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/fuzzy set, the membership function, reflecting the degree of truth that an element belongs…
We present a comprehensive study of two new Poisson-type algebras. Namely, we are working with $\delta$-Poisson and transposed $\delta$-Poisson algebras. Our research shows that these algebras are related to many interesting identities. In…
Continuous formal deformations of the Poisson superbracket defined on compactly supported smooth functions on R^2 taking values in a Grassmann algebra with N generating elements are described up to an equivalence transformation for N \ne 2.
This work explores the deformation theory of algebraic structures in a very general setting. These structures include commutative, associative algebras, Lie algebras, and the infinity versions of these structures, the strongly homotopy…
We review several procedures of quantization formulated in the framework of (classical) phase space M. These quantization methods consider Quantum Mechanics as a "deformation" of Classical Mechanics by means of the "transformation" of the…
We give a construction of a Poisson transform mapping density valued differential forms on generalized flag manifolds to differential forms on the corresponding Riemannian symmetric spaces, which can be described entirely in terms of finite…
On every split supermanifold equipped with the Rothstein even super-Poisson bracket we construct a deformation quantization by means of a Fedosov-type procedure. In other words, the supercommutative algebra of all smooth sections of the…
Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R = K[[h]]). We consider associative (resp. Poisson) R-deformations of the structure sheaf O_X. The set of…