Related papers: $L_\infty$-derivations and the argument shift meth…
The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L_\infty-algebra which induces a graded Lie bracket on…
By means of the Schouten calculus for contravariant antisymmetric tensor fields, we apply the Lie transform method to investigate smooth deformations of tensor fields and, in particular, to perturbations of Hamiltonian systems generated by…
We prove the conjecture that allows one extend the argument shifting procedure from symmetric algebra $S\mathfrak{gl}_d$ of the Lie algebra $\mathfrak{gl}_d$ to the universal enveloping algebra $U\mathfrak{gl}_d$. Namely, it turns out that…
We propose a simple approach to formal deformations of associative algebras. It exploits the machinery of multiplicative coresolutions of an associative algebra A in the category of A-bimodules. Specifically, we show that certain…
The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson-Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic…
Given a Leibniz algebra L with left center Z, we work on C(L,Z,S(Z)), the Z-standard complex of L with coefficients in S(Z). We construct the derived bracket for a fat Leibniz algebra in terms of a certain 3-cocycle and a Poisson algebra…
This paper is the sequel to [PTVV] (IHES Vol. 117, 2013). We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and polyvector fields. We then introduce the formalism of…
Any deformation of a Weyl or Clifford algebra can be realized through some change of generators in the undeformed algebra. Here we briefly describe and motivate our systematic procedure for constructing all such changes of generators for…
We give an $L_\infty$-interpretation of the classification, obtained in [AP2], of the formal deformations of a family of exact Poisson structures in dimension three. We indeed obtain again the explicit formulas for all the formal…
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables…
In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and describe some constructions of them. Then we study the formal distribution (noncommutative) Poisson algebras and coefficient (noncommutative)…
In this paper, we study a family of compatible quadratic Poisson brackets on gl(n), generalizing the Sklyanin one. For any of the brackets in the family, the argument shift determines the compatible linear bracket. The main interest for us…
Given a Lie algebra, there uniquely exists a Poisson algebra which is called a Lie-Poisson algebra over the Lie algebra. We will prove that given a Loday/Leibniz algebra there exists uniquely a noncommutative Poisson algebra over the Loday…
Our aim in this thesis is to use the language of deformation-quantization to understand certain quantized algebras by looking at properties of the corresponding commutative ones, and conversely to obtain results about the commutative…
In our earlier article [Lett. Math. Phys. 107 (2017), 475-503, arXiv:1409.8188], we explicitly described a topological Hopf algebroid playing the role of the noncommutative phase space of Lie algebra type. Ping Xu has shown that every…
Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie-Hamilton systems, to devise a novel formalism: the Poisson-Hopf algebra deformations of Lie-Hamilton systems. This approach applies to…
Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-like versions of usual deformations. We prove that…
In this paper we address the following question: is it always possible to choose a deformation quantization of a Poisson algebra A so that certain Poisson-commutative subalgebra C in it remains commutative? We define a series of…
In this paper we define Courant algebroids in a purely algebraic way and study their deformation theory by using two different but equivalent graded Poisson algebras of degree -2. First steps towards a quantization of Courant algebroids are…
We propose an algebraic viewpoint of the problem of deformation quantization of the so called almost Poisson algebras, which are algebras with a commutative associative product and an antisymmetric bracket which is a bi-derivation but does…