相关论文: A combinatorial approach to coefficients in deform…
We compute the homotopy derivations of the properads governing even and odd Lie bialgebras as well as involutive Lie bialgebras. The answer may be expressed in terms of the Kontsevich graph complexes. In particular, this shows that the…
In this article, we use Harrison cohomology to provide a framework for commutative deformations. In particular, Kontsevich's result that formality of (the Hochschild complex of) an associative algebra implies its deformability is adapted…
We study formal deformations of a crossed product $S(V)#_f G$, of a polynomial algebra with a group, induced from a universal deformation formula introduced by Witherspoon. These deformations arise from braided actions of Hopf algebras…
Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras…
In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes…
We develop methods for computing Hochschild cohomology groups and deformations of crossed product rings. We use these methods to find deformations of a ring associated to a particular orbifold with discrete torsion, and give a presentation…
For the Kirillov-Poisson structure on the vector space $\g^*$, where $\g$ is a finite-dimensional Lie algebra, it is known at least two canonical deformations quantization of this structure: they are the M. Kontsevich universal formula [K],…
In this paper, we introduce the cohomology theory of $\mathcal{O}$-operators on Hom-associative algebras. This cohomology can also be viewed as the Hochschild cohomology of a certain Hom-associative algebra with coefficients in a suitable…
This article gives an exposition of the deformation theory for pairs $(X, E)$, where $X$ is a compact complex manifold and $E$ is a holomorphic vector bundle over $X$, adapting an analytic viewpoint \`{a} la Kodaira-Spencer. By introducing…
The formality morphism $\boldsymbol{\mathcal{F}}=\{\mathcal{F}_n$, $n\geqslant1\}$ in Kontsevich's deformation quantization is a collection of maps from tensor powers of the differential graded Lie algebra (dgLa) of multivector fields to…
In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then…
We show that the zeroth cohomology of M. Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmueller Lie algebra grt_1. The map is explicitly described. This result has applications to deformation quantization and Duflo…
Let $X$ be a smooth complex algebraic variety and let $\operatorname{Coh} (X)$ denote its Abelian category of coherent sheaves. By the work of W. Lowen and M. Van den Bergh, it is known that the deformation theory of $\operatorname{Coh}…
Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebra $A$ together with a Lie algebra $L$ mapped into the derivations of $A$. A…
We extend the classical concept of deformation of an associative algebra, as introduced by Gerstenhaber, by using monoidal linear categories and cocommutative coalgebras as foundational tools. To achieve this goal, we associate to each…
Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain…
I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness --up to equivalence-- of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology…
The deformation quantization by Kontsevich [arXiv:q-alg/9709040] is a way to construct an associative noncommutative star-product $\star=\times+\hbar \{\ ,\ \}_{P}+\bar{o}(\hbar)$ in the algebra of formal power series in $\hbar$ on a given…
The main purpose of this paper is to study restricted formal deformations of restricted Lie-Rinehart algebras in positive characteristic $p$. For $p>2$, we discuss the deformation theory and show that deformations are controlled by the…
Let $k$ be a perfect field of positive characteristic and $Z$ an effective Cartier divisor in the projective line over $k$ with complement $U$. In this note, we establish some results about the formal deformation theory of overconvergent…