Related papers: On the perturbation algebra
Given a Hopf algebra $A$ graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of $A$. If the action is adjoint, this new Hopf algebra is…
We give a general treatment of deformation theory from the point of view of homotopical algebra following Hinich, Manetti and Pridham. In particular, we show that any deformation functor in characteristic zero is controlled by a certain…
We define the cohomology and formal deformation theories for algebra and bialgebra categories. We suggest some approaches to finding nontrivial deformations of the categories associated to the quantum groups by the work of Lusztig.
Let G denote a group and let W be an algebra over a commutative ring R. We will say that W is a G-graded twisted algebra (not necessarily commutative, neither associative) if there exists a G-grading W=\bigoplus_{g \in G}W_{g} where each…
We introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras based on the possibility to mix the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also…
We study modules over stacks of deformation quantization algebroids on complex Poisson manifolds. We prove finiteness and duality theorems in the relative case and construct the Hochschild class of coherent modules. We prove that this class…
The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category…
In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a…
The purpose of this paper is to study global deformations of Hom-Leibniz algebras. We introduce a cohomology for Hom-Leibniz algebras with values in a Hom-module, characterize versal deformations and provide examples.
We continue our work on variations of graded-polarized mixed Hodge structures by defining analogs of the harmonic metric equations for filtered bundles and proving a precise analog of Schmid's Nilpotent Orbit Theorem for 1-parameter…
This paper lays some of the foundations for working with not-necessarily-commutative bialgebras and their categories of comodules in $\infty$-categories. We prove that the categories of comodules and modules over a bialgebra always admit…
We describe a differential graded Lie algebra controlling infinitesimal deformations of triples $(X,\mathcal{F},\sigma)$, where $\mathcal{F}$ is a coherent sheaf on a smooth variety $X$ over a field of characteristic 0 and $\sigma\in…
In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one by considering the structure map. We develop…
The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the…
We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine an easy-to-use straightforward formula for the…
We construct a cohomology theory controlling the deformations of a general Drinfel'd algebra. The picture presented here has two sides -- the combinatorial one related with the fact of the existence of a graded Lie algebra structure on the…
The theory of modular deformations is generalized for the category of complex analytic polyhedra which includes germs of complex space as well as any compact complex analytic space. The objective of the theory is a construction of fine…
We develop an obstruction theory for Hirsch extensions of cbba's with twisted coefficients. This leads to a variety of applications, including a structural theorem for minimal cbba's, a construction of relative minimal models with twisted…
In the first part of the paper we define a perturbative (pre-formal) geometry and formulate a theorem on the relation between the construction of a perturbative neighborhood of affine varieties and the higher tangent bundles. In the second…
P. Aluffi introduced in [1] a new graded algebra in order to conveniently express characteristic cycles in the theory of singular varieties. This algebra is attached to a surjective ring homomorphism $A\surjects B$ by taking a suitable…