• Corpus ID: 59480893

Deformation theory of bialgebras, higher Hochschild cohomology and Formality

@article{Ginot2016DeformationTO,
  title={Deformation theory of bialgebras, higher Hochschild cohomology and Formality},
  author={Gr{\'e}gory Ginot and Sinan Yalin},
  journal={arXiv: Algebraic Topology},
  year={2016}
}
A first goal of this paper is to precisely relate the homotopy theories of bialgebras and E 2-algebras. For this, we construct a conservative and fully faithful ∞-functor from pointed conilpotent homotopy bialgebras to augmented E 2-algebras which consists in an appropriate " cobar " construction. Then we prove that the (derived) formal moduli problem of homotopy bial-gebras structures on a bialgebra is equivalent to the (derived) formal moduli problem of E 2-algebra structures on this " cobar… 

Derived deformation theory of algebraic structures

The main purpose of this article is to develop an explicit derived deformation theory of algebraic structures at a high level of generality, encompassing in a common framework various kinds of

Moduli Spaces of (Bi)algebra Structures in Topology and Geometry

After introducing some motivations for this survey, we describe a formalism to parametrize a wide class of algebraic structures occurring naturally in various problems of topology, geometry and

Hodge Filtration and Operations in Higher Hochschild (Co)homology and Applications to Higher String Topology

This paper is based on lectures given at the Vietnamese Institute for Advanced Studies in Mathematics and aims to present the theory of higher Hochschild (co)homology and its application to higher

Differentials forms on smooth operadic algebras

The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of

Higher Brackets on Cyclic and Negative Cyclic (Co)Homology

The purpose of this article is to embed the string topology bracket developed by Chas–Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket found by de Thanhoffer de

On deformation quantization of quadratic Poisson structures

. We study the deformation complex of the dg wheeled properad of Z -graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first

Braces and Poisson additivity

We relate the brace construction introduced by Calaque and Willwacher to an additivity functor. That is, we construct a functor from brace algebras associated to an operad ${\mathcal{O}}$ to

Automorphisms of the little disks operad with torsion coefficients

We compute the automorphisms of the Bousfield-Kan completion at a prime p of the little two-disks operads and show that they are given by the pro-p Grothendieck-Teichmuller group. We also show that

Cyclic Gerstenhaber–Schack cohomology

We show that the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad

An Explicit Two Step Quantization of Poisson Structures and Lie Bialgebras

We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra)

References

SHOWING 1-10 OF 79 REFERENCES

Quantization of quasi-Lie bialgebras

The main result of this paper is the construction of quantization functors of quasi-Lie bialgebras. We first recall the situation of this problem in the theory of quantum groups. This theory started,

Notes on A∞-Algebras, A∞-Categories and Non-Commutative Geometry

We develop a geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. The geometric approach clarifies several

Spaces of algebra structures and cohomology of operads

The aim of this paper is two-fold. First, we compare two notions of a "space" of algebra structures over an operad A: 1. the classification space, which is the nerve of the category of weak

A Formality Theorem for Poisson Manifolds

AbstractLet M be a differential manifold. Using different methods, Kontsevich and Tamarkin have proved a formality theorem, which states the existence of a Lie homomorphism 'up to homotopy' between

On the invertibility of quantization functors

Deformation theory of representations of prop(erad)s I

Abstract In this paper and its follow-up [Merkulov and Vallette, J. reine angew. Math.], we study the deformation theory of morphisms of properads and props thereby extending Quillen's deformation

Moduli stacks of algebraic structures and deformation theory

We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of

Function complexes in homotopical algebra

The cobar construction as a Hopf algebra

Abstract. We show that the integral cobar construction ΩC*X of a 1-reduced simplicial set X has the structure of a homotopy Hopf algebra. This leads to a rational differential Lie algebra ℒ(X) given

Iterated bar complexes of E -infinity algebras and homology theories

We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be
...