The Gerstenhaber-Schack complex for prestacks and non-cummutative deformation theory

Selecta Mathematica, 2016

We identify a class of quasi-compact semi-separated (qcss) twisted presheaves of algebras A for which well-behaved Grothendieck abelian categories of quasi-coherent modules Qch(A) are defined. This class is stable under algebraic deformation, giving rise to a 1-1 correspondence between algebraic deformations of A and abelian deformations of Qch(A). For a qcss presheaf A, we use the Gerstenhaber-Schack (GS) complex to explicitly parameterize the first order deformations. For a twisted presheaf A with central twists, we descibe an alternative category QPr(A) of quasi-coherent presheaves which is equivalent to Qch(A), leading to an alternative, equivalent association of abelian deformations to GS cocycles of qcss presheaves of commutative algebras. Our construction applies to the restriction O of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have Qch(O) ∼ = Qch(X)). Under a natural identification of Gerstenhaber-Schack cohomology of O and Hochschild cohomology of X, our construction is shown to be equivalent to Toda's construction from [48] in the smooth case.

2008

Deformation theory requires solving Maurer-Cartan equation (MCE) associated to an DGLA (L-infinity algebra). The universal solution of [HS] is obtained iteratively, as a fixed point of a contraction, analogous to the Picard method. The role of the Kuranishi functor in this construction is emphasized. The parallel with Lie theory suggests that deformation theory is a higher “dimensional” version. The deformation determined by the solution of the Maurer-Cartan equation associated to a contraction, splits the epimorphism, leading to a “doubling and gluing” interpretation. The *-operator associated to a contraction is introduced, and the connection with Hodge structures and generalized complex structures ( dd∗ -lemma) is established. The relations with bialgebra deformation quantization on one hand and ConnesKreimer renormalization on the other, are suggested.

arXiv (Cornell University), 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 bialgebras structures on a bialgebra is equivalent to the (derived) formal moduli problem of E 2-algebra structures on this "cobar" construction. We show consequently that the E 3-algebra structure on the higher Hochschild complex of this cobar construction, given by the solution to the higher Deligne conjecture, controls the deformation theory of this bialgebra. This implies the existence of an E 3-structure on the deformation complex of a dg bialgebra, solving a long-standing conjecture of Gerstenhaber-Schack. On this basis we solve a long-standing conjecture of Kontsevich, by proving the E 3-formality of the deformation complex of the symmetric bialgebra. This provides as a corollary a new proof of Etingof-Kazdhan deformation quantization of Lie bialgebras which extends to homotopy dg Lie bialgebras and is independent from the choice of an associator. Along the way, we establish new general results of independent interest about the deformation theory of algebraic structures, which shed a new light on various deformation complexes and cohomology theories studied in the literature. Contents 54 7. The E 3-formality Theorem 67 8. Etingof-Kazdhan deformation quantization 75 References 77

arXiv (Cornell University), 2019

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 algebras (associative, commutative, Poisson...) or bialgebras (associative and coassociative, Lie, Frobenius...), that is algebraic structures parametrized by props. A central aspect is that we define and study moduli spaces of deformations of algebraic structures up to quasi-isomorphisms (and not only up to isomorphims or ∞-isotopies). To do so, we implement methods coming from derived algebraic geometry, by encapsulating these deformation theories as classifying (pre)stacks with good infinitesimal properties and derived formal groups. In particular, we prove that the Lie algebra describing the deformation theory of an object in a given ∞-category of dg algebras can be obtained equivalently as the tangent complex of loops on a derived quotient of this moduli space by the homotopy automorphims of this object. Moreover, we provide explicit formulae for such derived deformation problems of algebraic structures up to quasi-isomorphisms and relate them in a precise way to other standard deformation problems of algebraic structures. This relation is given by a fiber sequence of the associated dg-Lie algebras of their deformation complexes. Our results provide simultaneously a vast generalization of standard deformation theory of algebraic structures which is suitable (and needed) to set up algebraic deformation theory both at the ∞categorical level and at a higher level of generality than algebras over operads. In addition, we study a general criterion to compare formal moduli problems of different algebraic structures and apply our formalism to En-algebras and bialgebras.

Journal of Generalized Lie Theory and Applications, 2011

The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of left alternative algebras. Connections to some other algebraic structures are given also.

Communications in Mathematical Physics, 2016

1999

We construct a differential and a Lie bracket on the space {Hom(A ⊗k , A ⊗l)}, k,l≥0 for any associative algebra A. The restriction of this bracket to the space {Hom(A ⊗k , A)}, k≥0 is exactly the Gerstenhaber bracket. We discuss some formality conjecture related with this construction. We also discuss some applications to deformation theory.

Eprint Arxiv Hep Th 0508225, 2005

We study an algebraic deformation problem which captures the data of the general deformation problem for a quantum vertex algebra. We derive a system of coupled equations which is the counterpart of the Maurer-Cartan equation on the usual Hochschild complex of an assocative algebra. We show that this system of equations results from an action principle. This might be the starting point for a perturbative treatment of the deformation problem of quantum vertex algebras. Our action generalizes the action of the Kodaira-Spencer theory of gravity and might therefore also be of relevance for applications in string theory.

International Journal of Modern Physics A, 2017

In this short review we describe some aspects of [Formula: see text]-deformation. After discussing the algebraic and geometric approaches to [Formula: see text]-Poincaré algebra we construct the free scalar field theory, both on noncommutative [Formula: see text]-Minkowski space and on curved momentum space. Finally, we make a few remarks concerning the interacting scalar field.

Physical Review D, 2005

We give a definition of admissible counterterms appropriate for massive quantum field theories on the noncommutative Minkowski space, based on a suitable notion of locality. We then define products of fields of arbitrary order, the so-called quasiplanar Wick products, by subtracting only such admissible counterterms. We derive the analogue of Wick's theorem and comment on the consequences of using quasiplanar Wick products in the perturbative expansion.