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Deformation Theory of Bialgebras, Higher Hochschild Cohomology and Formality Grégory Ginot, Sinan Yalin

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Deformation Theory of Bialgebras, Higher Hochschild Cohomology and Formality Grégory Ginot, Sinan Yalin Deformation theory of bialgebras, higher Hochschild cohomology and Formality Grégory Ginot, Sinan Yalin To cite this version: Grégory Ginot, Sinan Yalin. Deformation theory of bialgebras, higher Hochschild cohomology and Formality. 2018. hal-01714212 HAL Id: hal-01714212 https://hal.archives-ouvertes.fr/hal-01714212 Preprint submitted on 21 Feb 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. DEFORMATION THEORY OF BIALGEBRAS, HIGHER HOCHSCHILD COHOMOLOGY AND FORMALITY GRÉGORY GINOT, SINAN YALIN Abstract. A first goal of this paper is to precisely relate the homotopy the- ories of bialgebras and E2-algebras. For this, we construct a conservative and fully faithful ∞-functor from pointed conilpotent homotopy bialgebras to aug- mented E2-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 E2-algebra structures on this “cobar” construction. We show con- sequently that the E3-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 E3-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 E3-formality of the de- formation 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 in- dependent 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 Introduction 1 Notations 9 1. Recollections 10 2. Moduli problems of algebraic structures and deformation complexes 17 3. The “plus” construction and the corresponding deformation theory 30 4. Bialgebras versus iterated coalgebras 35 5. Bialgebras versus E2-algebras 48 6. Identification of Deformation complexes with higher Hochschild, Gerstenhaber-Schack and Tamarkin complexes of P oisn-algebras 54 7. The E3-formality Theorem 67 8. Etingof-Kazdhan deformation quantization 75 References 77 Introduction A deep interplay between bialgebras and quantum groups on one side and higher algebras on the other side (for instance E2 and E3-algebras) has been noticed for 1 BIALGEBRAS, HOCHSCHILD COHOMOLOGY AND FORMALITY 2 many years and more relations have been expected for a long time. This led people to investigate the relationship between these two kinds of structures with the hope to establish some equivalence between their deformation theories, which could help to understand various related problems on both sides. One of the first goals of this paper is to give a precise relation between their respective homotopy theories and to use it to study the deformation theory of dg-bialgebras. Algebras governed by En-operads and their deformation theory play a prominent role in a variety of topics such as the study of iterated loop spaces, Goodwillie-Weiss calculus for embedding spaces, deformation quantization of Poisson manifolds and Lie bialgebras, and factorization homology [47, 48, 52, 54, 8, 19, 20, 26, 30, 36, 39, 40, 44, 49, 55, 62, 69, 72, 77]. These algebras form a hierarchy of “more and more” commutative and homotopy associative structures, interpolating between homotopy associative or A∞-algebras (the E1 case) and E∞-algebras (the colimit of the En’s, which is homotopy equivalent to differential graded commutative algebras in characteristic zero). Their natural cohomology theory, the higher Hochschild cohomology, is a suitable generalization of the Hochschild cohomology of associative algebras. On the other hand, bialgebras are central in various topics of algebraic topology, representation theory and mathematical physics (often incarnated as Hopf alge- bras) [9, 10, 5, 16, 17, 29, 58, 59]. They consist of two structures, an associative algebra structure and a coassociative coalgebra structure, related by a compatibil- ity condition such that bialgebras can be alternately defined as algebras in coal- gebras or vice-versa. Their natural cohomology theory, the Gerstenhaber-Schack cohomology, is a suitable mixing between Hochschild cohomology of algebras and co-Hochschild cohomology of coalgebras. Two important and long-standing conjectures expected from the relationship be- tween these two kind of structures are the following. The first one, enunciated by Gerstenhaber and Schack (in a wrong way) at the beginning of the 90’s [29], charac- terizes the structure of the complex controlling the deformation theory of bialgebras, which remained quite mysterious for a while. It is a “differential graded bialgebra version” of the famous Deligne conjecture for associative differential graded alge- bras (see for instance [72] and [47]). The second one, enunciated by Kontsevich in his celebrated work on deformation quantization of Poisson manifolds [48], is a formality statement for the deformation complex of the symmetric bialgebra, which should imply as a corollary Drinfeld’s and Etingof-Kazdhan’s deformation quantization theory (see [9], [16] and [17]). This situation is somehow a generalization to bialgebras of the situation encoun- tered in deformation quantization of Poisson manifolds: the formality of the E2- operad implies the Deligne conjecture, which states the existence of an E2-algebra structure on the Hochschild complex of an associative algebra, and this allows to prove that the Hochschild complex of the ring of functions on a Poisson manifold is formal. This formality implies, in turn, the deformation quantization of such a man- ifold. Here, the formality of the E3-operad should allow to prove a bialgebra version of this Deligne conjecture, implying the formality of the Gerstenhaber-Schack com- plex of the symmetric bialgebra, inducing in turn Drinfeld and Etingof-Kazdhan deformation quantization of Lie bialgebras. In this paper, we solve this bialgebra version of the Deligne conjecture proposed by Gerstenhaber-Schack as well as Kontsevich’s E3-formality conjecture, both at a BIALGEBRAS, HOCHSCHILD COHOMOLOGY AND FORMALITY 3 greater level of generality than the original statements, by implementing methods coming from derived algebraic geometry (see in particular [52], [54] and [76]) as well as from previous results of the authors (see in particular [36] and [82]) and by solving partially a conjecture of Francis-Gaistgory. We deduce from it a generalization of Etingof-Kadhan’s celebrated deformation quantization. Moreover, we enlighten along the way the role of various versions of deformation complexes of algebraic structures considered in the literature, by characterizing their associated derived formal moduli problems in full generality. Our new methods have thus an independent interest going even further than the solutions to the two aforementioned conjectures. We now describe our main results and conclude with some perspectives for future works. 0.1. From bialgebras to E2-algebras. The cobar construction defining a Quillen equivalence between conilpotent dg coalgebras and augmented dg algebras can be lifted to a functor from the category of associative coassociative bialgebras to the category of E2-algebras [43]. A natural question [19] was thus to know whether this implies an equivalence of homotopy theories (in the sense of (∞, 1)-categories) between bialgebras and E2-algebras. The appropriate answer to this problem is more subtle, and we solve it by using an appropriate notion of cobar construction for bialgebras, which intertwines a bar construction on the algebra part of the structure with a cobar construction on the coalgebra part: Theorem 0.1. (1) There exists a bar-cobar adjunction Benh : E − Alg0−con(dgCogconil) E − Cogconil(dgCogconil):Ωenh E1 1 1 E1 inducing an equivalence of (∞, 1)-categories between 0-connected homotopy associa- tive algebras in conilpotent dg coalgebras (0-connected conilpotent homotopy asso- ciative bialgebras) and conilpotent homotopy coassociative coalgebras in conilpotent dg coalgebras. (2) The equivalence above induces an equivalence of (∞, 1)-categories (Benh,pt(−) ) : E −Algaug,con(dgCogconil) E −Cogconil,pt(dgCogconil) : (Ωenh,pt(−) ) E1 − + 1 1 E1 − + between connected augmented conilpotent homotopy associative bialgebras and pointed conilpotent homotopy coassociative coalgebras in conilpotent dg coalgebras. This solves a conjecture of Francis-Gaitsgory [19] in the case where the base category is the category of conilpotent dg coalgebras, respectively the category of pointed conilpotent dg coalgebras. Let us note that we work here with non- negatively graded chain complexes. Corollary 0.2. The left adjoint of Theorem 0.1(2) induces a fully faithful
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