Deformation theory of bialgebras, higher Hochschild cohomology and formality

Abstract

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.

Cyclic Gerstenhaber-Schack cohomology

“…The different behaviour of the Gerstenhaber bracket in the finite dimensional and infinite dimensional case may sound surprising, but it actually has a transparent explanation in terms of Hopf algebra theory: a finite dimensional Hopf algebra exhibits no nonzero primitive elements. The e 3 -algebra structure constructed this way on the Gerstenhaber-Schack cohomology of a finite dimensional Hopf algebra is presumably different from the ones obtained by higher categorical methods in [GiYa,Sh1,Sh2] as the latter do not rely on the finite dimensionality of H. Nevertheless, even our approach does not seem to work if H is merely a bialgebra, that is, without an antipode, the necessity of which was conjectured by Shoikhet in his approach [Sh2,p. 9].…”

“…Remark 5.3. The e 3 -algebra structure on H GS pH, Hq from Corollary 5.2 is presumably different from the one exhibited in [GiYa,Sh1,Sh2] as the latter does not rely on the finite dimensionality of H. In particular, the constructions in op. cit.…”

Derived deformation theory of algebraic structures

“…Further applications and perspectives. A first major application appeared earlier in our preprint [45], where some of the results of the present article were announced. Our article provides complete proofs of these results and add some new ones as well.…”

“…Our article provides complete proofs of these results and add some new ones as well. In this related work [45], the authors use them crucially to prove longstanding conjectures in deformation theory of bialgebras and E n -algebras as well as in deformation quantization. We prove a conjecture enunciated by Gerstenhaber and Schack (in a wrong way) in 1990 [37], whose correct version is that the Gerstenhaber-Schack complex forms an E 3 -algebra, hence unraveling the full algebraic structure of this complex which remained mysterious for a while.…”

On deformation quantization of quadratic Poisson structures

“…of the plus-extended prop with values in P (otherwise we shall loose an important and useful information about homotopy theory of the (wheeled) prop P, cf. [MW1,MW2,GY,MW3]). The dg props Hoqpois + c,d an Hoqpois + c,d is easy to describe explicitly -both are built from generators (5) and one extra (1, 1)-generator • which is assigned degree +1.…”