Classification of universal formality maps for quantizations of Lie bialgebras

Abstract

We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor ${\mathcal {D}}$ in the category of augmented props with the property that for any representation of a prop ${\mathcal {P}}$ in a vector space $V$ the associated prop ${\mathcal {D}}{\mathcal {P}}$ admits an induced representation on the graded commutative algebra $\odot ^\bullet V$ given in terms of polydifferential operators. Applying this functor to the minimal resolution $\widehat {\mathcal {L}\textit{ieb}}_\infty$ of the genus completed prop $\widehat {\mathcal {L}\textit{ieb}}$ of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props \[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \] satisfying certain boundary conditions, where $\mathcal {A}\textit{ssb}_\infty$ is a minimal resolution of the prop of associative bialgebras. We prove that the set of such formality morphisms is non-empty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator $\mathfrak{A}$ there is an associated ${\mathcal {L}} ie_\infty$ quasi-isomorphism between the ${\mathcal {L}} ie_\infty$ algebras $\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$ and $\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$ controlling, respectively, deformations of the standard bialgebra structure in $\odot V$ and deformations of any given Lie bialgebra structure in $V$ . We study the deformation complex of an arbitrary universal formality morphism $\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$ and prove that it is quasi-isomorphic to the full (i.e. not necessary connected) version of the graph complex introduced Maxim Kontsevich in the context of the theory of deformation quantizations of Poisson manifolds. This result gives a complete classification of the set $\{F_\mathfrak{A}\}$ of gauge equivalence classes of universal Lie connected formality maps: it is a torsor over the Grothendieck–Teichmüller group $GRT=GRT_1\rtimes {\mathbb {K}}^*$ and can hence can be identified with the set $\{\mathfrak{A}\}$ of Drinfeld associators.

Quantizations of Lie bialgebras, duality involution and oriented graph complexes

“…which is proven in [MW2] to be a quasi-isomorphism for any universal quantization Q, in particular for Q 0 . There is also a quasi-isomorphism of complexes…”

“…An application to the deformation quantization theory which has the property that for any dg prop P and its any representation, ρ : P → End V , in a dg vector space V the associated dg prop DP has an associated representation, Dρ : DP → End ⊙ • V , in the graded commutative algebra ⊙ • V given in terms of polydifferential (with respect to the standard multiplication in ⊙ • V) operators. We refer to §5.2 of [MW2] for full details and explain briefly only the explicit structure of the dg prop 3 DLieb. If elements of Lieb has all legs labelled differently, the elements of DLieb can be understood as graphs from Lieb whose different in-legs (or out-legs) may have identical numerical labels.…”

“…Question 2: According to the famous theorem of Etingof-Kazhdan [EK], any Lie bialgebra structure in a vector space V can be deformation quantized, that is, there is a continuous Hopf algebra structure in ⊙ • V[[ ]], being a formal parameter, which -at the infinitesimal in level -induces the given Lie bialgebra in V. Moreover, there is a deformation quantization procedure which is universal in the sense that it is applicable to any (possibly, infinite-dimensional) Lie bialgebra V. Such a universal quantization is best understood as a morphism props [MW2],…”

“…where Assb is the prop of associative bialgebras, and D is a polydifferential endofunctor [MW2] in the category of dg props applied to the genus completed prop(erad) Lieb of Lie bialgebras. It was proven in [MW2] that such morphisms (up to homotopy, or gauge equivalence) are classified by the set of V. Drinfeld associators.…”

“…where Assb is the prop of associative bialgebras, and D is a polydifferential endofunctor [MW2] in the category of dg props applied to the genus completed prop(erad) Lieb of Lie bialgebras. It was proven in [MW2] that such morphisms (up to homotopy, or gauge equivalence) are classified by the set of V. Drinfeld associators. Both sides of the above arrow admit a canonical direction involution as the relations defining associative bialgebras are also invariant under the direction (or duality) involution, in a full analogy to the case of Lie bialgebras.…”