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.

中文翻译：

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用于李双代数量化的通用形式映射的分类

我们通过将李双代数的完整分类归结为同伦等价，解决了关于李双代数的普遍变形量化理论的几个基本问​​题。此外，我们更广泛地解决了这些问题：我们给出了相关通用形式映射的完整分类。本文引入的一个重要的新技术成分是增强道具范畴中的多微分自函子 ${\mathcal {D}}$，其性质是对于向量中的道具 ${\mathcal {P}}$ 的任何表示空间 $V$ 相关的道具 ${\mathcal {D}}{\mathcal {P}}$ 承认分级交换代数 $\odot ^\bullet V$ 上的归纳表示以多微分算子给出。将此函子应用于李双代数的属完成道具 $\widehat {\mathcal {L}\textit{ieb}}$ 的最小分辨率 $\widehat {\mathcal {L}\textit{ieb}}_\infty$我们表明李双代数量化的通用形式映射与 dg props 的态射一一对应 \[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}} \widehat{\mathcal{L}\textit{ieb}}_\infty \] 满足某些边界条件，其中 $\mathcal {A}\textit{ssb}_\infty$ 是结合双代数的 prop 的最小分辨率. 我们证明了这种形式态射的集合是非空的。我们研究任意通用形式态射的变形复形 $\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat { \mathcal {L}\textit{ieb}}_\infty )$ 并证明它与在变形量化理论的上下文中引入 Maxim Kontsevich 的图复数的完整（即不需要连接）版本是准同构的泊松流形。该结果给出了通用李连接形式映射的规范等价类集合 $\{F_\mathfrak{A}\}$ 的完整分类：它是 Grothendieck-Teichmüller 群 $GRT=GRT_1\rtimes {\ mathbb {K}}^*$，因此可以用 Drinfeld 关联子的集合 $\{\mathfrak{A}\}$ 识别。不需要连接）图复合体的版本在泊松流形变形量化理论的背景下介绍了 Maxim Kontsevich。该结果给出了通用李连接形式映射的规范等价类集合 $\{F_\mathfrak{A}\}$ 的完整分类：它是 Grothendieck-Teichmüller 群 $GRT=GRT_1\rtimes {\ mathbb {K}}^*$，因此可以用 Drinfeld 关联子的集合 $\{\mathfrak{A}\}$ 识别。不需要连接）图复合体的版本在泊松流形变形量化理论的背景下介绍了 Maxim Kontsevich。该结果给出了通用李连接形式映射的规范等价类集合 $\{F_\mathfrak{A}\}$ 的完整分类：它是 Grothendieck-Teichmüller 群 $GRT=GRT_1\rtimes {\ mathbb {K}}^*$，因此可以用 Drinfeld 关联子的集合 $\{\mathfrak{A}\}$ 识别。