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Classification of universal formality maps for quantizations of Lie bialgebras

Published online by Cambridge University Press:  27 November 2020

Sergei Merkulov
Affiliation:
Mathematics Research Unit, University of Luxembourg, Grand Duchy of Luxembourg sergei.merkulov@uni.lu
Thomas Willwacher
Affiliation:
Department of Mathematics, ETH Zurich, Zurich, Switzerland thomas.willwacher@math.ethz.ch
Corresponding

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.

Type
Research Article
Copyright
© The Author(s) 2020

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Footnotes

S.M. has been partially supported by the Swedish Vetenskaprådet, grant 2012-5478. T.W. has been partially supported by the Swiss National Science foundation, grant 200021_150012, and the SwissMAP NCCR funded by the Swiss National Science foundation.

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