Classification of universal formality maps for quantizations of Lie bialgebras

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2020 • In *Compositio Mathematica, 156* (10), p. 2111-2148

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Keywords :

Quantum algebra; Hopf algebra; Lie bialgebra

Abstract :

[en] We introduce an endofunctor D in the category of augmented props with the property that for any representation of a prop P in a vector space V the associated prop DP admits an induced representation on the graded commutative tensor algebra S(V) given in terms of polydifferential operators.
Applying this functor to the prop LieB of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in in 1-1 correspondence with prop morphisms from the minimal resolution AssB_infty of the prop of associative bialgebras to the polydifferential prop DLieB_infty satisfying certain boundary conditions. We prove that the set of such formality morphisms (having an extra property of being Lie connected) 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 there is an associated Lie_infty quasi-isomorphism between the Lie_infty algebras controlling, respectively, deformations of the standard bialgebra structure in S(V) and deformations of any given Lie bialgebra structure in V.
We study the deformation complex of an arbitrary universal formality morphism and show that it is quasi-isomorphic (up to one class corresponding to the standard rescaling automorphism of the properad LieB) to the oriented graph complex GC or 3 studied earlier in \cite{Wi2}. This result gives a complete classification of the set of gauge equivalence classes of universal Lie connected formality maps --- it is a torsor over the Grothendieck-Teichm\"uller group GRT and can hence can be identified with the set of Drinfeld associators.

Disciplines :

Mathematics

Merkulov, Sergei ^{}; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit

Willwacher, Thomas; ETH, Zurich > Mathematics

External co-authors :

yes

Language :

English

Title :

Classification of universal formality maps for quantizations of Lie bialgebras

Publication date :

October 2020

Journal title :

Compositio Mathematica

ISSN :

1570-5846

Publisher :

Cambridge University Press, United Kingdom

Volume :

156

Issue :

10

Pages :

2111-2148

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBilu :

since 09 May 2016

Scopus citations^{®}

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Scopus citations^{®}

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WoS citations^{™}

6