Classification of universal formality maps for quantizations of Lie bialgebras
METADATA ONLY
Loading...
Author / Producer
Date
2020-10
Publication Type
Journal Article
ETH Bibliography
yes
Citations
Altmetric
METADATA ONLY
Data
Rights / License
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 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 algebra circle dot V-center dot given in terms of polydifferential operators. Applying this functor to the minimal resolution (Lieb) over cap (infinity) of the genus completed prop (Lieb) over cap 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 : Assb(infinity) -> D (Lieb) over cap (infinity) satisfying certain boundary conditions, where Assb(infinity) is a minimal resolution of the prop of associative bialgebras. We prove that the set of such formality morphisms is nonempty. 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 A there is an associated Lie(infinity) quasi-isomorphism between the Lie(infinity) algebras Def(AssB(infinity) -> End(circle dot center dot V)) and Def(LieB -> End(V)) controlling, respectively, deformations of the standard bialgebra structure in circle dot V and deformations of any given Lie bialgebra structure in V. We study the deformation complex of an arbitrary universal formality morphism Def(Assb(infinity) -> F D (Lieb) over cap (infinity)) 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 {FA} of gauge equivalence classes of universal Lie connected formality maps: it is a torsor over the Grothendieck-Teichmuller group GRT = GRT1 (sic) K* and can hence can be identified with the set {U} of Drinfeld associators.
Permanent link
Publication status
published
External links
Editor
Book title
Journal / series
Volume
156 (10)
Pages / Article No.
2111 - 2148
Publisher
Cambridge University Press
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Organisational unit
09577 - Willwacher, Thomas / Willwacher, Thomas
Notes
Funding
150012 - Graphical Models, Quantization, and the Grothendieck-Teichmüller Group (SNF)