Reference : Classification of universal formality maps for quantizations of Lie bialgebras
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Classification of universal formality maps for quantizations of Lie bialgebras
Merkulov, Sergei mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Willwacher, Thomas [ETH, Zurich > Mathematics]
Compositio Mathematica
Cambridge University Press
Yes (verified by ORBilu)
United Kingdom
[en] Quantum algebra ; Hopf algebra ; Lie bialgebra
[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.
Researchers ; Students

File(s) associated to this reference

Fulltext file(s):

Open access
16 Classification.pdfPublisher postprint524.49 kBView/Open

Bookmark and Share SFX Query

All documents in ORBilu are protected by a user license.