The Drinfeld associator of gl(1 | 1)

2009 ◽  
pp. 39-80 ◽  
Author(s):  
Jens Lieberum
Keyword(s):  
Drinfeld Associator

The Drinfeld associator

2012 ◽  
pp. 265-309
Author(s):  
S. Chmutov ◽  
S. Duzhin ◽  
J. Mostovoy
Keyword(s):  
Drinfeld Associator

scholarly journals Classification of universal formality maps for quantizations of Lie bialgebras

2020 ◽  
Vol 156 (10) ◽  
pp. 2111-2148
Author(s):  
Sergei Merkulov ◽  
Thomas Willwacher
Keyword(s):  
Complete Classification ◽  
Poisson Manifolds ◽  
Minimal Resolution ◽  
Lie Bialgebras ◽  
Induced Representation ◽  
Gauge Equivalence ◽  
Image Position ◽  
Drinfeld Associator ◽  
Deformation Complex

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.

scholarly journals The Grothendieck–Teichmüller group action on differential forms and formality morphisms of chains

2017 ◽  
Vol 2017 (726) ◽  
Author(s):  
Thomas Willwacher
Keyword(s):  
Group Action ◽  
Differential Forms ◽  
Drinfeld Associator

AbstractIt is known that one can associate a Kontsevich-type formality morphism to every Drinfeld associator. In the present paper, we show that this morphism may be extended to a Kontsevich–Shoikhet formality morphism of cochains and chains, by describing the action of the Grothendieck–Teichmüller group on such objects (up to homotopy).

scholarly journals All orderα′-expansion of superstring trees from the Drinfeld associator

2014 ◽  
Vol 89 (6) ◽  
Author(s):  
Johannes Broedel ◽  
Oliver Schlotterer ◽  
Stephan Stieberger ◽  
Tomohide Terasoma
Keyword(s):  
Drinfeld Associator
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