2 results
Classification of universal formality maps for quantizations of Lie bialgebras
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 10 / October 2020
- Published online by Cambridge University Press:
- 27 November 2020, pp. 2111-2148
- Print publication:
- October 2020
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- Article
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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
satisfying certain boundary conditions, where
\[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \]
$\mathcal {A}\textit{ssb}_\infty$ 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.
On the Simple ℤ2-homotopy Types of Graph Complexes and Their Simple ℤ2-universality
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- Journal:
- Canadian Mathematical Bulletin / Volume 51 / Issue 4 / 01 December 2008
- Published online by Cambridge University Press:
- 20 November 2018, pp. 535-544
- Print publication:
- 01 December 2008
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- Article
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We prove that the neighborhood complex
$\text{N}\left( G \right)$, the box complex 
$\text{B}\left( G \right)$, the homomorphism complex
$\text{Hom}\left( {{K}_{2}},\,G \right)$
and the Lovász complex 
$\text{L}\left( G \right)$ have the same simple
${{\mathbb{Z}}_{2}}$
-homotopy type in the sense of Whitehead. We show that these graph complexes are simple
${{\mathbb{Z}}_{2}}$
-universal.
