An Explicit Two Step Quantization of Poisson Structures and Lie Bialgebras

Merkulov, Sergeï; Willwacher, Thomas

doi:10.1007/s00220-018-3267-9

Cited by 8 publications

(32 citation statements)

References 33 publications

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“…In the proof of part (i) we used the Etingof–Kazhdan theorem [EK96] which says that any Lie bialgebra can be deformation quantized in the sense introduced by Drinfeld in [Dri92]. Later different proofs of this theorem have been given by Tamarkin [Tam07], Severa [Sev16] and the authors [MW18b] (the latest proof in [MW18b] gives an explicit formula for a universal quantization of Lie bialgebras). Part (ii) is a Lie bialgebra analogue of the famous Kontsevich formality theorem for deformation quantizations of Poisson manifolds (it was first proved in [Mer11] with the help of a more complicated method).…”

Section: Introductionmentioning

confidence: 99%

Classification of universal formality maps for quantizations of Lie bialgebras

Merkulov

Willwacher

2020

Compositio Math.

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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 \[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \] 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.

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Section: Introductionmentioning

confidence: 99%

Classification of universal formality maps for quantizations of Lie bialgebras

Merkulov

Willwacher

2020

Compositio Math.

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“…Recently, some other papers making use and further developing the ideas and results of this paper have appeared, see e.g. [KMW], [MW1], [MW2]. The author believes that the constructions introduced in this paper may find more fruitful applications in near future.…”

Section: 2mentioning

confidence: 88%

An $$L_\infty $$L∞ algebra structure on polyvector fields

Shoikhet

2018

Sel. Math. New Ser.

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It is well-known that the Kontsevich formality [K97] for Hochschild cochains of the polynomial algebra A = S(V * ) fails if the vector space V is infinite-dimensional. In the present paper, we study the corresponding obstructions. We construct an L ∞ structure on polyvector fields on V having the even degree Taylor components. The degree 2 component is given by the Schouten-Nijenhuis bracket, but all its higher even degree components are nonzero. We prove that this L ∞ algebra is quasi-isomorphic to the corresponding Hochschild cochain complex. We prove that our L ∞ algebra is L ∞ quasi-isomorphic to the Lie algebra of polyvector fields on V with the Schouten-Nijenhuis bracket, if V is finite-dimensional.

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“…In particular we have isomorphisms of Lie algebras, (26) H 0 (GC d-or d+2 , δ 0 ) = H 0 (GC 0↑1 2 , δ 0 ) = grt 1 , for any d ≥ 0. For d = 2 and d = 3 the algebro-geometric meanings of the associated graph complex incarnations of the Grothendieck-Teichmüller group GRT 1 are clear: the d = 2 case corresponds to the action of GRT 1 (through cocycle representatives in GC 0↑1 2 ) on universal Kontsevich formality maps associated with the deformation quantization of Poisson structures (given explicitly with the help of suitable configuration spaces in the two dimensional upper half-plane [Ko]), while the case d = 3 corresponds to the action of GRT 1 (through cocycle representatives in GC 1-or 3 ) on universal formality maps associated with the deformation quantization of Lie bialgebras (see [MW3] where compactified configuration spaces in three dimensions have been used). The above results tell us that the Grothendieck-Teichmüller group survives in any geometric dimension ≥ 4 but now in the multi-oriented graph complex incarnation.…”

Section: Theoremmentioning

confidence: 99%

Multi-oriented props and homotopy algebras with branes

Merkulov

2020

Lett Math Phys

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We introduce a new category of differential graded multi-oriented props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of k linear subspaces in that space, k being the number of extra directions (if k = 0 this structure recovers an ordinary prop); symplectic vector spaces equipped with k Lagrangian subspaces play a distinguished role in this theory. Manin triples is a classical example of an algebraic structure (concretely, a Lie bialgebra structure) given in terms of a vector space and its subspace; in the context of this paper Manin triples are precisely symplectic Lagrangian representations of the 2-oriented generalization of the classical operad of Lie algebras. In a sense, the theory of multi-oriented props provides us with a far reaching strong homotopy generalization of Manin triples type constructions.The homotopy theory of multi-oriented props can be quite non-trivial (and different from that of ordinary props). The famous Grothendieck-Teichmüller group acts faithfully as homotopy non-trivial automorphisms on infinitely many multioriented props, a fact which motivated much the present work as it gives us a hint to a non-trivial deformation quantization theory in every geometric dimension d ≥ 4 generalizing to higher dimensions Drinfeld-Etingof-Kazhdan's quantizations of Lie bialgebras (the case d = 3) and Kontsevich's quantizations of Poisson structures (the case d = 2). 1 2 ... k and defines rules for multi-oriented prop compositions via contractions along admissible multi-oriented subgraphs. However, it is much less evident how to transform that more or less standard rules into non-trivial and interesting representations (i.e. examples) -the intuition from the theory of ordinary (wheeled) props does not help much. Adding new k directions to each edges of a decorated graph of an ordinary prop can be naively understood as extending that ordinary prop into a 2 k -coloured one, but then the requirement that the new directions on graph edges do not create "wheels" (that is, closed directed paths of edges with respect to any of the new orientations) kills that naive picture immediately -the elements of the set of 2 k new colours start interacting with each other in a non-trivial way. We know which structure distinguishes ordinary props (the ones with no wheels in the given single orientation, i.e. the ones which are 1-oriented in the terminology of this paper) from the ordinary wheeled props (that is, 0-oriented 1directed props in the terminology of this paper) in terms of representations in, say, a graded vector space V -it is the 1 dimension of V. In general, a wheeled prop can have well-defined representations in V only in the case dim V < ∞ as graphs with wheels generate the trace operation V × V * → K which explodes in the case dim V = ∞; this phenomenon explains the need for 1-oriented props. How to explain the need for all (or some) of the extra k directions to be oriented? Which structure on a graded vector space V can be used to...

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An Explicit Two Step Quantization of Poisson Structures and Lie Bialgebras

Cited by 8 publications

References 33 publications

Classification of universal formality maps for quantizations of Lie bialgebras

Classification of universal formality maps for quantizations of Lie bialgebras

An $$L_\infty $$L∞ algebra structure on polyvector fields

Multi-oriented props and homotopy algebras with branes

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