A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids

Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichmül...

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2022
Автор:	Morand, Kevin
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Мова:	Англійська
Опубліковано:	Інститут математики НАН України 2022
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Цитувати:	A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids. Kevin Morand. SIGMA 18 (2022), 020, 38 pages

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author	Morand, Kevin
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citation_txt	A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids. Kevin Morand. SIGMA 18 (2022), 020, 38 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichmüller group via one-colored directed and oriented graphs, respectively. In this note, we study the action of multi-oriented graph complexes on Lie bialgebroids and their ''quasi'' generalisations. Using results due to T. Willwacher and M. Zivković on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck-Teichmüller group on Lie bialgebras and quasi-Lie bialgebras can be generalised to quasi-Lie bialgebroids via graphs with two colors, one of them being oriented. However, this action generically fails to preserve the subspace of Lie bialgebroids. By resorting to graphs with two oriented colors, we instead show the existence of an obstruction to the quantization of a generic Lie bialgebroid in the guise of a new Lie∞-algebra structure non-trivially deforming the ''big bracket'' for Lie bialgebroids. This exotic Lie∞-structure can be interpreted as the equivalent in = 3 of the Kontsevich-Shoikhet obstruction to the quantization of infinite-dimensional Poisson manifolds (in = 2). We discuss the implications of these results with respect to a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 020, 38 pages A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids Kevin MORAND ab a) Department of Physics, Sogang University, Seoul 04107, South Korea E-mail: morand@sogang.ac.kr b) Center for Quantum Spacetime, Sogang University, Seoul 04107, South Korea Received July 07, 2021, in final form March 09, 2022; Published online March 20, 2022 https://doi.org/10.3842/SIGMA.2022.020 Abstract. Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck–Teichmüller group via one-colored directed and oriented graphs, respectively. In this note, we study the action of multi-oriented graph complexes on Lie bialgebroids and their “quasi” generalisations. Using results due to T. Willwacher and M. Živković on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck–Teichmüller group on Lie bialgebras and quasi-Lie bialgebras can be generalised to quasi-Lie bialgebroids via graphs with two colors, one of them being oriented. However, this action generically fails to preserve the subspace of Lie bialgebroids. By re- sorting to graphs with two oriented colors, we instead show the existence of an obstruction to the quantization of a generic Lie bialgebroid in the guise of a new Lie∞-algebra structure non-trivially deforming the “big bracket” for Lie bialgebroids. This exotic Lie∞-structure can be interpreted as the equivalent in d = 3 of the Kontsevich–Shoikhet obstruction to the quantization of infinite-dimensional Poisson manifolds (in d = 2). We discuss the implica- tions of these results with respect to a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids. Key words: deformation quantization; Kontsevich’s graphs; Lie bialgebroids; Grothendieck– Teichmüller group 2020 Mathematics Subject Classification: 53D55; 18G85; 17B62 1 Introduction Graph complexes play an essential rôle in the understanding of the deformation quantiza- tion of various algebraic and geometric structures, the paradigmatic example thereof being the Kontsevich graph complex and its relation to the deformation quantization problem for (finite- dimensional) Poisson manifolds [32]. In particular, the space of Kontsevich quantization maps is acted upon by the pro-unipotent group exponentiating the zeroth cohomology of the Kontsevich graph complex of directed graphs [12]. As shown by T. Willwacher [71], the latter is isomorphic to the Grothendieck–Teichmüller group GRT1 – introduced by V. Drinfeld1 [17] in the context of the absolute Galois group Gal ( Q̄/Q ) and the theory of quasi-Hopf algebras – so that the space of Kontsevich maps2 is a GRT1-torsor [33]. Since its inception, the Grothendieck–Teichmüller group appeared in a variety of mathematical contexts such as the Kashiwara–Vergne conjecture, multiple zeta values, rational homotopy of the E2-operad, etc. 1Based on a suggestion due to A. Grothendieck in his Esquisse d’un Programme [27] who proposed to study the combinatorial properties of the absolute Galois group Gal ( Q̄/Q ) via its natural action on the tower of Teichmüller groupoids. 2More precisely, the space of homotopy classes [11] of stable [12] formality morphisms is a GRT1-torsor. mailto:morand@sogang.ac.kr https://doi.org/10.3842/SIGMA.2022.020 2 K. Morand In the present context of graph complexes, another incarnation of the Grothendieck–Teich- müller group can be found in relation to the deformation quantization problem for Lie bialgebras via the action of the graph complex of oriented graphs [72] in dimension d = 3.3 The latter action generalises to Lie-quasi bialgebras4 and furthermore provides a rationale for the classi- fying rôle played by the Grothendieck–Teichmüller group on the space of quantization maps for Lie and Lie-quasi bialgebras à la Etingof–Kazhdan [20, 61]. The oriented graph complex also plays a crucial rôle regarding the obstruction theory to the existence of a universal quan- tization of infinite-dimensional Poisson manifolds [72]. The corresponding obstruction lives in the first order cohomology of the oriented graph complex in d = 2 which is a one-dimensional space spanned by the so-called Kontsevich–Shoikhet cocycle. When represented on the space of (infinite-dimensional) polyvector fields, the latter yields an exotic Lie∞-structure [65] deform- ing non-trivially the Schouten bracket. Further, the zeroth order cohomology of the oriented graph complex (in d = 2) vanishes thus preventing the Grothendieck–Teichmüller group to play a classifying rôle for quantizations of infinite-dimensional Poisson manifolds. The deformation quantization problem for infinite-dimensional Poisson manifolds thus differs essentially from the finite-dimensional case and this discrepancy can be traced back to the fact that their respective deformation theory is acted upon by a different graph complex (directed vs. oriented). Defor- mation quantization problems can then be partitioned into different classes according to the cohomology of the graph complex acting on them. We distinguish between three main classes: � A no-go class comprising the deformation quantization problems for infinite-dimensional Poisson manifolds: 1. The Grothendieck–Teichmüller group plays no classifying rôle regarding the universal deformations (and hence quantizations). 2. There exists a potential obstruction to the existence of universal quantizations. � A yes-go class comprising the deformation quantization problems for (finite-dimensional) Poisson manifolds and Lie-(quasi) bialgebras for which: 1. The Grothendieck–Teichmüller group plays a classifying rôle. 2. There is (conjecturally) no generic obstruction to the existence of universal quanti- zations. � A middle way class comprising the deformation quantization problems for Courant alge- broids: 1. The Grothendieck–Teichmüller group plays no classifying rôle; rather deformations are generated by the triangle cocycle as well as by conformal rescalings associated with trivalent graphs. 2. There is no generic obstruction to the existence of universal quantizations. In the present note, we add two threads to this on-going story by introducing some new universal models for the deformation theory of Lie bialgebroids and their “quasi” versions. Lie bialge- broids have been introduced by Mackenzie–Xu [42] as linearisations of Poisson groupoids and constitute a common generalisation of the notions of Poisson manifolds and Lie bialgebras. The 3Recall that the parameter d corresponds to the dimension of the source manifold of the relevant AKSZ σ- model [1]. The latter is related to the degree n of the corresponding target manifold via d = n+1 and is therefore independent of the dimension of the associated algebro-geometric structure (Poisson manifold, Lie bialgebra, etc.). Consistently, it relates to the dimension of the compactified configuration spaces of points of the associated de Rham field theories [45, 46]. Therefore, any graph complex related to Poisson manifolds has dimension d = 2 while the ones related to Lie bialgebras and generalisations thereof have dimension d = 3. 4As well as their dual, referred to as quasi-Lie bialgebras in the following, see footnote 14 for terminology. A Note on Multi-Oriented Graph Complexes 3 corresponding quantization problem unifies the quantization problems for (finite-dimensional) Poisson manifolds and Lie bialgebras. It was spelled out by P. Xu [74, 76] who then conjectured that any Lie bialgebroid is quantizable. The main result of this note consists in providing some arguments for the non-existence of universal quantizations of Lie bialgebroids. This is done by exhibiting a potential obstruction to the existence of a universal quantization map for Lie bial- gebroids in the guise of an exotic Lie∞-structure on the deformation complex of Lie bialgebroids. The latter is a non-trivial deformation of the so-called “big bracket” for Lie bialgebroids and can be considered as an avatar in d = 3 of the Kontsevich–Shoikhet obstruction to the quantization of infinite-dimensional Poisson manifolds. Our main result is stated as follows: Theorem 1.1 (no-go). The deformation complex of Lie bialgebroids is endowed with an exotic Lie∞-structure deforming non-trivially the big bracket of Lie bialgebroids. It follows from the above considerations that the deformation quantization problem for Lie bialgebroids differs essentially from its Lie bialgebra counterpart and is in fact more akin to the one for infinite-dimensional Poisson manifolds, i.e., it belongs to the no-go class. The origin of this obstruction can be traced back to an action of the graph complex of bi-oriented graphs (i.e., graphs with two oriented colors) on the deformation theory of Lie bialgebroids. Relaxing the orientation of one of the colors yields an action on the deformation theory of Lie-quasi bialgebroids (and their dual). As a corollary, we find an action of the Grothendieck–Teichmüller group on Lie-quasi bialgebroids generalising the one on Lie-(quasi) bialgebras. Theorem 1.2 (yes-go). The Grothendieck–Teichmüller group acts via Lie∞-automorphisms on the deformation complex of both Lie-quasi bialgebroids and quasi-Lie bialgebroids. Hence, the deformation quantization problem for Lie-quasi bialgebroids differs from its Lie bialgebroid counterpart and resembles more closely the one for Lie bialgebras, i.e., it belongs to the Yes-go class. We conjecture on the basis of these results the existence – given a Drinfeld associator – of a universal quantization for Lie-quasi bialgebroids (and their dual). The graph complex approach to deformation quantization5 is summed up in Table 1, where the original contribution of the present paper lies at c = 2.6 Organisation of this paper. The original universal model introduced by M. Kontsevich [32] takes advantage of the graded geometric interpretation of Poisson manifolds as7 dg symplectic manifolds of degree 1. Correspondingly, Section 2 reviews the formulation of Lie bialgebras and Lie bialgebroids (as well as their generalisations Lie-quasi, quasi-Lie and proto-Lie) as particular dg symplectic manifolds of degree 2. This graded geometric description of the deformation theory of Lie bialgebroids and generalisations thereof will be instrumental in formulating associated universal models in Section 4. The class of universal models introduced in this note involves multi-oriented graphs, as intro- duced in [77] and studied in [47] in the context of multi-oriented props and their representations on homotopy algebras with branes. The main definitions and results regarding the cohomology of multi-oriented graph complexes are reviewed in Section 3. Following these two review sections, we introduce our main results in Section 4. We start by reviewing the known action of the (one-colored) oriented graph complex on Lie-(quasi) bial- gebras in Section 4.2 and then move on to the Lie bialgebroid case in Section 4.3. Using the 5Note that the case d = 1 is somehow special among Table 1 as there is no associated quantization problem. The relevant cohomological class H1(fcGC1) ≃ K is therefore not viewed as an obstruction to quantization but rather as a non-trivial deformation of the symplectic Poisson bracket as a Lie algebra, yielding the Moyal bracket on symplectic manifolds (cf. Section 3.2 for additional details). 6The graph complex oidj fGCd featured in Table 1 lives in dimension d (cf. footnote 3) and involves graphs with c = i+ j directed colors, i of them are oriented, cf. Section 3 for details. 7In the remaining of the text, we use the prefix dg to refer to differential graded objects. 4 K. Morand Table 1. Classification of deformation quantization problems via graph cohomology. Graph complexes oidjfGCd Class name no-go yes-go middle way Model cohomology H0(fcGC1) ≃ 0 H0(fcGC2) ≃ grt1 H0(fcGC3) ≃ K H1(fcGC1) ≃ K H1(fcGC2) ?≃ 0 H1(fcGC3) ≃ 0 Oriented directions i d− 1 d− 2 d− 3 d c = i+ j Actions d = 1 c = 1 symplectic manifolds d = 2 Poisson (dim = ∞) Poisson (dim <∞) d = 3 Lie bialgebras proto-Lie bialgebras Lie-quasi bialgebras quasi-Lie bialgebras Courant algebroids c = 2 Lie bialgebroids Lie-quasi bialgebroids proto-Lie bialgebroids quasi-Lie bialgebroids cohomological results reviewed in Section 3, we prove our main results regarding the existence of an exotic Lie∞-structure for Lie bialgebroids (Theorem 1.1) and the action of the Grothendieck– Teichmüller group on Lie-quasi bialgebroids (Theorem 1.2). In view of the results of Section 4.3, we formulate two conjectures in Section 4.4: a no-go (Conjecture 4.11) regarding the existence of (universal) quantizations for Lie bialgebroids and a yes-go (Conjecture 4.12) regarding the one of Lie-quasi bialgebroids (and their dual). Two appendices conclude the present note. Appendix A reviews the (ungraded) geometric formulation of Lie bialgebroids and related notions, to be compared with the graded geomet- ric interpretation of Section 2. Appendix B contains explicit formulae and additional results regarding the graph cocycle generating the exotic Lie∞-structure for Lie bialgebroids. Conventions. Throughout the text, we work over a ground field K of characteristic zero. The operads introduced in the text live in the category of (graded) vector spaces over K. Given a graded K-vector space g := ⊕ k∈Z g k, the n-suspended graded vector space g[n] is defined via its homogeneous components g[n]k := gk+n. We will denote s : g[n] → g the corresponding suspension map of intrinsic degree n. 2 Graded geometry It is a well-known result that a Lie algebra structure on a vector space g yields a differential structure on the exterior algebra ∧•g∗ in the guise of the Chevalley–Eilenberg differential. The exterior algebra can equivalently be recast as the algebra of functions on the shifted vector space g[1] seen as a graded manifold of degree 1 on which the Chevalley–Eilenberg differential defines a homological vector field. Such a supergeometric formulation of Lie algebras was gener- alised by A.Yu. Vaintrob [67] who showed a bijective correspondence between dg manifolds (or NQ-manifolds) of degree 1 and Lie algebroids. On the other hand, it was shown by D. Royten- berg [58, 59] that dg symplectic manifolds (or NPQ-manifolds) of degree 1 (resp. of degree 2) are A Note on Multi-Oriented Graph Complexes 5 in bijective correspondence with Poisson manifolds (resp. Courant algebroids). Our aim in this section is to review how Lie bialgebra and Lie bialgebroid structures (and generalisations8) can be naturally recast as Hamiltonian functions for a specific graded Poisson algebra of functions on a graded manifold (as pioneered in [39], cf. [37, 58] for details and related constructions). We start by reviewing this graded geometric construction for Lie bialgebras (including proto- Lie, Lie-quasi and quasi-Lie bialgebras) in Section 2.1 and then move on to the Lie bialgebroid counterparts of these notions in Section 2.2. 2.1 Lie bialgebras Lie bialgebra structures (and generalisations thereof) on a vector space g can be conveniently encoded into particular Hamiltonian functions on the graded manifold T ∗(g[1]) ≃ (g⊕g∗)[1] with homogeneous coordinates9 { ξa 1 , ζa 1 } , with a ∈ {1, . . . ,dim g}. The latter is a graded symplectic manifold with symplectic 2-form Ω = dξa ∧ dζa of degree 2. The associated Poisson bracket of degree −2 acts on homogeneous functions in C∞((g⊕ g∗)[1]) as { f, g }g Ω = (−1)\|f \| ( ∂f ∂ξa ∂g ∂ζa + ∂f ∂ζa ∂g ∂ξa ) . (2.1) The graded Poisson bracket (2.1) can be seen as the graded geometric formulation of the “big bracket” (introduced by Y. Kosmann–Schwarzbach [35]) acting on ∧•(g⊕g∗) ≃ C∞((g⊕ g∗)[1]). Upgrading the graded symplectic manifold (g⊕ g∗)[1] to a dg symplectic manifold (or NPQ- manifold)10 allows to define various algebraic structures. A differential structure on a graded symplectic manifold is given by a vector field Q of degree 1 being homological (i.e., [Q,Q]Lie = 0) with respect to the graded Lie bracket of vector fields and compatible with the symplectic 2-form (i.e., LQΩ = 0). This last compatibility relation ensures11 that Q is necessarily a Hamiltonian vector field, i.e., there exists a function of degree 3 called the Hamiltonian satisfying the structure equation { H ,H }g Ω = 0 and such that Q := { H , · }g Ω . The most general function of degree 3 on (g⊕ g∗)[1] reads explicitly as12 H = −1 2fab cξaξbζc − 1 2Cc abζaζbξ c + 1 6φ abcζaζbζc + 1 6ψabcξ aξbξc, (2.2) where13 fab c = f[ab] c, Cc ab = Cc [ab], φabc = φ[abc] and ψabc = ψ[abc]. The Hamiltonian condition { H ,H }g Ω = 0 translates as a set of 5 constraints on the defining maps {f, C, φ, ψ}: • D1abc d := −fe[adfbc]e − ψe[abCc] ed = 0, (2.3) • D2d abc := −Cd e[aCe bc] − φe[abfed c] = 0, (2.4) • D3ab cd := 2fe[a [cCb] d]e − 1 2fab eCe cd − 1 2ψeabφ ecd = 0, (2.5) • D4 abcd := 1 2φ e[abCe cd] = 0, (2.6) • D5abcd := 1 2ψe[abfcd] e = 0. (2.7) A set of maps {f, C, φ, ψ} satisfying the constraints (2.3)–(2.7) form the components of a proto- Lie bialgebra on (g, g∗) (cf. Appendix A for a definition) whose deformation theory is therefore 8As reviewed in Appendix A. 9The subscript denotes the corresponding degree. 10Or equivalently, endowing the graded Poisson algebra of functions ( C ∞((g⊕ g∗)[1]), ·, {·, ·}gΩ ) with a com- patible differential. 11Via Cartan’s homotopy formula, cf. [59, Lemma 2.2]. 12The signs and coefficients are chosen for later convenience. 13Here and in the following, round (resp. square) brackets of indices will denote (skew)symmetrisation. 6 K. Morand controlled by the dg Lie algebra14 ( C∞((g⊕ g∗)[1]),Q, {·, ·}gΩ ) . Proto-Lie bialgebras thus con- stitute the most general notion in the bialgebra realm and other structures (Lie-quasi, quasi-Lie and Lie bialgebras) will be defined as particular cases thereof. The remainder of this section will therefore introduce several particular graded Poisson sub- algebras of C∞((g⊕ g∗)[1]) whose Hamiltonian functions will encode various sub-classes of proto-Lie bialgebras. Let us start by defining the subspace Ag Lie-quasi ⊂ C∞((g⊕ g∗)[1]) as Ag Lie-quasi := { f ∈ C∞((g⊕ g∗)[1]) ∣∣ f \|ζ=0 = 0 } . In plain words, the subspace Ag Lie-quasi is ob- tained by discarding all functions of the form ψa1···amξ a1 · · · ξam , for arbitrary values of m ≥ 0. It can be easily checked that Ag Lie-quasi is preserved by both the pointwise product of func- tions and the graded Poisson bracket (2.1) and thus defines a graded Poisson subalgebra of C∞((g⊕ g∗)[1]). The most general Hamiltonian function of Ag Lie-quasi reads as (2.2) with ψ ≡ 0, where the maps {f, C, φ} satisfy (2.3)–(2.6) with ψ ≡ 0. In particular, imposing ψ ≡ 0 in equation (2.3) ensures that the map f defines a genuine Lie algebra structure on g (while the structure defined on g∗ is still “quasi” due to the presence of φ). The resulting equations repro- duce the defining conditions of a Lie-quasi bialgebra on (g, g∗) as introduced by Drinfeld in [16] as semi-classicalisation of the notion of quasi-bialgebra.15 Dually to the previous case, one defines the graded Poisson subalgebra Ag quasi-Lie as the sub- space obtained by discarding all functions of the form φa1···anζa1 · · · ζan , for all n ≥ 0, i.e., Ag quasi-Lie := { f ∈ C∞((g⊕ g∗)[1]) ∣∣ f \|ξ=0 = 0 } ⊂ C∞((g⊕ g∗)[1]). The most general Hamil- tonian function of Ag quasi−Lie reads as (2.2) with φ ≡ 0, where the functions {f, C, ψ} satisfy (2.3)–(2.5) and (2.7) with φ ≡ 0. Dually to the Lie-quasi case, setting φ ≡ 0 in equation (2.4) ensures that the map C defines a genuine Lie algebra structure on g∗ (while the structure de- fined on g is only “quasi” Lie due to the presence of ψ). The resulting equations reproduce the defining conditions of a quasi-Lie bialgebra on (g, g∗) as introduced and studied in [3, 35]. Finally, let us define the subspace Ag Lie := { f ∈ C∞((g⊕ g∗)[1]) ∣∣ f \|ξ=0 = 0 and f \|ζ=0 = 0 } , i.e., Ag Lie is defined as the intersection Ag Lie := Ag Lie-quasi ∩ Ag quasi-Lie between the two previous Poisson subalgebras. The latter subspace can again be checked to be a graded Poisson sub- algebra of C∞((g⊕ g∗)[1]) obtained by discarding all functions of the form ψa1···amξ a1 · · · ξam and φa1···anζa1 · · · ζan for all m,n ≥ 0. In particular, the most general Hamiltonian function of Ag Lie reads as (2.2) with φ ≡ 0 and ψ ≡ 0, where the functions {f, C} satisfy (2.3)–(2.5) with φ ≡ 0 and ψ ≡ 0. In particular, constraint (2.3) ( resp. (2.4) ) ensures that the map f (resp. C) defines a genuine Lie algebra structure on g (resp. g∗). These two Lie algebras are furthermore compatible with each other due to (2.5) and hence define a Lie bialgebra on (g, g∗) (cf. (A.1)). We sum up the previous discussion in the following proposition: Proposition 2.1. Let g be a vector space. The following correspondences hold: � Hamiltonians in C∞((g⊕ g∗)[1]) are in bijective correspondence with proto-Lie bialgebra structures on (g, g∗). � “ Ag Lie-quasi “ Lie-quasi bialgebra “ . � “ Ag quasi-Lie “ quasi-Lie bialgebra “ . � “ Ag Lie “ Lie bialgebra “ . This interpretation of the deformation theory for Lie bialgebras and generalisations as graded Poisson algebras will be put to use in Section 4 where will be discussed universal models thereof. 14Note that the graded Poisson bracket has intrinsic degree −2 on C ∞((g⊕ g∗)[1]). To recover the usual grading, one needs to consider the 2-suspension C ∞((g⊕ g∗)[1])[2]. 15Remark that Lie-quasi bialgebras were denoted “quasi-Lie bialgebras” in [16]. We follow the terminology used in [37] where the term Lie-quasi bialgebras is used for Lie algebras which fail to be Lie bialgebras (so that they are only “quasi” bialgebras) while the term quasi-Lie was reserved for the dual counterpart (not considered in [16]) where the Jacobi identity for f is only “quasi” satisfied. A Note on Multi-Oriented Graph Complexes 7 In the next section, we turn to the generalisation of this graded geometric interpretation to the larger class of Lie bialgebroids and variations thereof. 2.2 Lie bialgebroids Letting E π→ M be a vector bundle over the smooth (finite-dimensional) manifold16M , the relevant graded Poisson algebra is the algebra of functions of the graded symplectic manifold T ∗[2]E[1], defined as the (2-shifted) cotangent bundle of the (1-shifted17) vector bundle E, and denoted E ≡ T ∗[2]E[1] in the following. The graded manifold E is of degree 2 and can be locally18 coordinatised by the set of homogeneous coordinates { xµ 0 , ξa 1 , ζa 1 , pµ 2 } so that the symplectic 2-form of degree 2 can be written as Ω = dxµ ∧ dpµ + dξa ∧ dζa. The associated Poisson bracket of degree −2 acts as follows on homogeneous functions f, g ∈ C∞(E ):{ f, g }E Ω = ∂f ∂xµ ∂g ∂pµ − ∂f ∂pµ ∂g ∂xµ + (−1)\|f \| ( ∂f ∂ξa ∂g ∂ζa + ∂f ∂ζa ∂g ∂ξa ) . (2.8) The latter is sometimes referred to as the “big bracket” for Lie bialgebroids. Upgrading the graded symplectic manifold E to a dg symplectic manifold will allow to define various geomet- ric structures. Following the same path as in the bialgebra case, we introduce a compatible differential through a Hamiltonian function. The most general function of degree 3 on E reads H = ρa µ(x)ξapµ − 1 2fab c(x)ξaξbζc +Ra\|µ(x)ζapµ − 1 2Cc ab(x)ζaζbξ c + 1 6φ abc(x)ζaζbζc + 1 6ψabc(x)ξ aξbξc, (2.9) where {ρ, f,R,C, φ, ψ} are functions on the base space M , with symmetries fab c = f[ab] c, Cc ab = Cc [ab], φabc = φ[abc] and ψabc = ψ[abc]. Imposing the Hamiltonian constraint { H ,H }E Ω = 0 yields a set of 9 conditions on the defining functions {ρ, f,R,C, φ, ψ} that we denote as follows • C1abµ := 2ρ[a λ∂λρb] µ − ρc µfab c +Rc\|µψcab = 0, (2.10) • C2abcd := ρ[a λ∂λfbc] d − fe[a dfbc] e + 1 3R d\|λ∂λψabc − ψe[abCc] ed = 0, (2.11) • C3ab\|µ := 2R[a\|λ∂λR b]\|µ −Rc\|µCc ab + ρc µφcab = 0, (2.12) • C4dabc := R[a\|λ∂λCd bc] − Cd e[aCe bc] + 1 3ρd λ∂λφ abc − φe[abfed c] = 0, (2.13) • C5µν := Ra(µρa ν) = 0, (2.14) • C6ab\|µ := ρa λ∂λR b\|µ −Rb\|λ∂λρa µ − ρc µCa bc −Rc\|µfca b = 0, (2.15) • C7abcd := ρ[a λ∂λCb] cd +R[c\|λ∂λfab d] + 2fe[a [cCb] d]e − 1 2fab eCe cd − 1 2ψeabφ ecd = 0, (2.16) • C8abcd := 1 3R [d\|λ∂λφ abc] + 1 2φ e[abCe cd] = 0, (2.17) • C9abcd := 1 3ρ[d λ∂λψabc] + 1 2ψe[abfcd] e = 0. (2.18) The latter constraints identify with the component expressions of the defining conditions of a proto-Lie bialgebroid on (E,E∗) (compare with (A.2)–(A.6)). The graded Poisson alge- bra C∞(E ) admits several subalgebras defining in turn various sub-classes of proto-Lie bial- gebroids. A convenient way to characterise these Poisson subalgebras is as vanishing ideals of 16The restriction from the “algebroid” case to the “algebra” case can be done by assuming that M is the one-point manifold so that E ≃ g becomes a K-vector space. 17Here and in the following, A[n] will denote the vector bundle obtained by shifting the grading of the fiber of the vector bundle A by n. 18Correspondingly, all formulae appearing in this note will be local. 8 K. Morand particular Lagrangian submanifolds.19 As noted by D. Roytenberg [60], both E[1] and E∗[1] are Lagrangian submanifolds of E , thus motivating to consider the Poisson subalgebras of func- tions vanishing on them. We start by defining the Poisson subalgebra AE Lie-quasi ⊂ C∞(E ) as the vanishing ideal of E[1]. In plain words, the subalgebra AE Lie-quasi is obtained by discarding all functions of the form ψa1···am(x)ξ a1 · · · ξam , for all m ≥ 0. The most general Hamiltonian function of AE Lie-quasi reads as (2.9) with ψ ≡ 0, where the functions {ρ, f,R,C, φ} satisfy (2.10)– (2.17) with ψ ≡ 0. In particular, setting ψ ≡ 0 in equations (2.10)–(2.11) ensures that the pair {ρ, f} defines a genuine Lie algebroid structure on E. The resulting equations reproduce the defining conditions of a Lie-quasi bialgebroid on (E,E∗). The pair ( (E ,Ω,H ), E[1] ) defines a Manin pair,20 in the terminology of Roytenberg [60]. Dually to the previous case, we define the graded Poisson subalgebra AE quasi-Lie ⊂ C∞(E ) as the vanishing ideal of E∗[1], i.e., as the subspace obtained by discarding all functions of the form φa1···an(x)ζa1 · · · ζan with n ≥ 0. The most general Hamiltonian function of AE quasi-Lie reads as (2.9) with φ ≡ 0, where the functions {ρ, f,R,C, ψ} satisfy (2.10)–(2.16) and (2.18) with φ ≡ 0. Dually to the Lie-quasi case, setting φ ≡ 0 in equations (2.12)–(2.13) ensures that the pair {R,C} defines a genuine Lie algebroid structure on E∗ (while the structure defined on E by {ρ, f} is still “quasi” due to the presence of ψ). The resulting equations reproduce the defining conditions of a dual structure on (E,E∗), dubbed quasi-Lie bialgebroid in [37]. Quasi-Lie bialgebroids are then equivalently characterised as Manin pairs of the form ( (E ,Ω,H ), E∗[1] ) . We conclude by defining the subspace AE Lie := AE Lie-quasi ∩ AE quasi-Lie. The latter subspace can again be checked to be a graded Poisson subalgebra of C∞(E ) obtained by discarding all functions of the form ψa1···am(x) ξ a1 · · · ξam or φa1···an(x)ζa1 · · · ζan , for allm,n ≥ 0. In particular, the most general Hamiltonian function of AE Lie reads as (2.9) with φ ≡ 0 and ψ ≡ 0, where the functions {ρ, f,R,C} satisfy (2.10)–(2.16) with φ ≡ 0 and ψ ≡ 0. Constraints (2.10)–(2.11) (resp. (2.12)–(2.13)) ensure that the pair {ρ, f} (resp. {R,C}) defines a genuine Lie algebroid structure on E (resp. E∗). These two Lie algebroids are furthermore compatible with each other due to (2.13)–(2.16) and hence define a Lie bialgebroid structure on (E,E∗) (cf. Appendix A). Lie bialgebroids are furthermore equivalently characterised21 as Manin triples22 of the form ((E ,Ω,H ), E[1], E∗[1]). As usual, gauge transformations for the Hamiltonian function (2.9) are generated by func- tions23 of degree 2 in C∞(E ) reading explicitly: X = Xµpµ + λabξ bζa − 1 2Λ abζaζb − 1 2ωabξ aξb, (2.19) 19Letting N be a (graded) manifold, the vanishing ideal of a (graded) submanifold C ⊂ N is defined as the subalgebra of functions IC := { f ∈ C ∞ (N ) ∣∣ f \|C = 0 } . Moreover, a (graded) submanifold C of a (graded) symplectic manifold (N ,Ω) is said to be Lagrangian if it is maximally isotropic, i.e., 1. The restriction of the symplectic form Ω to C vanishes, i.e., Ω\|C = 0. 2. The submanifold C has maximal dimension dim C = 1 2 dimN . For graded Lagrangian submanifolds, it will be assumed that the underlying bosonic manifold of C (whose algebra of functions is coordinatised by coordinates of degree 0) identifies with the one of N . Lagrangian submanifolds are coisotropic, so that the corresponding vanishing ideal is closed under the (graded) Poisson bracket associated to Ω and hence is a Poisson subalgebra of the algebra of functions on N . 20A Manin pair of degree n is defined as a dg symplectic manifold (N ,Ω,H ) of degree n supplemented with a dg Lagrangian submanifold (also called a Λ-structure [62]), i.e., a graded Lagrangian submanifold C ⊂ N such that H ∈ IC , with IC the vanishing ideal of C. Manin pairs of degree 0 correspond to a pair formed by a (bosonic) symplectic manifold endowed with a Lagrangian submanifold. Manin pairs of degree 1 are in bijective correspondence with pairs composed of a Poisson manifold together with a coisotropic submanifold while Manin pairs of degree 2 identify with Courant algebroids endowed with a Dirac structure [62]. 21We refer to Appendix A for analogues of these statements within the framework of Courant algebroids. 22A Manin triple is a dg symplectic manifold supplemented with two transverse dg Lagrangian submanifolds C,D ⊂ N such that H ∈ IC and H ∈ ID. 23Gauge transformations for Lie-quasi (resp. quasi-Lie) bialgebroids are generated by (2.19) with ω ≡ 0 (resp. Λ ≡ 0), and with Λ ≡ 0 ≡ ω for Lie bialgebroids. A Note on Multi-Oriented Graph Complexes 9 so that the gauge algebra for proto-Lie bialgebroids is isomorphic to Γ(TM ) ⊕ End ( Γ(E) ) ⊕ Γ ( ∧2E ) ⊕ Γ ( ∧2E∗) as a vector space. We refer to [60, Section 5] for a careful treatment of the group structure of these gauge transformations. The various parameters in (2.19) can be interpreted24 in terms of infinitesimal morphisms of proto-Lie bialgebroids as follows: � X ∈ Γ(TM ): diffeomorphism of M , � λ ∈ End ( Γ(E) ) : rotation of the fibers of (E,E∗), � Λ ∈ Γ ( ∧2E ) twist of Lie-quasi bialgebroids, � ω ∈ Γ ( ∧2E∗) twist of quasi-Lie bialgebroids, where the denomination twist refers to the following construction: given a proto-Lie bialgebroid on (E,E∗) defined by the Hamiltonian function H , one can define a new proto-Lie bialgebroid structure on (E,E∗) via twisting25 by an arbitrary bivector Λ ∈ Γ ( ∧2E ) . Explicitly, the twisted proto-Lie bialgebroid HΛ is obtained by performing a canonical transformation generated by the flow of the Hamiltonian vector field XΛ := { −Λ, · }E Ω where Λ := 1 2Λ abζaζb is a function of degree 2 in C∞(E ). Explicitly, a twist by Λ corresponds to the canonical transformation26 xµ Λ−→ xµ, pµ Λ−→ pµ − 1 2∂µΛ abζaζb, ξa Λ−→ ξa − Λabζb, ζa Λ−→ ζa, which amounts to a shift of the components of H (cf. equation (A.7) in Appendix A for explicit expressions). The previous canonical transformation maps Lie-quasi bialgebroids to Lie-quasi bialgebroids but generically fails to preserve quasi-Lie bialgebroids, as is consistent with the fact that Λ /∈ AE quasi-Lie so that Γ ( ∧2E ) is not part of the gauge algebra for quasi-Lie bialgebroids.27 We refer to Appendix A for more details on the twisting procedure. We sum up the previous discussion in the following proposition, generalising Proposition 2.1 to bialgebroids: Proposition 2.2. Let E π→ M be a vector bundle. The following correspondences hold: � Hamiltonians in C∞(E ) are in bijective correspondence with proto-Lie bialgebroid struc- tures on (E,E∗). � “ AE Lie-quasi “ Lie-quasi bialgebroid “ . � “ AE quasi-Lie “ quasi-Lie bialgebroid “ . � “ AE Lie “ Lie bialgebroid “ . As noted previously, the graded geometric interpretation of algebro-geometric structures is instrumental to the construction of corresponding universal models, formulated in terms of graph complexes. The next section will introduce the relevant graph complexes which will be shown in Section 4 to act on the various (sub)-algebras previously introduced. 24We refer to Appendix A for more details on this interpretation. 25The twisting procedure was introduced in [16] for quasi-Hopf algebras, in [35] for quasi-Lie bialgebras and in [60] for quasi-Lie bialgebroids. 26Dually, one can consider twisting by a 2-form field ω ∈ Γ ( ∧2E∗) corresponding to the canonical transformation generated by the flow of the Hamiltonian vector field Xω := { −ω, · } , where ω := 1 2 ωab ξ aξb: xµ ω−→ xµ, pµ ω−→ pµ − 1 2 ∂µωabξ aξb, ξa ω−→ ξa, ζa ω−→ ζa − ωabξ b (2.20) which amounts to a shift of the components of H as in equation (A.8). 27Dually, the canonical transformation (2.20) preserves the space of quasi-Lie bialgebroids, but generically maps Lie-quasi bialgebroids to proto-Lie bialgebroids. This is consistent with the fact that ω /∈ AE Lie-quasi so that Γ ( ∧2E∗) is not part of the gauge algebra for quasi-Lie bialgebroids. 10 K. Morand 3 (Multi)-oriented graph complexes The aim of this section is to review the definition and main results regarding multi-oriented graph complexes and their cohomology, as introduced and studied in [47, 77, 78] (cf. also [48] for a review). Graph complexes are most clearly defined as deformation complexes of a suitable morphism of operads [49].28 We start by introducing the relevant graph operads of multi-directed and multi-oriented graphs from a combinatorial point of view (Section 3.1) before moving to the definition of the associated graph complexes. We conclude by discussing known results regarding the cohomology of multi-oriented graph complexes (Section 3.2) by putting the emphasis on some particular classes relevant for our purpose (cf. Section 4). 3.1 Directed, oriented and sourced graphs We will denote graN,k (resp. dgraN,k) the set of multi(di)graphs29 with N vertices and k directed edges. The set dcgraN,k of multi-directed graphs with c colors is defined as the set of ordered pairs (γ, c) where: � γ ∈ dgraN,k is a multidigraph. We will denote Vγ (resp. Eγ) the set of vertices (resp. edges) of γ. � c stands for a map c : Eγ × [c − 1] → {+,−} where c ∈ N stands for the total number of colors30 and [c− 1] := {1, 2, . . . , c− 1}. A pictorial representation of a multi-directed graph in dcgraN,k can be given by decorating each directed edge of the underlying multidigraph in dgraN,k with c− 1 additional arrows of different colors (cf. Figure 1 for an example). 1 2 34 i ii iii ivv Figure 1. Example of a multi-directed graph in d4gra4,5. The direction of the arrow of color i on the edge e is aligned with the one of the black arrow if c(e, i) = + and opposite to it if c(e, i) = −. There is a natural right-action of the permutation group SN (resp. Sk) on elements of dcgraN,k by permutation of the labeling of vertices (resp. edges). Operads. For all d ∈ N∗, we define the collection {dcGrad(N)}N≥1 of SN -modules: dcGrad(N) := ⊕ k≥0 ( K 〈 dcgraN,k 〉 ⊗Sk sgn ⊗\|d−1\| k ) [k(d− 1)], (3.1) where sgnk denotes the 1-dimensional sign representation of Sk. The subscript stands for taking coinvariants with respect to the diagonal right action of Sk and the term between brackets denotes degree suspension. In plain words, this means that edges carry an intrinsic degree 28Or equivalently, as convolution Lie algebras constructed from suitable graph operads. 29Recall that multi(di)graphs are undirected (resp. directed) graphs allowed to contain both loops and multiple edges. 30That is, including the underlying (black) arrow of the multidigraph. A Note on Multi-Oriented Graph Complexes 11 1 − d and are bosonic for d odd and fermionic for d even. The S-module {dcGrad(N)}N≥1 can further be given the structure of an operad31 by endowing it with the usual equivariant partial composition operations ◦i : dcGrad(M) ⊗ dcGrad(N) → dcGrad(M + N − 1), cf. Figure 2 for an example and, e.g., of [53, Section 4] for more details. 1 2 3 i iiiii ◦2 1 2 i = 1 4 i iv iii 2 3 ii = 4 3 21 i ii iii iv + 4 2 31 i ii iii iv + 1 2 4 3 i iiiii iv + 1 3 4 2 i iiiii iv Figure 2. Example of partial composition ◦2 : d2Grad(3)⊗ d2Grad(2) → d2Grad(4). There is a natural sequence of embeddings of operads32 Grad O⃗r ↪−→ dGrad O⃗r ↪−→ d2Grad O⃗r ↪−→ d3Grad O⃗r ↪−→ · · · (3.2) given by mapping each graph in dcGrad to a sum of graphs in dc+1Grad where the summation runs over all the possible ways to orient the arrow of the additional direction. We call such mapping the orientation morphism O⃗r : dcGrad ↪−→ dc+1Grad (cf. Figure 3 for an example). O⃗r ( 1 2 i ii ) = 1 2 i ii + 1 2 i ii + 1 2 i ii + 1 2 i ii Figure 3. The orientation morphism O⃗r : dGrad ↪−→ d2Grad. Denoting Lie {1− d} the (1− d)-suspended Lie operad,33 there is an operad morphism γ0 : Lie {1− d} → dcGrad sending the generator ∈ Lie {1− d} (2) to the graph 1 2 := 1 2 + (−1)d 2 1 where the graph 1 2 ∈ dcGrad(2) is obtained by decorating 1 2 with c − 1 additional colors and summing over all the possible orientations.34 A multidigraph in dgraN,k will be said oriented (or acyclic) if it does not contain cycles.35 Contrariwise, it will be said non-oriented (or cyclic) if it contains at least one cycle, cf. Figure 4. The subset of oriented multidigraphs will be denoted ograN,k ⊂ dgraN,k. This definition can 31The identity element id ∈ dcGrad(1) is defined as the graph id := 1 . 32Here, Grad ≡ d0Grad (resp. dGrad ≡ d1Grad) stands for the operad of one-colored undirected (resp. directed) graphs. 33Recall that representations of Lie {1− d} on a vector space g are in bijective correspondence with Lie algebra structures on g[1− d], hence the graded Lie bracket on g has intrinsic degree 1− d. 34In other words, 1 2 := (O⃗r)c−1 ( 1 2 ) . For example, 1 2 := 1 2 + 1 2 in d2Grad. 35Recall that a cycle (or wheel) is a (non-trivial) directed path from a vertex to itself. 12 K. Morand 21 3 , 21 3 Figure 4. Example of oriented (left) and cyclic (right) graph in dgra3,3. be extended to multi-directed graphs by defining the subset36 oidjgraN,k ⊆ dcgraN,k of multi- directed graphs with c = i + j directions for which there exists a subset of i directions – black and/or colored – such that there are no cycles made of the corresponding arrows.37 Substi- tuting oidjgraN,k in place of dcgraN,k in (3.1) allows to define the collection of SN -modules {oidjGrad(N)}N≥1 for all i, j, d ≥ 0. It is easy to check that the latter is closed under partial compositions and hence defines a suboperad oidjGrad ⊆ dcGrad of multi-oriented graphs. Note that the graph 1 2 is (trivially) multi-oriented and hence defines a morphism of operads γ0 : Lie {1− d} → oidjGrad for all i, j, d ≥ 0. Oriented graphs belong to the larger subset of multidigraphs possessing a source, i.e., a vertex admitting only outgoing arrows. More generally, the suboperad of multi-directed graphs with c = \|k\|+ j directions such that \|k\| directions are sourced will be denoted skdjGrad ⊆ dcGrad for all k ∈ Z and d, j ≥ 0, where negative values of k correspond to directions admitting a sink, i.e., a vertex with only ingoing arrows. Finally, the suboperad of graphs such that j directions admit at least one source and one sink will be denoted si,−idjGrad, for all i, j, d ≥ 0. It is a well-known result in graph theory that oriented graphs admit at least one source and one sink (see, e.g., [78] for a statement) so that we have a sequence of inclusions o\|k\|djGrad ⊆ s\|k\|,−\|k\|djGrad ⊆ skdjGrad for all k ∈ Z and d, j ≥ 0. Graph complexes. Given the multi-oriented graph operad oidjGrad, one defines the dg Lie algebra of multi-oriented graphs oidjfGCd as the deformation complex oidjfGCd := Def ( Lie {1− d} γ0→ oidjGrad ) of the morphism of operads γ0. 38 As a graded vector space, oidjfGCd is defined as • d even: oidjfGCd := ⊕ N≥1 ( oidjGrad(N)[d(1−N)] )SN , (3.3) • d odd: oidjfGCd := ⊕ N≥1 ( oidjGrad(N)⊗ sgnN [d(1−N)] )SN , (3.4) where the terms between brackets denote degree suspension while the superscript stands for taking invariants with respect to the right action of SN , with sgnN the 1-dimensional signature representation of SN . In other words, vertices are bosonic for d even and fermionic for d odd. According to the degree suspension in (3.1) and (3.3)–(3.4), the degree of an element39 γ ∈ oidjfGCd with N vertices and k edges is given by \|γ\| = d(N − 1) + k(1− d). The graded Lie bracket [·, ·] of degree 0 on oidjfGCd is defined as usual in terms of the partial composition operations (cf., e.g., [53, Section 4.2] for details). The differential δ := [ΥS, ·] is 36By convention, we identify o0d0graN,k ≡ graN,k, o0d1graN,k ≡ dgraN,k and o1d0graN,k ≡ ograN,k. 37As an example, the graph of Figure 1 belongs to o2d2gra4,5 as it does not contain cycles for black and yellow arrows (although it does so for red and blue arrows). In the right-hand side of Figure 3, the first and fourth graphs belong to o2d0gra2,2 while the second and third graphs belong to o1d1gra2,2. 38We refer to references [41, 49] for details. 39Note that the graph degree is insensitive to the number i of oriented directions. A Note on Multi-Oriented Graph Complexes 13 defined by taking the adjoint action with respect to the Maurer–Cartan element40 ΥS := 1 2 = 1 2 + (−1)d 2 1 . Note to conclude that the dg Lie algebra o\|k\|djfGCd of multi-oriented graphs is a sub-dg Lie algebra of the dg Lie algebra of multi-sourced/sinked graphs skdjfGCd defined for all k ∈ Z and d, j ≥ 0 by41 substituting skdjGrad(N) in place of okdjGrad(N) in (3.3)–(3.4). The various graph operads and complexes discussed in the present section are summarised in Table 2. Table 2. Summary of graph operads and complexes (and their connected variants) in dimension d. Undirected graphs with one color (Kontsevich’s graphs) Grad cGrad fGCd fcGCd d ≥ 0 Multi-directed graphs with c colors dcGrad dccGrad dcfGCd dcfcGCd c, d ≥ 0 Multi-oriented graphs with c = i+ j colors and i oriented directions oidjGrad oidjcGrad oidjfGCd oidjfcGCd i, j, d ≥ 0 Multi-sourced/sinked graphs with c = \|k\|+ j colors and \|k\| sourced/sinked directions skdjGrad skdjcGrad skdjfGCd skdjfcGCd j, d ≥ 0, k ∈ Z Multi-sourced/sinked graphs with c = i+ j colors and i directions being both sourced and sinked si,−idjGrad si,−idjcGrad si,−idjfGCd si,−idjfcGCd i, j, d ≥ 0 3.2 Cohomology Having introduced the graph complex of multi-directed graphs as well as its sub-complexes of multi-sourced and multi-oriented graphs, we conclude this review section by collecting some known facts about their respective cohomology.42 We start by introducing the suboperad oidjcGrad ⊂ oidjGrad spanned by connected graphs, which in turns yields a sub-dg Lie alge- bra oidjfcGCd ⊂ oidjfGCd. 43 This is justified by the fact that the cohomology of the graph complex oidjGrad is captured by its connected part,44 as follows from H•(oidjfGCd) = Ô ( H•(oidjfcGCd)[−d] ) [d], where Ô(g) denotes the completed symmetric algebra associated with the graded vector space g. As far as cohomology is concerned, one can therefore restrict the analysis to the connected part of the above complexes. Moreover, a further simplification comes from the fact that the 40For example, the graph ΥS := 2 3 = 1 2 + 1 2 +(−1)d ( 2 1 + 2 1 ) is a Maurer– Cartan element in oidjfGCd with i, j, d ≥ 0 and i+ j = 2. 41Equivalently, one can define skdj fGCd := Def ( Lie {1− d} γ0→ skdjGrad ) . 42See [47, Section 5] and [48, Section 7] for reviews. 43Similarly, one can introduce the suboperads of connected multi-directed graphs djcGrad and connected multi- sourced graphs sidjcGrad as well as their corresponding sub-dg Lie algebras dj fcGCd and sidj fcGCd. 44We refer to [71] and [72] for the cases i = 0, 1 respectively and to [47] for a general statement. 14 K. Morand cohomology of the various complexes previously introduced can be related to one another. This is embodied by the following important theorem, due to T. Willwacher45 for the case i = 0 and to M. Živković [77, 78] for its generalisation to arbitrary i ≥ 0. Theorem 3.1. For all integers i, j, d ≥ 0 and k ∈ Z: 1. The inclusion oidjfcGCd ↪−→ oidj+1fcGCd is a quasi-isomorphism. 2. There is a quasi-isomorphism oidjfcGCd −→ oi+1djfcGCd+1. 3. The inclusion o\|k\|djfcGCd ↪−→ skdjfcGCd is a quasi-isomorphism. The quasi-isomorphism of the second item preserves the additional grading provided by the first Betti number.46 A few comments are in order. We start by noting that the sequence of embeddings of operads (3.2) induces a sequence47 of injective quasi-isomorphisms of complexes48 fcGCd ∼ ↪−→ O⃗r dfcGCd ∼ ↪−→ O⃗r d2fcGCd ∼ ↪−→ O⃗r d3fcGCd ∼ ↪−→ O⃗r · · · . Hence, adding extra colored direction does not affect the cohomology. The first item of Theo- rem 3.1 asserts that this result generalises to multi-oriented graphs where the number i of oriented directions is kept fixed. The situation gets more interesting when orienting extra directions since adding oriented directions does change the cohomology. More precisely, the second item of the above theorem relates the cohomology of a given multi-oriented graph complex to the one of a (less oriented) complex in higher dimension. We will comment more on this important result in the next paragraph. Before doing so, let us note that the third item asserts that sourcing directions also affects the cohomology, but in a way that is completely captured by the cohomology of the (smaller) multi-oriented graph complex. Hence, the computation of the cohomology of the multi-sourced/sinked graph complex can always be reduced to computing the cohomology of the multi-oriented graph complex. In the next section, we will therefore express our results in terms of the smaller multi-oriented graph complex when possible. For later use, Table 3 summarises the cohomology in degrees 0, 1 and 2 of the (undirected) graph complex in dimensions d = 1, 2, 3.49 Table 3. Cohomology in low dimension and degree. H0(fcGCd) H1(fcGCd) H2(fcGCd) d = 1 0 K ⟨Θ⟩ K ⟨L3⟩ d = 2 grt1 0? ? d = 3 K ⟨L3⟩ 0 0 45See [71, Appendix K] for the first item (cf. also [13]), [72] for the second and [73] for the third. 46The first Betti number is defined as b := k −N + 1 for a connected graph with N vertices and k edges. 47Here, the dg Lie algebra fcGCd ≡ d0fcGCd (resp. dfcGCd ≡ d1fcGCd) stands for the usual Kontsevich graph complex of connected undirected (resp. directed) graphs. 48See [13, 71] for the first arrow and [47] for a general statement. 49The cocycle L3 stands for the triangle loop 21 3 . Regarding the Θ-cocycle and the Grothendieck– Teichmüller algebra grt1, see below. A Note on Multi-Oriented Graph Complexes 15 Climbing the dimension ladder. Informally, the second item of Theorem 3.1 allows to map familiar structures in low dimensions50 to novel incarnations thereof in higher dimen- sions.51 The most striking example of such hierarchy of structures stems from another impor- tant theorem of T. Willwacher [71] showing the existence of an isomorphism of Lie algebras H0(fcGC2) ≃ grt1 where grt1 denotes the infinite-dimensional Grothendieck–Teichmüller alge- bra. Combined with the second item of Theorem 3.1, we obtain a sequence H0(fcGC2) ≃ H0(o1d•fcGC3) ≃ H0(o2d•fcGC4) ≃ · · · ≃ grt1 of incarnations of grt1 in arbitrary dimensions d ≥ 2.52 Different incarnations yield different actions of the Grothendieck–Teichmüller group53 on various algebro-geometric structures. In its original incarnation via directed cocycles in H0(d1fGCd), the Grothendieck–Teichmüller group naturally acts via Lie∞-automorphisms on the Schouten Lie algebra of polyvector fields Tpoly on (finite-dimensional) manifolds. This yields an action on the space of universal formality maps hence on the space of universal quantization maps for finite-dimensional Poisson manifolds [12, 29, 32, 34, 45, 71]. In dimension 3, there is a natural action of GRT1 via oriented cocycles in H0(o1d0fcGC3) on the deformation complex of Lie bialgebras54 hence on the space of universal formality maps related to the deformation quantization of Lie bialgebras [51, 50, 72].55 Another important result for our story concerns the manifold incarnations of the Θ-cocycle ∈ dgra2,3 spanning the first cohomology class of fcGC1, i.e., H 1(fcGC1) ≃ K ⟨Θ⟩. Again, applying the second item of Theorem 3.1 results in a sequence of isomorphisms H1(fcGC1) ≃ H1(o1d•fcGC2) ≃ H1(o2d•fcGC3) ≃ · · · ≃ K. In its original incarnation as the cocycle spanning H1(fcGC1), the Θ-graph can be recursively extended56 to a non-trivial Maurer–Cartan element ΥΘ := • •+ + · · · in the graded Lie algebra ( fcGC1, [·, ·] ) , see, e.g., [30]. This Maurer–Cartan element is mapped to the Moyal star- commutator [26, 54] via the natural action of fcGC1 on the algebra of functions of a (bosonic) symplectic manifold. Considering the case d = 2 yields another important incarnation of the Θ- graph as the oriented Kontsevich–Shoikhet cocycle – denoted Θ2 in the following, cf. Figure 5 – and spanning H1(o1d•fcGC2). The latter first appeared implicitly in [56] (see also [10, 70]) as the obstruction to the existence of a cycle-less universal quantization of Poisson manifolds beyond order ℏ3. It then appeared explicitly in [65] as the obstruction to formality in infinite dimension while its graph theoretical interpretation – as the avatar of the Θ-cocycle in dimension 2 – has been elucidated in [72]. The corresponding Maurer–Cartan element ΥΘ2 induces an exotic (and essentially unique) universal Lie∞-structure on polyvector fields deforming non-trivially57 the 50See footnote 3 for the interpretation of the dimension d. 51Since the quasi-isomorphism of the second item of Theorem 3.1 preserves both the graph degree and the first Betti number, a connected cocycle γ ∈ dgraN,k in dimension d is mapped to a cocycle γ′ ∈ dgraN′,k′ in dimension d+ 1 with N ′ = k + 1 and k′ = 2k −N + 1. More generally, the incarnation of a cocycle γ ∈ dgraN,k of H•(oidj fcGCd) in dimension d′ > d is a graph γ′ ∈ dgraN′,k′ with N ′ = N + (d′ − d)b and k′ = k + (d′ − d)b with b the first Betti number. 52For example, the tetrahedron graph t2 ∈ H0(fcGC2) is mapped to an oriented cocycle t3 ∈ H0(o1d•fcGC3) with N = 7 vertices and k = 9 edges (see footnote 51). 53Recall that the Grothendieck–Teichmüller group is defined by exponentiation of the pro-nilpotent Grothen- dieck–Teichmüller algebra, i.e., GRT1 ≡ exp(grt1), see [69] for a review. 54As reviewed in Section 4.2. 55The fact that quantization of Lie bialgebras involves oriented graphs was already recognised in [18]. 56Potential obstructions in promoting the Θ-graph to a full Maurer–Cartan element lie in H2(fcGC1) ≃ K ⟨L3⟩, cf. Table 3. Since the obstruction to the prolongation of the Θ-graph at order k ≥ 2 has Betti number k + 2, it never hits the loop graph L3 (of Betti number 1) so that the prolongation is unobstructed at all orders and can be performed recursively. The argument carries identically to incarnations of the Θ-graph in higher dimensions. 57For finite-dimensional manifolds, this exotic Lie∞-structure can be shown to be isomorphic to the standard Schouten bracket, although in a highly non-trivial way, cf. [50] for explicit transcendental formulae. 16 K. Morand Schouten algebra on infinite-dimensional manifolds [65]. The latter can then be considered as the avatar in d = 2 of the Moyal star-commutator in d = 1. 21 3 4 − 2 21 3 4 + 21 3 4 Figure 5. Kontsevich–Shoikhet cocycle Θ2 ∈ H1(o1d0fcGC2). Of special interest for our purpose is the incarnation of the Θ-cocycle in dimension 3, dubbed Θ3 in the following. The latter is a combination of bi-oriented graphs with N = 6 and k = 758 (see Appendix B) that will be argued to provide an obstruction to the universal quantization of Lie bialgebroids in Section 4. 4 Universal models The aim of the present section is to introduce universal models of multi-oriented graphs (cf. Section 3) for Lie bialgebroids – and variations thereof – using the graded geometric picture reviewed in Section 2. We start by providing an abstract characterisation of universal models and emphasise their relevance to address questions related to formality theory and deformation quantization. We then review universal models for Lie bialgebras (and their “quasi” versions) before moving on to the Lie bialgebroid case. We conclude by discussing the implications of our results regarding the deformation quantization problem for Lie bialgebroids. 4.1 Armchair formality theory Let (g, δ, [·, ·]g) be a dg Lie algebra and denote (H(g), 0, [·, ·]H(g)) the associated cohomology endowed with the canonical dg Lie algebra structure inherited from g (with trivial differen- tial). Let furthermore Φ: (H(g), 0) ∼−→ (g, δ) be a quasi-isomorphism of complexes. Generically, Φ fails to preserve the additional graded Lie structures – i.e., Φ([x, y]H(g)) ̸= [Φ(x),Φ(y)]g with x, y ∈ H(g) – so that Φ is not a morphism of dg Lie algebras. According to the ho- motopy transfer theorem,59 any dg Lie algebra g is quasi-isomorphic (as a Lie∞-algebra) to its cohomology H(g) endowed with a certain Lie∞-structure (H(g), l) deforming the canoni- cal dg Lie algebra structure (H(g), 0, [·, ·]H(g)), i.e., l1 = 0, l2 = [·, ·]H(g) and the higher order brackets l>2 are transferred from the dg Lie algebra structure on g. In other words, any quasi- isomorphism of complexes Φ: (H(g), 0) ∼−→ (g, δ) can be upgraded to a quasi-isomorphism of Lie∞-algebras U : (H(g), l) ∼−→ (g, δ, [·, ·]g) with U1 = Φ. If the higher brackets l>2 vanish, then (H(g), 0, [·, ·]H(g)) and (g, δ, [·, ·]g) are quasi-isomorphic as Lie∞-algebras and g is said to be formal. The homotopy transfer theorem thus allows to reduce questions regarding formal- ity (such as existence of formality maps and their classification) to the study of the space of Lie∞-structures deforming the canonical dg Lie structure (H(g), 0, [·, ·]H(g)). The relevant defor- mation theory is controlled by the Chevalley–Eilenberg dg Lie algebra CE ( H(g) ) endowed with 58More generally, the Maurer–Cartan element ΥΘd associated with the incarnation Θd of the Θ-graph in di- mension d is a sum ΥΘd = 1 2 +Θd + · · · = ∑ p≥0 Υ p Θd where the graph Υp Θd has N = 2p (d− 1)+ 2 vertices and k = 2pd+ 1 edges. For d = 1, we recover the sum of graphs ΥΘ1 := ∑ p≥0 1 (2p+1)! 1 ... 2p+ 1 edges 2 yielding the Moyal star-commutator when represented on the algebra of functions on a symplectic manifold [30]. 59See, e.g., [41, Theorem 10.3.1] for a statement as well as Chapter 10 therein for details and history. A Note on Multi-Oriented Graph Complexes 17 the Nijenhuis–Richardson bracket [·, ·]NR and the differential δS := [ [·, ·]H(g) , · ] NR . Since we are interested in formality maps given by universal formulae, our aim is to introduce – for each de- formation quantization problem at hand – a universal model for the deformation theory of H(g) in the guise of a dg Lie algebra of graphs (collectively denoted GC) together with a morphism of dg Lie algebras GC → CE ( H(g) ) . Example 4.1 (universal model for polyvector fields). The paradigmatic example of the above construction is due to M. Kontsevich [32] in the context of the deformation quantization problem for Poisson manifolds. In this context, the quasi-isomorphism of complexes is provided by the HKR map ΦHKR : Tpoly ∼−→ Dpoly between: � Tpoly: the Schouten graded Lie algebra of polyvector fields on the affine space Rm, � Dpoly: the Hochschild dg Lie algebra of multidifferential operators on Rm. According to the previous reasoning, the existence of a formality map U : Tpoly ∼−→ Dpoly can be probed by studying the deformation theory of the Schouten algebra (Tpoly, 0, [·, ·]S), controlled by the Chevalley–Eilenberg dg Lie algebra CE(Tpoly). In the formulation of his Formality con- jecture [32], M. Kontsevich introduced a dg Lie algebra of graphs – denoted fGC2 – together with a morphism of dg Lie algebras fGC2 → CE(Tpoly) given by explicit local formulae.60 The dg Lie algebra fGC2 can therefore be interpreted as a universal model for CE(Tpoly) allowing to reduce important questions related to formality theory to the cohomology of fGC2: � Existence: Obstructions to the existence of universal formality maps live in H1(fGC2). � Classification: The space of universal formality maps is classified by H0(fGC2). This characterisation of the universal solutions to deformation quantization problems via the cohomology of suitable graph complexes can be generalised to other algebro-geometric structures. In the next sections, we will mimic the Kontsevich construction for Lie bialgebras and Lie bialgebroids by resorting to models of (multi)-oriented graphs. 4.2 Lie bialgebras We start by reviewing some known results regarding universal models on Lie bialgebras, see, e.g., [50, 51, 72]. Let us come back to the graded symplectic manifold T ∗(g[1]) ≃ (g ⊕ g∗)[1] of Section 2.1 endowed with a set of homogeneous local coordinates { ξa 1 , ζa 1 } , with a ∈ {1, . . . ,dim g}. Using this set of coordinates, one can (locally) endow the graded algebra of functions C∞((g⊕ g∗)[1]) with a natural structure of dGra3-algebra, with dGra3 the operad of 1-directed graphs in dimension 3. Explicitly, we define a morphism of operads61 Repg : dGra3 → EndC∞((g⊕g∗)[1]) via the following sequence of morphisms of graded62 vector spaces, for N ≥ 1: RepgN : dGra3(N)⊗ C∞((g⊕ g∗)[1])⊗N → C∞((g⊕ g∗)[1]), RepgN (γ)(f1 ⊗ · · · ⊗ fN ) = µN ( ∏ e∈Eγ ∆e(f1 ⊗ · · · ⊗ fN ) ) , (4.1) where 60The explicit formulae defining the morphism fGC2 → CE(Tpoly) take advantage of the graded geometric formulation of Poisson manifolds as dg symplectic manifolds of degree 1, see [71] for the affine space case (cf. also the earlier work [45] as well as [53] for a generalisation to dg symplectic manifolds of arbitrary degree), [29] for a globalisation to any smooth manifolds and [14] for a generalisation to the sheaf of polyvector fields on any smooth algebraic variety. 61Here EndV stands for the endomorphism operad associated with the (graded) vector space V . 62Recall that the grading of a graph in oidjGrad is given by \|γ\| = k(1− d) with k the number of edges. 18 K. Morand � The fi’s are functions on (g⊕ g∗)[1]. � The symbol µN denotes the multiplication map on N elements: µN : C∞((g⊕ g∗)[1])⊗N → C∞((g⊕ g∗)[1]), f1 ⊗ · · · ⊗ fN 7→ f1 · · · fN . � The product is performed over the set Eγ of edges of the graph γ ∈ dGra3(N). � For each edge e ∈ Eγ connecting vertices labeled by integers i and j, the operator ∆e is defined as ∆ i j = ∂ ∂ξa(i) ∂ ∂ζ (j) a , where the sub(super)scripts (i) or (j) indicate that the derivative acts on the i-th or j-th factor in the tensor product. Note that \|∆e\| = −2, consistently with the grading of edges in dGra3, and that ∆e∆e′ = ∆e′∆e as is consistent with the fact that edges are bosonic for d odd. We refer to [53, Proposition 5.1] for a proof (in a slightly more general context) that the map Repg is well-defined and satisfies the axioms of a morphism of operads. The representation Repg yields a sequence63 of morphisms of operads Lie {−2} γ0 ↪−→ dGra3 Repg−→ EndC∞((g⊕g∗)[1]) mapping the generator of Lie {−2} to the graded Poisson bracket (2.1) via the graph 1 2 := 1 2 − 2 1 . Although this action of dGra3 on C∞((g⊕ g∗)[1]) is well-defined, it generically fails to preserve the various subalgebras introduced in Section 2.1. Example 4.2. Consider the following action of a cycle graph Repg2 ( 1 2 ) (f1 ⊗ f2) = ∂2f1 ∂ξa∂ζb ∂2f2 ∂ζa∂ξb , � on f1 = f2 ∼ ξξζ ∈ Ag Lie-quasi yields Rep g 2 ( 1 2 ) (f1 ⊗ f2) ∼ ξξ /∈ Ag Lie-quasi, � on f1 = f2 ∼ ξζζ ∈ Ag quasi-Lie yields Repg2 ( 1 2 ) (f1 ⊗ f2) ∼ ζζ /∈ Ag quasi-Lie, � A fortiori, the action of 1 2 fails to preserve Ag Lie := Ag Lie-quasi ∩ Ag quasi-Lie. This defect can be cured by carefully restricting the space of graphs, as embodied in the following lemma. Lemma 4.3. Let g be a vector space. � The graded algebra of functions C∞((g⊕ g∗)[1]) is endowed with a structure of dGra3- algebra via the action of 1-directed graphs. � The graded subalgebra Ag Lie-quasi is endowed with a structure of s1d0Gra3-algebra via the action of sourced 1-directed graphs. � The graded subalgebra Ag quasi-Lie is endowed with a structure of s−1d0Gra3-algebra via the action of sinked 1-directed graphs. 63More generally, the action of the 3-Gerstenhaber operad Ger3 on the algebra of functions factors through dGra3 as Ger3 ↪−→ dGra3 Repg−→ EndC∞((g⊕g∗)[1]), see footnote 64 for more details. A Note on Multi-Oriented Graph Complexes 19 � The graded subalgebra Ag Lie is endowed with a structure of s1,−1d0Gra3-algebra via the action of 1-directed graphs being both sourced and sinked. Proof. Let us first consider the action of a graph containing a source vertex and let f denote the function decorating the source upon the action of Repg. Then the differential operator acting on f is of the form ∂ ∂ξ · · · ∂ ∂ξf . Assuming that f belongs to the subalgebra Ag Lie-quasi ensures that f is at least linear in ζ (by definition of Ag Lie-quasi, cf. Section 2.1). Hence the function obtained as the outcome of the action of a sourced graph on Ag Lie-quasi either vanishes or is at least linear in ζ, so that it belongs to Ag Lie-quasi. The subalgebra Ag Lie-quasi is therefore closed under the action of sourced graphs. A similar reasoning shows that Ag quasi-Lie is closed under the action of sinked graphs. It follows that the intersection Ag Lie := Ag Lie-quasi ∩ Ag quasi-Lie is closed under the action of graphs possessing at least one source and one sink. ■ As noted previously (see Section 3.1) oriented graphs necessarily possess at least one source and one sink. Denoting collectively by Ag sub the three subalgebras Ag Lie-quasi, Ag quasi-Lie and Ag Lie ⊂ C∞((g⊕ g∗)[1]), the previous lemma thus ensures that Ag sub is endowed with a structure of o1d0Gra3-algebra via the action of oriented 1-directed graphs, i.e., the representation Repg induces morphisms of operads Repg : o1d0Gra3 → EndAg sub . Since we are primarily interested in the cohomology of the associated graph complexes, it is sufficient for our purpose to resort to the smaller operad of oriented graphs as this is where the cohomology lies (cf. the third item of Theorem 3.1). Applying Def(Lie {−2} → ·) on both sides of the morphism Repg yields the following proposition: Proposition 4.4. � The morphism of operads Repg : dGra3 → EndC∞((g⊕g∗)[1]) induces a morphism of dg Lie algebras( d1fGC3, δ, [·, ·] ) → ( CE ( C∞((g⊕ g∗)[1])[2] ) , δS, [·, ·]NR ) . � The morphisms of operads Repg : o1d0Gra3 → EndAg sub induce morphisms of dg Lie algebras( o1d0fGC3, δ, [·, ·] ) → ( CE(Ag sub[2]), δS, [·, ·]NR ) , (4.2) where Ag sub collectively denotes the subalgebras Ag Lie-quasi,A g quasi-Lie,A g Lie⊂C∞((g⊕ g∗)[1]). Proof. The proposition follows straightforwardly from Lemma 4.3 together with the equivari- ance of Repg. ■ We denoted CE(g) the Chevalley–Eilenberg cochain space (in the adjoint representation) associated with the vector space g while [·, ·]NR stands for the Nijenhuis–Richardson bracket and δS := [ {·, ·}gΩ , · ] NR for the Chevalley–Eilenberg differential associated with the graded Poisson bracket (2.1). In plain words, Proposition 4.4 states that the graph complex d1fcGC3 provides a universal model for the deformation theory of proto-Lie bialgebras while o1d0fcGC3 can be seen as a universal model for the deformation theory of Lie-quasi, quasi-Lie and Lie bialgebras. This last fact combined with the cohomology computations reviewed in Section 3.2 yields the following well-known result: Corollary 4.5. The Grothendieck–Teichmüller group acts via Lie∞-automorphisms on the de- formation complexes of Lie-quasi, quasi-Lie and Lie bialgebras. 20 K. Morand Proof. Going to the zeroth cohomology in (4.2) yields a map H0 ( o1d0fGC3\|δ ) → H0 ( CE(Ag sub[2])\|δS ) . Hence any non-trivial zero degree cocycle γ in o1d0fGC3 is mapped to a Lie∞-derivation Repg(γ) of the dg Lie algebras ( Ag sub, {·, ·} g Ω ) , thus ensuring that exp ( Repg(γ) ) is a Lie∞-automorphism thereof. Therefore the pro-unipotent group exp ( H0 ( o1d0fGC3 )) acts on the deformation com- plexes ( Ag sub, {·, ·} g Ω ) via Lie∞-automorphisms. Focusing on connected graphs, the second item of Theorem 3.1 ensures that H0(o1d0fcGC3) ≃ H0(fcGC2) ≃ grt1, where the last equivalence is T. Willwacher’s theorem [71]. Exponentienting thus yields an action of the pro-unipotent Grothendieck–Teichmüller group GRT1 ≡ exp(grt1) on the deformation complexes ( Ag sub, {·, ·} g Ω ) of Lie-quasi, quasi-Lie and Lie bialgebras via Lie∞-automorphisms. ■ This result is consistent with the known action of GRT1 (via the GRT1-torsor of Drinfeld associators) on quantization maps for Lie bialgebras [20, 25, 28, 50, 51, 66] and Lie-quasi bial- gebras [19, 61]. In comparison, there is no such action of GRT1 on the deformation com- plex of proto-Lie bialgebras as the above Lie∞-automorphisms become trivial when acting on C∞((g⊕ g∗)[1]) (consistently with the fact that H0(d1fcGC3) ≃ H0(fcGC3) ≃ K, cf. Table 3). In that sense, there seems to be no true “intermediate case” between the proto-Lie bialgebra case and the Lie bialgebra case since restricting to oriented graphs allows to preserve all three subalgebras at once. As we will see, this fact is in contradistinction with the “bialgebroid” case in which Lie-quasi and quasi-Lie bialgebroids provide a true intermediate case between proto-Lie and Lie bialgebroids. 4.3 Lie bialgebroids We now move on to the main result of this note by introducing novel universal models for the deformation complexes of the family of variations on Lie bialgebroids reviewed in Section 2.2 and Appendix A. Let E π→ M be a vector bundle and consider the graded symplectic manifold( E ,Ω ) , with E ≡ T ∗[2]E[1], coordinatised by the set of homogeneous coordinates { xµ 0 , ξa 1 , ζa 1 , pµ 2 } . The algebra of functions on E carries a natural action RepE : d2Gra3 → EndC∞(E ) of the operad d2Gra3 of bi-directed graphs containing both black and red directions. The action RepE is defined similarly as the action (4.1) where for each edge e ∈ Eγ connecting vertices labeled by integers i and j, the operator ∆e is defined as ∆ i j = ∂ ∂xµ(i) ∂ ∂p (j) µ , ∆ i j = ∂ ∂ξa(i) ∂ ∂ζ (j) a , where the sub(super)scripts (i) or (j) indicate that the derivative acts on the i-th or j-th factor in the tensor product. As in the Lie bialgebra case, we note that the operator ∆e has grading \|∆e\| = −2, consistently with the grading of edges in dGra3, and that ∆e∆e′ = ∆e′∆e as is consistent with the fact that edges are bosonic for d odd. The representation RepE maps the graph 2 3 = 1 2 + 1 2 − ( 2 1 + 2 1 ) towards the graded Poisson bracket (2.8) and furthermore yields a sequence64 of morphisms of 64Similarly as in the Lie bialgebra case, the action of the 3-Gerstenhaber operad Ger3 (also called e3) on the algebra of functions C ∞(E ) factors through d2Gra3 as Ger3 i3 ↪−→ d2Gra3 RepE−→ EndC∞(E ) where the embedding is explicitly given by the following action on generators a1 ∧ a2, { a1, a2 } ∈ Ger3(2): � i3(a1 ∧ a2) = 1 2 with ∧ the graded commutative associative product of degree 0 � i3( { a1, a2 } ) = 2 3 with {·, ·} the graded Lie bracket of degree −2. A Note on Multi-Oriented Graph Complexes 21 operads Lie {−2} γ0 ↪−→ d2Gra3 RepE−→ EndC∞(E ). Although the action of d2Gra3 on C∞(E ) is well-defined and satisfies the axioms of a morphism of operads (cf. the discussion around (4.1)), it generically fails to preserve the various subalgebras introduced in Section 2.2. Example 4.6. Considering the following action of a cycle graph RepE2 ( 1 2 ) (f1 ⊗ f2) = ∂2f1 ∂xµ∂pν ∂2f2 ∂xν∂pµ , � on f1 = f1(x) µ apµξ a and f2 = f2(x) µ apµξ a ∈ AE Lie-quasi yielding RepE2 ( 1 2 ) (f1 ⊗ f2) = ∂µf1(x) ν [a∂νf2(x) µ b]ξ aξb /∈ AE Lie-quasi, � on f1 = f1(x) µ\|apµζa and f2 = f2(x) µ\|apµζa ∈ AE quasi-Lie yielding RepE2 ( 1 2 ) (f1 ⊗ f2) = ∂µf1(x) ν\|[a∂νf2(x) µ\|b] ζaζb /∈ AE quasi-Lie. A fortiori, the previous graph fails to preserve the intersection AE Lie := AE Lie-quasi∩AE quasi-Lie. Hence AE Lie-quasi, AE quasi-Lie and AE Lie generically fail to be d2Gra3-algebras. Similarly to the Lie bialgebra case, this apparent defect can be cured via a suitable restriction of the class of graphs, as performed in the following lemma: Lemma 4.7. Let E π→ M be a vector bundle. � The graded algebra of functions C∞(E ) is endowed with a structure of d2Gra3-algebra via the action of bi-directed graphs. � The graded subalgebra AE Lie-quasi is endowed with a structure of s1d1Gra black 3 -algebra via the action of bi-directed graphs with a black source. � The graded subalgebra AE quasi-Lie is endowed with a structure of s1d1Gra red 3 -algebra via the action of bi-directed graphs with a red source. � The graded subalgebra AE Lie is endowed with a structure of s2d0Gra3-algebra via the action of bi-directed graphs with a black source and a red source. Proof. We mimic the proof in the Lie bialgebra case by considering first the action of a graph containing a black source vertex. Letting f denote the function decorating the black source upon the action of RepE , the differential operator acting on f is of the form ∂ ∂x · · · ∂ ∂x ∂ ∂ξ · · · ∂ ∂ξf . Assuming that f belongs to the subalgebra AE Lie-quasi ensures that f is at least linear either in p or ζ (by definition of AE Lie-quasi, cf. Section 2.2). Hence the function obtained as the outcome of the action of a black sourced graph on AE Lie-quasi cannot be of the form ξ · · · ξ so that it belongs to AE Lie-quasi. The subalgebra AE Lie-quasi is therefore closed under the action of black sourced graphs. A similar reasoning shows that AE quasi-Lie is closed under the action of red sourced graphs.65 It follows that the intersection AE Lie := AE Lie-quasi ∩AE quasi-Lie is closed under the action of graphs possessing both a black and red source. ■ 65Note that graphs with (black or red) sinks preserve neither AE Lie-quasi nor AE quasi-Lie. 22 K. Morand As in the Lie bialgebra case, one can make use of the inclusion o•d•Gra3 ⊂ s•d•Gra3 (see Section 3.1) to extract from the previous lemma an action of (suitable) operads of multi-oriented graphs on the three subalgebras at hand. Explicitly, the subalgebra AE Lie-quasi (resp. AE quasi-Lie) is acted upon by the operad of bi-directed graphs with oriented black (resp. red) arrows, denoted o1d1Gra black 3 (resp. o1d1Gra red 3 ) in the following.66 Crucially, preserving the intersection AE Lie := AE Lie-quasi ∩ AE quasi-Lie requires to orient both directions, thus yielding an action of o2d0Gra3 on AE Lie. As pointed out in Section 3.2 (see Theorem 3.1), the number of oriented colors (in contradistinction with the number of directed colors) is the relevant factor to compute the respective cohomology. From this simple observation will then follow that the Lie bialgebroid case differs essentially from the dual cases of Lie-quasi and quasi-Lie bialgebroids. Applying Def(Lie {−2} → ·) on both sides of RepE : o•d•Gra3 → EndAE yields the following proposition: Proposition 4.8. � The morphism of operads RepE : d2Gra3 → EndC∞(E ) induces a morphism of dg Lie algebras( d2fGC3, δ, [·, ·] ) → ( CE ( C∞(E )[2] ) , δS, [·, ·]NR ) . (4.3) � The morphisms of operads RepE : o1d1Gra3 → EndAE quasi induce morphisms of dg Lie alge- bras ( o1d1fGC3, δ, [·, ·] ) → ( CE ( AE quasi[2] ) , δS, [·, ·]NR ) , (4.4) where AE quasi stands for AE Lie-quasi, AE quasi-Lie ⊂ C∞(E ). � The morphism of operads RepE : o2d0Gra3 → EndAE Lie induces a morphism of dg Lie alge- bras ( o2d0fGC3, δ, [·, ·] ) → ( CE ( AE Lie[2] ) , δS, [·, ·]NR ) . Proof. The proposition is a direct consequence of Lemma 4.7 and the equivariance of the morphism RepE . ■ The conventions used are as in the bialgebra case and δS := [ {·, ·}EΩ , · ] NR stands for the Chevalley–Eilenberg differential associated with the graded Poisson bracket (2.8). Going into co- homology and using Theorem 3.1 allows to compute the relevant cohomology groups, as summed up in Table 4. Table 4. Cohomology groups for Lie bialgebroid structures. Structure Black Red Cohomology proto-Lie bialgebroids directed directed H•(d2fGC3) ≃ H•(fGC3) Lie-quasi bialgebroids oriented directed H•(o1d0fGC black 3 ) ≃ H•(fGC2) quasi-Lie bialgebroids directed oriented H•(o1d0fGC red 3 ) ≃ H•(fGC2) Lie bialgebroids oriented oriented H•(o2d0fGC3) ≃ H•(fGC1) 66Alternatively, both actions could be written in terms of o1d1Gra black 3 (say) by making use of ∆ i j = ∂ ∂ξa (i) ∂ ∂ζ (j) a for AE Lie-quasi and ∆′ i j = ∂ ∂ζ (i) a ∂ ∂ξa (j) for AE quasi-Lie. A Note on Multi-Oriented Graph Complexes 23 The morphism (4.3) on the deformation complex of proto-Lie bialgebroids can be seen as a particular subcase of the action of fGC3 on the deformation complex of Courant algebroids [53] when restricted to the split case.67 The latter does not yield interesting structures in degrees 0 and 1 as the dominant level of the relevant cohomology H•(fcGC3) is located in degree −3. The corresponding cohomology space H−3(fcGC3) is a unital commutative algebra spanned by trivalent graphs modulo the IHX relation where the rôle of the unit is played by the Θ-graph (see, e.g., [4]). Given a proto-Lie bialgebroid E π→ M represented by the Hamiltonian function H( see (2.9) ) , each trivalent graph γ ∈ H−3(fcGC3) yields a cocycle function Ωγ ∈ C∞ (M ) ( i.e., such that { H ,Ωγ }E Ω = 0 ) thus yielding a conformal flow on the space of proto-Lie bialgebroids on E (cf. [53] for details). Turning to Lie-quasi and quasi-Lie bialgebroids, the morphism (4.4) yields a natural exten- sion of the action of the Grothendieck–Teichmüller group exp ( H0(o1d1fcGC3) ) ≃ GRT1 on the deformation complexes of Lie-quasi and quasi-Lie bialgebras to the “bialgebroid” case. This is the essence of Theorem 1.2. Proof of Theorem 1.2. The proof is identical to the one of Corollary 4.5 upon substituting o1d0fcGC3 with o1d1fcGC3, both admitting the same cohomology thanks to the first item of Theorem 3.1. ■ We refer to Section 4.4 for a discussion of the consequences of this action in the context of deformation quantization. Explicitly, the action of GRT1 is through graphs with one oriented color (either black or red) and as such generically fails to preserve the sub-deformation com- plex AE Lie for Lie bialgebroids. This is in contradistinction with the Lie bialgebra case whose deformation complex Ag Lie does carry a representation of GRT1. Rather the action of o2d0fcGC3 endows AE Lie with a new Lie∞-structure deforming non-trivially the big bracket (2.8), as captured by Theorem 1.1, for which we are now in position to articulate the following proof: Proof of Theorem 1.1. It follows from the third item of Proposition 4.8 that Maurer–Cartan elements for the dg Lie algebra ( o2d0fGC3, δ, [·, ·] ) are mapped via RepE to universal defor- mations of the graded Lie algebra ( AE Lie, {·, ·} E Ω ) as a Lie∞-algebra. As recalled in Section 3.2, applying the second item of Theorem 3.1 results in a sequence of isomorphisms H1(o2d0fcGC3)≃ H1(o1d0fcGC3)≃H1(fcGC1)≃K, thus providing a non-trivial cocycle graph Θ3∈H1(o2d0fcGC3) of degree 1. The latter can be recursively extended to a non-trivial Maurer–Cartan element ΥΘ3 := 1 2 + Θ3 + · · · = ∑ p≥0 Υp Θ3 in the graded Lie algebra ( o2d0fGC3, [·, ·] ) – where the graph Υp Θ3 possesses N = 4p+2 vertices and k = 6p + 1 edges – as the corresponding obstructions vanish, cf. footnote 56. Mapping the Maurer–Cartan element ΥΘ3 via the representation RepE yields an exotic Lie∞-structure deforming non trivially the deformation complex ( AE Lie, {·, ·} E Ω ) as a Lie∞-algebra. ■ The non-vanishing brackets of the (essentially unique68) exotic Lie∞-structure of Theorem 1.1 – denoted θ in the following – take the form69 θ2 = RepE2 (ΥS) = {·, ·}EΩ , θ6 = RepE6 (Θ3), θ10 = RepE10 ( Υ2 Θ3 ) , . . . , θ4p+2 = RepE4p+2 ( Υp Θ3 ) , . . . . 67Here by split Courant algebroids we mean Courant algebroids whose underlying vector bundle is a Whitney sum E ⊕ E∗. 68Up to gauge transformations and rescalings. 69The intrinsic degree carried by each bracket is given by \|θ4p+2\| = −12p− 2. Pulling back the brackets along the suspension map s : AE Lie[2] → AE Lie of degree 2 yields a series of brackets θ̃4p+2 on AE Lie[2] with the usual degree −4p. 24 K. Morand The minimal70 Lie∞-structure (AE Lie, θ) can be interpreted as the avatar in dimension d = 3 both of the Moyal star-commutator in d = 1 and the Kontsevich–Shoikhet exotic Lie∞-structure on infinite-dimensional manifolds in d = 2. It relies on bi-oriented graphs and as such possesses no counterpart in the “bialgebra” realm where only one orientable direction is available. In fact, one can explicitly check that the first non-trivial deformed bracket θ6 = RepE6 (Θ3) vanishes identically on the graded Poisson subalgebra71 Ag Lie ⊂ AE Lie controlling deformations of Lie bialgebras, cf. Proposition B.3. Remark 4.9. � In the next section, we will consider Maurer–Cartan elements in the formal extension AE Lie[[ℏ]] of AE Lie by a formal parameter ℏ. By analogy with the d = 2 case [50], we will refer to Maurer–Cartan elements of ( AE Lie[[ℏ]], θ ) as formal “quantizable Lie bialgebroids”, to contrast with the Maurer–Cartan elements of ( AE Lie[[ℏ]], {·, ·} E Ω ) referred to simply as formal Lie bialgebroids. Formal Lie bialgebroids being linear in ℏ are just Lie bialge- broids and accordingly, we will refer to formal “quantizable Lie bialgebroids” linear in ℏ as “quantizable Lie bialgebroids”. � Note that “quantizable Lie bialgebroids” are in particular Lie bialgebroids as they satisfy{ H ,H }E Ω = 0 on top of some higher consistency conditions ( θ6 ( H ∧6) = 0, etc. ) . The distinction between Lie bialgebroids and “quantizable Lie bialgebroids” will become salient when applied to the quantization problem for Lie bialgebroids in Section 4.4. 4.4 Application to quantization and future directions The quantization problem for Lie-(quasi) bialgebras was formulated by V. Drinfeld (cf. [18, Question 1.1] for Lie bialgebras and Section 5 for Lie-quasi bialgebras) and solved in [20] by Etingof–Kazhdan for the Lie bialgebra case72 and in [19, 61] for the Lie-quasi bialgebra case. In both cases, the solution is universal and involves the use of a Drinfeld associator, yielding an action of the Grothendieck–Teichmüller group GRT1 on the set of inequivalent universal quantization maps. The latter can be traced back to the action of H0(o1d0fcGC3) ≃ grt1 on the deformation complex of Lie and Lie-quasi bialgebras, as reviewed in Section 4.2. The situation for Lie bialgebras (and their quasi versions) is therefore much akin to the situation for finite- dimensional Poisson manifolds, in that both cases share the following important features (cf. the Introduction and Table 1): 1. The Grothendieck–Teichmüller group plays a classifying rôle. 2. There is (conjecturally) no generic obstruction to the existence of universal quantizations. As for Lie bialgebroids, the corresponding quantization problem was formulated by P. Xu in [74, 76] as follows: Given a Lie bialgebroid structure on (E,E∗), the associated quantum object is a topological deformation (called quantum groupoid) of the standard (associative) bialgebroid structure on the universal enveloping algebra UR(E) associated with the Lie algebroid structure on E [52] – with R ≡ C∞ (M ) – whose semi-classicalisation reproduces the original Lie bial- gebroid structure. The quantization problem for Lie bialgebroids then consists in associating to each Lie bialgebroid a quantum groupoid quantizing it. Although the quantization problem for a generic Lie bialgebroid remains open, several explicit examples of quantizations for par- ticular Lie bialgebroids have been exhibited in the literature. Apart from the above mentioned 70Recall that minimal Lie∞-algebras are characterised by a vanishing differential θ1 ≡ 0. 71See footnote 16. 72See also [25, 28, 50, 51, 66]. A Note on Multi-Oriented Graph Complexes 25 quantization of Lie bialgebras [20], it was shown in [76] that M. Kontsevich’s solution to the quantization problem for (finite-dimensional) Poisson manifolds [34] ensures that Lie bialge- broids associated with Poisson manifolds (cf. Example A.2) constitute another example of Lie bialgebroids admitting a quantization.73 This result was shown to extend to regular triangular Lie bialgebroids in [76] (see also [55]) using methods à la Fedosov [21] and to generic triangular Lie bialgebroids in [6] using a generalisation of Kontsevich’s formality theorem for Lie algebroids. In this context, the following natural conjecture was formulated by P. Xu: Conjecture 4.10 (Xu [76, Section 6]). Every Lie bialgebroid admits a quantization as a quantum groupoid. Although Conjecture 4.10 might still hold true in the most general setting, we would like to argue for the non-existence of universal74 quantizations of Lie bialgebroids, on the basis of the following results from Section 4.3: 1. The Grothendieck–Teichmüller group plays no classifying rôle regarding the universal de- formations (and hence quantizations) of Lie bialgebroids. 2. There exists a potential obstruction to the existence of universal quantizations of Lie bialgebroids. Contrasting these two features with their above mentioned counterparts for Lie bialgebras, one is led to conclude that the quantization problem for Lie bialgebroids differs essentially from its Lie bialgebra analogue and is in fact more akin to the quantization problem for infinite- dimensional manifolds. In view of this analogy, the obstruction appearing in Theorem 1.1 can be understood as the avatar in d = 3 of the Kontsevich–Shoikhet obstruction in d = 2. As shown in [10, 56, 65, 70], the latter obstruction is hit in d = 2 and thus prevents the existence of an oriented star product, thereby yielding a No-go result regarding the existence of universal quantizations for infinite-dimensional Poisson manifolds. Pursuing the analogy with the d = 2 case, we conjecture the following: Conjecture 4.11 (no-go). There are no universal quantizations of Lie bialgebroids as quantum groupoids. To explicitly show that the obstruction is hit would require a better understanding of the deformation theory of (associative) bialgebroids, which goes beyond the ambition of the present note.75 We nevertheless conclude the present discussion by outlining a strategy of proof for Conjecture 4.11 by mimicking the two-steps procedure of [50] for Poisson manifolds and Lie bialgebras and adapting it to the case at hand. Denoting C• GS(OE ,OE ) the equivalent of the Gerstenhaber–Schack complex for the standard commutative co-commutative bialgebroid struc- ture on the symmetric algebra OE associated to the vector bundle E, the former should be 73More precisely, the Kontsevich star product ∗ quantizing the Poisson bivector π provides a Drinfeld twistor J∗ ∈ ( UR(E) ⊗R UR(E) ) [[ℏ]] for the standard bialgebroid UR(E), where R ≡ C ∞ (M ) and E ≡ TM . Twisting UR(E) by J∗ then provides a quantization of the Lie bialgebroid associated with π on (TM , T ∗M ), see [76]. 74Recall that a universal quantization admits formulae given by expansions in terms of graphs with universal coefficients. 75While the deformation theory for associative algebras [23] and bialgebras [49] are well understood using the frameworks of operads and properads respectively, the deformation theory for bialgebroids has for now been evad- ing such (pr)operadic formulation. The underlying reason is that the relevant category to deal with bialgebroids is the one of ( Re, Re ) -bimodules – where R is a ring and Re denotes its enveloping ring Re := R⊗Rop – which is not symmetric monoidal, as usually required to work with (pr)operads. Rather, the category of ( Re, Re ) -bimodules is naturally endowed with a structure of lax-oplax duoidal category (more precisely, a lax-strong duoidal category, that is the oplax structure is strong monoidal) whose bimonoids are bialgebroids, see, e.g., [5]. As such, the deformation theory for bialgebroids cannot be described using the theory of properads (at least in its standard form). We are grateful to T. Basile and D. Lejay for clarifications regarding this fact. 26 K. Morand endowed with a Lie∞-algebra structure µ – generalising the one of [43, 49] for the bialgebra case – whose corresponding Maurer–Cartan elements are quantum groupoids. By analogy with the bialgebra case, the cohomology of ( C• GS(OE ,OE), µ1 ) should be isomorphic as a graded Lie algebra to the deformation complex ( AE Lie, {·, ·} E Ω ) of Lie bialgebroids on E. Although these two Lie∞-algebras should coincide in cohomology, we do not expect them to be quasi-isomorphic as Lie∞-algebras, i.e., ( C• GS(OE ,OE ) , µ ) is not formal. To show explicitly that the obstruction to formality is hit would require computing the Lie∞-algebra structure obtained by transfer of µ on H•(C• GS(OE ,OE), µ1 ) and showing that the latter coincides with the exotic Lie∞-structure θ on AE Lie, 76 as is the case in d = 2 [50, 65]. This would provide a trivial (in the sense that no Drinfeld associator is needed) formality Lie∞quasi-isomorphism ( AE Lie, θ ) ∼−→ ( C• GS(OE ,OE), µ ) , yielding in turn a quantization map for (formal) “quantizable Lie bialgebroids” (the Maurer– Cartan elements of θ in AE Lie[[ℏ]], cf. Remark 4.9). Finally, the fact that θ is not Lie∞-isomorphic to the big bracket in AE Lie (cf. Theorem 1.1) would prevent the existence of a formality mor- phism for Lie bialgebroids. We sum up these (non)-formality conjectures for Lie bialgebroids in Figure 6. ( AE Lie, {·, ·} E Ω ) ( AE Lie, θ ) ( C• GS(OE ,OE), µ ) Lie bialgebroids “Quantizable Lie bialgebroids” Quantum groupoids × ∼ × ∼ Figure 6. Conjectural (non)-formality maps for Lie bialgebroids. Note that the situation is markedly different in the Lie-quasi (and quasi-Lie) bialgebroid case.77 Firstly, recall that the Maurer–Cartan element ΥΘ3 is not gauge-related to the Maurer– Cartan element ΥS in o2d0fcGC3 – since Θ3 is a non-trivial cocycle in o2d0fcGC3 – hence θ is indeed a non-trivial deformation of the big bracket in AE Lie. However, the cocycle Θ3 is a coboundary in o1d1fcGC3 (cf. Appendix B) so that there exists a combination of graphs ϑ3 ∈ o1d1fcGC3 such that Θ3 = −δϑ3 ∈ o2d0fcGC3. In order for ΥΘ3 and ΥS to be gauge- related in o1d1fcGC3, one needs to find a degree 0 element78 ϑ = ϑ3 + ϑ23 + · · · + ϑp3 of o1d1fcGC3 such that ΥΘ3 = eadϑΥS. Contrarily to the problem of prolongating the cocycle Θ3 to the Maurer–Cartan element ΥΘ3 – which can be solved by a trivial induction – to dis- play an explicit gauge map ϑ is a highly non-trivial task79 as the higher obstructions80 live in H1(o1d1fcGC3) ≃ H1(fcGC2), i.e., the recipient of the obstructions to the universal quantization of finite-dimensional Poisson manifolds. Although this cohomological space conjecturally van- ishes81 (Drinfeld–Kontsevich), maps allowing to convert cocycles into coboundaries are highly non-trivial and necessarily involve the choice of a Drinfeld associator (consistently with the fact that two coboundaries differ by the choice of an element in H0(fcGC2) ≃ grt1). Up to the Drinfeld–Kontsevich conjecture, it is nevertheless expected that, given a Drinfeld associator, one can define a Lie∞-isomorphism ( AE Lie-quasi, {·, ·} E Ω ) ∼−→ ( AE Lie-quasi, θ ) . Repeating the argu- ment laid down in the Lie bialgebroid case, one needs to find a (trivial) Lie∞quasi-isomorphism( AE Lie-quasi, θ ) ∼−→ ( C• quasi−GS(OE ,OE), µ ) , where the right-hand side stands for the deforma- tion complex of the quantum object associated to Lie-quasi bialgebroids. The relevant category 76This is only possible thanks to the fact that the exotic Lie∞-structure θ on AE Lie of Theorem 1.1 is minimal and hence is a potential candidate for being the cohomology of another Lie∞-structure. 77For definiteness, we will focus on the Lie-quasi case, keeping in mind that the arguments apply similarly to the dual case. 78The graph ϑp 3 has N = 4p+ 1 vertices and k = 6p edges so that to have degree 0 in d = 3. 79We refer to [50] for an explicit construction in d = 2. 80The first obstruction vanishes since Θ3 is exact in o1d1fcGC3. The second obstruction Υ2 Θ3 − 1 2 [ϑ3,Θ3] can be checked to live in H1(o1d1fcGC3). 81See Table 3. A Note on Multi-Oriented Graph Complexes 27 here is the one of quasi-bialgebroids, a common generalisation of the notions of bialgebroids and quasi-bialgebras, allowing to define the associated notion of quasi-quantum groupoid as topological deformation of the standard bialgebroid UR(E) as a quasi-bialgebroid. Compos- ing with the (non-trivial) Lie∞-isomorphism ( AE Lie-quasi, {·, ·} E Ω ) ∼−→ ( AE Lie-quasi, θ ) would yield a formality morphism for Lie-quasi bialgebroids. We sum up the discussion by formulating the following conjecture and recap the corresponding conjectural formality maps in the Lie-quasi case in Figure 7. Conjecture 4.12 (yes-go). Given a Drinfeld associator, one can define a universal quantization of Lie-quasi bialgebroids as quasi-quantum groupoids.( AE Lie-quasi, {·, ·} E Ω ) ( AE Lie-quasi, θ ) ( C• quasi−GS(OE ,OE), µ ) Lie-quasi bialgebroids “Quantizable Lie-quasi bialgebroids” Quasi-quantum groupoids ∼ ⟲GRT1 ∼ ∼ ⟲GRT1 ∼ Figure 7. Conjectural formality maps for Lie-quasi bialgebroids. A dual conjecture can be made about quasi-Lie bialgebroids. Of particular interest in this context would be to investigate the quantization of quasi-Lie bialgebroids associated to twisted Poisson manifolds (cf. Example A.4) by means of a Drinfeld twistor [76] for the corresponding non-associative Kontsevich star product [7]. To conclude, let us note that particularising the conjectural quantization map of Conjec- ture 4.12 to Lie bialgebroids ensures that every Lie bialgebroid admits a quantization as a quasi- quantum groupoid (but generically not as a quantum groupoid as stated in Conjecture 4.10). However, for the particular subclass of “quantizable Lie bialgebroids” (which are in particular Lie bialgebroids, see Remark 4.9) – such as Lie bialgebras and coboundary Lie bialgebroids (cf. Appendix B) – the associated quantization should yield a quantum groupoid. According to the above picture, the exotic Lie∞-structure θ of Theorem 1.1 can therefore be seen as a concrete means to delineate the subclass of Lie bialgebroids susceptible to be quantized as quantum groupoids. A Geometry of Lie bialgebroids Lie bialgebras. A Lie bialgebra is a vector space g endowed with a Lie algebra structure on both g and its dual g∗ such that the cobracket ∆g : g → ∧2g is a cocycle for the Lie algebra (g, [·, ·]g), i.e., ∆g([x, y]) = adx∆g(y)− ady∆g(x) where the representation used is the extension ad : g⊗(∧2g) → ∧2g of the adjoint action of (g, [·, ·]g) on ∧2g as adx(y∧z) = [x, y]g∧z+y∧[x, z]g. Letting {ea} \|a∈{1,...,dim g} be a basis of g, one denotes [ea, eb]g = fab cec and ∆g(ec) = Cc abea⊗eb. The three defining conditions of a Lie bialgebra read fe[a dfbc] e = 0, Cd e[aCe bc] = 0, fab eCe cd − 4fe[a [cCb] d]e = 0, (A.1) where fab c = f[ab] c and Cc ab = Cc [ab] where square brackets denote skewsymmetrisation. Lie algebroids. A Lie algebroid is a triplet (E, ρ, [·, ·]E) where: � E π→ M is a vector bundle over the manifold M , � ρ : E → TM is a morphism of vector bundles called the anchor,82 � [·, ·]E : Γ(E)⊗ Γ(E) → Γ(E) is a K-bilinear map called the bracket, 82We will use the same symbol ρ to denote the induced map of sections ρ : Γ(E) → Γ(TM ). 28 K. Morand such that the following conditions are satisfied for all f ∈ C∞ (M ) and X,Y, Z ∈ Γ(E): 1. skewsymmetry : [X,Y ]E = − [Y,X]E , 2. Leibniz rule: [X, f · Y ]E = ρX [f ] · Y + f · [X,Y ]E , 3. Jacobi identity : [X, [Y,Z]E ]E + [Y, [Z,X]E ]E + [Z, [X,Y ]E ]E = 0. The previous conditions ensure that the map ρ : Γ(E) → Γ(TM ) defines a morphism of Lie algebras between the Lie algebra (Γ(E), [·, ·]E) and the Lie algebra of vector fields on M , i.e., ρ[X,Y ]E = [ρX , ρY ] for all X,Y ∈ Γ(E). Proposition A.1. Let (E, ρ, [·, ·]E) be a Lie algebroid. The following statements hold: 1. Let {xµ} \|µ∈{1,...,dimM } be a set of coordinates of M and {ea} \|a∈{1,...,dimE} be a basis of Γ(E). Setting ρea [f ] = ρa µ(x)∂µf and [ea, eb]E = fab c(x)ec, the defining conditions of a Lie algebroid can be expressed in components as fab c = −fbac, 2ρ[a ν∂νρb] µ = ρc µfab c, ρ[c ν∂νfab] d = fe[c dfab] e. Acting on generic sections of E, the Lie algebroid bracket reads [X,Y ]E = ( ρX [Y c]− ρY [X c] + fab cXaY b ) ec. 2. The exterior algebra Γ(∧•E∗) is naturally endowed with a structure of dg commutative algebra with differential dE : Γ(∧•E∗) → Γ(∧•+1E∗) defined by � (dEf)(X) = ρX [f ], � (dEη)(X,Y ) = ρX [η(Y )]− ρY [η(X)]− η([X,Y ]E), � dE(α ∧ β) = (dEα) ∧ β + (−1)\|α\|α ∧ (dEβ), for all X,Y ∈ Γ(E), f ∈ C∞ (M ), η ∈ Γ(E∗) and α, β ∈ Γ(∧•E∗). 3. The dual exterior algebra Γ(∧•E) is endowed with a structure of Gerstenhaber algebra with graded bracket {·, ·}E : Γ(∧•E)⊗ Γ(∧◦E) → Γ(∧•+◦−1E) defined as follows{ f, g } E = 0, { X, f } E = ρX [f ], { X,Y } E = [X,Y ]E ,{ P,Q } E = −(−1)(\|p\|−1)(\|q\|−1) { Q,P } E ,{ P,Q ∧R } E = { P,Q } E ∧R+ (−1)\|q\|(\|p\|−1)Q ∧ { P,R } E , for all f, g ∈ C∞ (M ), X,Y ∈ Γ(E) and P,Q,R ∈ Γ(∧•E). These conditions can be checked to ensure the graded Jacobi identity:{{ P,Q } E , R } E + (−1)(\|P \|−1)(\|Q\|+\|R\|){{Q,R} E , P } E + (−1)(\|R\|−1)(\|P \|+\|Q\|){{R,P} E , Q } E = 0. Example A.2. � A Lie algebra is a Lie algebroid whose base manifold is a point. � Given a manifold M , the standard Lie algebroid is defined as the tangent bundle TM to- gether with the identity map as anchor and the usual Lie bracket of vector fields as bracket. The corresponding differential on the space of differential forms Ω•(M ) ≃ Γ(∧•T ∗M ) coin- cides with the de Rham differential while the induced Gerstenhaber bracket on Γ(∧•TM ) identifies with the Schouten bracket on polyvector fields. A Note on Multi-Oriented Graph Complexes 29 � Let (M , π) be a Poisson manifold. The dual tangent bundle T ∗M is naturally endowed with a Lie algebroid structure with anchor π♯ : Γ(T ∗M ) → Γ(TM ) : αµdx µ 7→ πµναν∂µ and bracket [α, β]π = 1 2 ( Lπ♯(α)β − Lπ♯(β)α+ iπ♯(α)dβ − iπ♯(β)dα ) for all α, β ∈ Γ(T ∗M ). Lie bialgebroids. The concept of Lie bialgebroid was introduced by Mackenzie–Xu in [42] as the infinitesimal variant of a Poisson groupoid. We follow the modern definition of [36] and define a Lie bialgebroid as a vector bundle E π→ M endowed with two dual Lie algebroid structures satisfying a natural compatibility condition. Denoting (ρ, [·, ·]E) the Lie algebroid structure on E and (R, [·, ·]E∗) the one on E∗, the pair (E,E∗) is a Lie bialgebroid if dE∗ is a derivation of {·, ·}E , where we denoted {·, ·}E the Gerstenhaber bracket on Γ(∧•E) induced by (ρ, [·, ·]E) and dE∗ the differential on Γ(∧•E) induced by (R, [·, ·]E∗).83 Example A.3. � A Lie bialgebra is a Lie bialgebroid whose base manifold is a point. � Letting M be a Poisson manifold, the two Lie algebroid structures on TM and T ∗M as defined in Example A.2 are compatible in the above sense and hence define a Lie bialgebroid structure on (TM , T ∗M ). Lie bialgebroids thus generalise both Lie bialgebras and Poisson manifolds. In fact, the base manifold M of any Lie bialgebroid is endowed with a canonical Poisson bracket84 defined as{ f, g } = ⟨dE∗f, dEg⟩ ( or in components as { f, g } = Ra[µρa ν]∂µf∂νg ) . The exterior algebra ( Γ(∧•E),∧ ) of a Lie bialgebroid is endowed with both a structure of Gerstenhaber bracket {·, ·}E and of a differential dE∗ being a derivation for both the graded com- mutative product ∧ and the Gerstenhaber bracket. Such a quadruplet ( Γ(∧•E),∧, dE∗ , {·, ·}E ) is called a strong differential Gerstenhaber algebra. It was in fact shown in [75] (see also [36]) that Lie bialgebroid structures on a vector bundle E π→ M are in one-to-one correspondence with strong differential Gerstenhaber structures on ( Γ(∧•E),∧ ) ( or equivalently with dg Poisson structures on (Γ(∧•E)[1],∧) ) . Coboundary Lie bialgebroids. Letting E be a Lie algebroid, an r-matrix is a section Λ ∈ Γ ( ∧2E ) satisfying { X, { Λ,Λ } E } E = 0 for all X ∈ Γ(E). An r-matrix endows E with a structure of Lie bialgebroid by defining dE∗ = { Λ, · } E . The defining condition on Λ is necessary and sufficient to ensure that the inner derivation dE∗ squares to zero. A Lie bialgebroid defined in this way is called a coboundary Lie bialgebroid. Whenever the stronger condition { Λ,Λ } E = 0 holds, the induced Lie bialgebroid is said to be triangular [42] (cf. for example the Lie bialgebroid on Poisson manifolds defined in Example A.2). If furthermore, Λ is of constant rank, the triangular Lie bialgebroid is said to be regular. Quasi-Lie, Lie-quasi and proto-Lie bialgebroids. The above mentioned characteri- sation of Lie bialgebroids as dg Poisson structures on ( Γ(∧•E)[1],∧ ) calls for several natural generalisations, as summarised in the following table85 (see, e.g., [2, 22, 37, 60]): � Lie bialgebroids on (E,E∗) ⇔ dg Poisson algebras on Γ(∧•E)[1], 83Note that the defining condition of a Lie bialgebroid can equivalently be stated as the fact that the differential dE on Γ(∧•E∗) induced by (ρ, [·, ·]E) is a derivation of the Gerstenhaber bracket {·, ·}E∗ induced by (R, [·, ·]E∗), i.e., the notion of Lie bialgebroid is self-dual. 84More generally, for a proto-Lie bialgebroid, the Jacobi identity for the bivector is deformed as πρ[λ∂ρπ µν] = 1 3 Ra[λRb\|µRc\|ν]ψabc + 1 3 ρa [λρb µρc ν]φabc. 85As is transparent from the correspondence below, there is a series of inclusions of bialgebroids: Lie ⊂ quasi-Lie ⊂ proto-Lie and Lie ⊂ Lie-quasi ⊂ proto-Lie. Note furthermore that the notions of Lie bialgebroids and proto- Lie bialgebroids are self-dual ( and thus can be defined on both Γ(∧•E)[1] and Γ(∧•E∗)[1] ) while the notions of Lie-quasi bialgebroids and quasi-Lie bialgebroids are dual to each other. 30 K. Morand � quasi-Lie bialgebroids on (E,E∗) ⇔ homotopy Poisson algebras86 on Γ(∧•E)[1], � Lie-quasi bialgebroids on (E,E∗) ⇔ homotopy Poisson algebras on Γ(∧•E∗)[1], � proto-Lie bialgebroids on (E,E∗) ⇔ curved homotopy Poisson algebras on Γ(∧•E)[1]. We now focus on the most general case, namely proto-Lie bialgebroids. Apart from the usual data (two anchors ρ, R and two brackets [·, ·]E , [·, ·]E∗), a proto-Lie bialgebroid contains two elements φ ∈ Γ ( ∧3E ) and ψ ∈ Γ ( ∧3E∗) which play the rôle of various obstructions to the usual Lie bialgebroid identities. Letting E∗[1] be the shifted dual bundle coordinatised by {xµ, ζa} of respective degree 0 and 1, the graded commutative algebra ( C∞ (E∗[1]) , · ) is isomorphic to the exterior algebra of sections ( Γ(∧•E),∧ ) . The most general curved homotopy Poisson structure l on Γ(∧•E)[1] thus takes the form87 • l0 := 1 6 φabcζaζbζc, • l1(f) := dE∗f = Ra\|µζa ∂f ∂xµ − 1 2 Cc abζaζb ∂f ∂ζc , • l2(f, g) := { f, g } E = −ρaµ ( ∂f ∂ζa ∂g ∂xµ + (−1)\|f \| ∂f ∂xµ ∂g ∂ζa ) + (−1)\|f \|fab cζc ∂f ∂ζa ∂g ∂ζb , • l3(f, g, h) := (−1)\|g\|ψabc ∂f ∂ζa ∂g ∂ζb ∂h ∂ζc . Imposing the defining quadratic condition [l, l]NR = 0 of a (curved) Lie∞-algebra yields a series of identities which precisely reproduce the components conditions (2.10)–(2.18) as • [l0, l1]NR = 0 ⇔ C8 = 0, (A.2) • [l0, l2]NR + 1 2 [l1, l1]NR = 0 ⇔ C3 = C4 = 0, (A.3) • [l0, l3]NR + [l1, l2]NR = 0 ⇔ C5 = C6 = C7 = 0, (A.4) • [l1, l3]NR + 1 2 [l2, l2]NR = 0 ⇔ C1 = C2 = 0, (A.5) • [l2, l3]NR = 0 ⇔ C9 = 0. (A.6) Imposing ψ ≡ 0 (resp. φ ≡ 0) yields a Lie-quasi (resp. quasi-Lie) bialgebroid and Lie bialgebroids are recovered by setting ψ ≡ 0, φ ≡ 0. Assuming that the base manifold M is the one-point manifold and denoting the vector space Γ(E) as g allows to define the counterparts of these notions in the “bialgebra” realm, cf. Section 2.1. Twisting. The previous formulation allows us to introduce the notion of twist of a proto- Lie bialgebroid in terms of twist of (curved) Lie∞-algebras.88 Given a proto-Lie bialgebroid on (E,E∗) defined by the curved homotopy Poisson algebra ( Γ(∧•E)[1],∧, l ) where l := {lp}0≤p≤3, 86Recall that a (curved) homotopy Poisson structure on a graded commutative algebra (g,∧) is a (curved) Lie∞-structure on g such that all brackets are multi-derivations with respect to (g,∧), see, e.g., [38, 44]. A dg Poisson algebra is thus a (flat) homotopy Poisson algebra for which the brackets of arity above 2 vanish. 87The bracket lp being of degree 3−2p on Γ(∧•E) ≃ C ∞ (E∗[1]), the fact that each bracket is a multi-derivation for the underlying graded commutative algebra constrains all brackets of arity higher than 3 to vanish. Pulling back the brackets along the suspension map s : Γ(∧•E)[1] → Γ(∧•E) of degree 1 yields a series of brackets on Γ(∧•E)[1] with the usual degree 2− p. 88The twisting procedure has been introduced by Quillen [57] for dg Lie algebras and later generalised by Getzler [24] to Lie∞-algebras. Letting (g, l) be a nilpotent (curved) Lie∞-algebra and m ∈ g1 be an arbitrary element of degree 1, one defines the twisted brackets lmn : g∧n → g of degree 2− n for n ≥ 0 as lmn (a1, . . . , an) := ∑ k≥0 1 k! lk+n(m ∧k, a1, . . . , an), where ai ∈ g. The twisted brackets can be checked to endow g with a structure of curved Lie∞-algebra. Whenever the curvature lm0 ∈ g2 vanishes, the element m ∈ g1 is called a Maurer–Cartan element and (g, lm) is then a flat Lie∞-algebra. A Note on Multi-Oriented Graph Complexes 31 one can define a new proto-Lie bialgebroid structure on (E,E∗) via twisting the (curved) Lie∞- algebra structure l by an arbitrary bivector Λ ∈ Γ ( ∧2E ) ( being the most general element of degree 1 in Γ(∧•E)[1] ) . In components, such a twist amounts to the following shift of the components of the proto-Lie bialgebroid: ρa µ Λ−→ ρa µ, fab c Λ−→ fab c + ψabeΛ ec, ψabc Λ−→ ψabc, Ra\|µ Λ−→ Ra\|µ − ρb µΛba, Cc ab Λ−→ Cc ab + ρc µ∂µΛ ab + 2Λe[afec b] − ψcefΛ eaΛfb, φabc Λ−→ φabc − 3R[c\|µ∂µΛ ab] + 3Λe[cCe ab] + 3ρd µΛd[a∂µΛ bc] − 3fef [cΛe\|aΛf \|b] − ψdefΛ daΛebΛfc. (A.7) The latter can be checked to form the components of a proto-Lie bialgebroid without extra assumption on the bivector Λ. Note that, while a twist by a generic Λ does preserve the subspace of Lie-quasi bialgebroids (for which ψ ≡ 0), only bivectors satisfying the Maurer– Cartan equation l1(Λ) + 1 2 l2(Λ,Λ) + 1 6 l3(Λ,Λ,Λ) = 0 in Γ(∧•E)[1] preserve the subspace of quasi-Lie bialgebroids89 (for which φ ≡ 0), see [40] for the Lie bialgebroid case and [60] for quasi-Lie bialgebroids. Example A.4 (twisted Poisson manifolds). Let M be a manifold and H ∈ Ω3(M ) be a closed 3-form. There is a quasi-Lie bialgebroid structure on (TM , T ∗M ) defined in components as ρν µ = δν µ, fµν λ = 0, Rν\|µ = 0, Cλ µν = 0, ψλµν = Hλµν . Letting π ∈ Γ ( ∧2TM ) be a bivector, the components of the twisted proto-Lie bialgebroid read ρ̃ν µ = δν µ, f̃µν λ = Hµνγπ γλ, R̃ν\|µ = −πµν , C̃λ µν = ∂λπ µν −Hλρσπ ρµπσν , φ̃λµν = 3πρ[λ∂ρπ µν] −Hρσγπ ρλπσµπγν , ψ̃λµν = Hλµν . The resulting proto-Lie bialgebroid is again a quasi-Lie bialgebroid if and only φ̃ ≡ 0, that is, if π is a twisted Poisson bivector [60].90 Whenever H vanishes, we recover the Lie bialgebroid structure on the (co)-tangent bundle of a Poisson manifold, cf. Example A.3. Dually, one can consider twisting the curved homotopy Poisson algebra structure on Γ(∧•E∗)[1] by a 2-form field ω ∈ Γ ( ∧2E∗) which amounts to the following shift of the components of the proto-Lie bialgebroid ρa µ ω−→ ρa µ −Rb\|µωba, fab c ω−→ fab c +Rc\|µ∂µωab − 2ωd[aCb] cd − φcdeωdaωeb, Ra\|µ ω−→ Ra\|µ, Cc ab ω−→ Cc ab + φabdωdc, φabc ω−→ φabc, ψabc ω−→ ψabc − 3ρ[a µ∂µωbc] + 3ωe[cfab] e + 3Rd\|µωd[a∂µωbc] − 3C[c deωd\|aωe\|b − φdefωdaωebωfc. (A.8) Twisting by a 2-form field does preserve quasi-Lie bialgebroids but fails to preserve Lie-quasi bialgebroids unless ω satisfies the associated Maurer–Cartan equation on Γ(∧•E∗)[1]. Courant algebroids. A Courant algebroid structure on a pseudo-Euclidean vector bundle91( E , ⟨·, ·⟩E ) is a pair ( ρE , [·, ·]E ) where ρE : Γ(E) → Γ(TM ) is a C∞(M )-linear map called the 89Equivalently, the Maurer–Cartan condition ensures that the twisting preserves the flatness of the Lie∞-algebra associated to a given quasi-Lie bialgebroid, see footnote 88. 90Recall that twisted Poisson bivectors [31, 64] are bivectors satisfying the twisted Jacobi identity [π, π]S = 1 3 H(π, π, π). 91We remind the reader that a pseudo-Euclidean vector bundle is a vector bundle E → M endowed with a symmetric, non-degenerate and C ∞ (M )-bilinear form on the space of sections of E , denoted ⟨·, ·⟩E : Γ(E) ∨ Γ(E) → C ∞(M ) and referred to as the fiber-wise metric 32 K. Morand anchor while [·, ·]E : Γ(E) ⊗ Γ(E) → Γ(E) is a K-bilinear form on the fibers of E referred to as the Dorfman bracket. The latter satisfy the following minimal set of axioms: 1. The Dorfman bracket satisfies the Jacobi identity in its Leibniz form [e1, [e2, e3]E ]E = [[e1, e2]E , e3]E + [e2, [e1, e3]]E for all e1, e2, e3 ∈ Γ(E). 2. The symmetric part of the Dorfman bracket is controlled by the anchor ⟨[e1, e1]E , e2⟩E = 1 2ρE \|e2 [ ⟨e1, e1⟩E ] for all e1, e2 ∈ Γ(E). 3. The fiber-wise metric is compatible with the Courant algebroid structure ρE \|e1 [ ⟨e2, e3⟩E ] = ⟨[e1, e2]E , e3⟩E + ⟨e2, [e1, e3]E⟩E for all e1, e2, e3 ∈ Γ(E). Letting ( E , ⟨·, ·⟩E , ρE , [·, ·]E ) and ( E ′, ⟨·, ·⟩E ′ , ρE ′ , [·, ·]E ′ ) be two Courant algebroids over the same manifold92 M , a morphism of Courant algebroids is defined as a morphism of the underlying vector bundles F ∈ Hom(E , E ′) preserving the additional structures,93 i.e., ⟨e1, e2⟩E = ⟨F(e1),F(e2)⟩E ′ , ρE = ρE ′ ◦ F , F([e1, e2]E) = [F(e1),F(e2)]E ′ for all e1, e2 ∈ Γ(E). Whenever the morphism F is invertible, it is called an isomorphism of Courant algebroids. Courant algebroids first appeared implicitly in the work of I. Dorfman [15] and T. Cou- rant [8, 9] on integrable Dirac structures before their precise geometric structure was abstracted away in [40] to account for the concept of double of Lie bialgebroids. More generally, the double E := E ⊕ E∗ of a proto-Lie bialgebroid (E,E∗) carries a natural structure of pseudo- Euclidean vector bundle with fiber-wise metric ⟨e1, e2⟩E := α(Y ) + β(X), where e1 := X ⊕ α and e2 := Y ⊕ β for all X,Y ∈ Γ(E) and α, β ∈ Γ(E∗). One can furthermore endow E with the anchor ρE(e1) := ρ(X) ⊕ R(α) while the Dorfman bracket is defined through the following explicit expression [e1, e2]E := ( [X,Y ]E + LE∗ α Y − ιβdE∗X − φ(α, β, ·) ) ⊕ ( [α, β]E∗ + LE Xβ − ιY dEα− ψ(X,Y, ·) ) , where LE X stands for the unique derivative operator extending the action of [X, ·]E to the tensor algebra of E ( and similarly for LE∗ α ) . The axioms of a proto-Lie bialgebroid thus ensure that the pair ( ρE , [·, ·]E ) defines a Courant algebroid structure on ( E , ⟨·, ·⟩E ) , cf. [58, Theorem 3.8.2]. Example A.5 (exact Courant algebroids). A Courant algebroid such that the following se- quence94 0 −→ T ∗M ρ∗E−→ E ρE−→ TM −→ 0 92More general notions of morphisms of Courant algebroids over different base manifolds can be defined, see [68] for a thorough treatment. 93A morphism of vector bundles F ∈ Hom(E , E ′) satisfying ⟨e1, e2⟩E = ⟨F(e1),F(e2)⟩E′ will be called a mor- phism of pseudo-vector bundles. 94The vector bundle morphism ρ∗E : T ∗M → E is defined via ⟨ρ∗E \|α, e⟩E = ⟨α, ρE \|e⟩ for all α ∈ Ω1 (M ) and e ∈ Γ(E), and satisfies ρE ◦ ρ∗E = 0. A Note on Multi-Oriented Graph Complexes 33 is exact is called an exact Courant algebroid. It is an important result due to P. Ševera [63] that exact Courant algebroids over a fixed base manifold M are classified95 by the third de Rham cohomology H3 dR(M ) of M . Dirac structures. A subbundle L ⊂ E of a Courant algebroid E is called a Dirac structure if: 1. L is maximally isotropic with respect to the fiber-wise metric ⟨·, ·⟩E , i.e., ⟨Γ(L),Γ(L)⟩E = 0, dimL = 1 2 dim E . 2. The space of sections Γ(L) is closed under the Courant bracket [·, ·]E , i.e., [Γ(L),Γ(L)]E ⊆ Γ(L). These conditions ensure in particular that the restrictions of the Courant anchor and bracket to L endow the vector bundle L with a structure of Lie algebroid. Example A.6 (Dirac structure). Let M be a manifold and [H] ∈ H3 dR(M ). Given a repre- sentative H ∈ Ω3(M ) and a twisted Poisson structure π with respect to H, the graph of the map π♯ : T ∗M → TM can be checked to be a Dirac structure for the exact Courant algebroid associated to ( M , [H] ) [63, 64]. A pair (E , L) where L is a Dirac structure for the Courant algebroid E is referred to as aManin pair. A triplet (E , L,M) where L, M are two Dirac structures of E such that E = L ⊕M is referred to as a Manin triple. Both the notions of Manin pairs and triples recover their usual extension in the context of Lie algebras when the base manifold M is a point. Letting ( E , L ) and ( E ′, L′) be two Manin pairs over the same manifold M , a morphism of Manin pairs96 is a morphism of Courant algebroids F ∈ Hom(E , E ′) such that F(L) ⊆ L′. Letting E = E⊕E∗ be the pseudo-Euclidean vector bundle associated to the vector bundle E over M and endowed with the canonical fiber-wise metric ⟨·, ·⟩E , we have the following hierarchy of identifications: � proto-Lie bialgebroids on (E,E∗) ⇔ Courant algebroids structures on ( E , ⟨·, ·⟩E ) , � Lie-quasi bialgebroids on (E,E∗) ⇔ Manin pairs ( (E , ⟨·, ·⟩E), E ) , � quasi-Lie bialgebroids on (E,E∗) ⇔ Manin pairs ( (E , ⟨·, ·⟩E), E∗), � Lie bialgebroids on (E,E∗) ⇔ Manin triples ( (E , ⟨·, ·⟩E), E,E∗). Such identification allows to define morphisms of proto/Lie-quasi/quasi-Lie/Lie bialgebroids as morphisms of Courant algebroids preserving additional Dirac structures. Explicitly, morphisms of pseudo-vector bundles E⊕E∗ F−→ E′⊕E′∗ with canonical fiber-wise metrics generically take the form F(e)a ′ = Fa′ bX b + Fa′bαb, F(e)a′ = Fa′bX b + Fa′ bαb satisfying Fc′(aFc′ b) = 0, Fc′(aFc′ b) = 0, Fc′ aFc′ b + Fc′aFc′b = δab. 95More precisely, any exact Courant algebroid is isomorphic to E := TM⊕T ∗M with canonical fiber-wise metric, projection TM ⊕ T ∗M → TM as anchor and Dorfman bracket [e1, e2]E := [X,Y ]⊕ ( LXβ − ιY dα−H(X,Y, ·) ) , where H ∈ Ω3(M ) is a closed 3-form. The latter Courant algebroid structure coincides with the one induced by the quasi-bialgebroid structure of Example A.4. Letting ω ∈ Γ ( ∧2E∗), the isomorphismX⊕α ω−→ X⊕ ( α+ 1 2 ιXω ) corresponding to the twist (2.20) amounts to shift the closed 3-form H by an exact 3-form as H ω−→ H − dω. Hence isomorphism classes of exact Courant algebroids over M are in bijective correspondence with elements of H3 dR(M ). 96Similarly, letting ( E , L,M ) and ( E ′, L′M ′) be two Manin triples over the same manifold M , a morphism of Manin triples is a morphism of Courant algebroids F ∈ Hom(E , E ′) such that F(L) ⊆ L′ and F(M) ⊆M ′. 34 K. Morand Such morphisms map the Dirac structure E to E′ if and only if Fa′b = 0 and the Dirac struc- ture E∗ to E′∗ if and only if Fa′b = 0. Infinitesimal endomorphisms of the pseudo-Euclidean vector bundle (E ⊕ E∗, ⟨·, ·⟩E) read97 δF(e) = ( λ(X) + 1 2 ιαΛ ) ⊕ ( −λT (α) + 1 2 ιXω ) , where � λ ∈ End(Γ(E)) generates infinitesimal rotations of the fibers of (E,E∗), � Λ ∈ Γ ( ∧2E ) generates the infinitesimal version of the twist of Lie-quasi bialgebroids (A.7), � ω ∈ Γ ( ∧2E∗) generates the infinitesimal version of the twist of quasi-Lie bialgebroids (A.8). B Incarnation of the Θ-graph in d = 3 The present appendix is devoted to collect some additional results regarding the exotic Lie∞- structure θ of Theorem 1.1 generated by the cocycle class [Θ3] ∈ H1(o2d0fGC3). For concreteness, we fix a representative of the class [Θ3] as follows: Proposition B.1. There is a unique pair of combination of graphs Θ3 ∈ o2d0fGC3 and ϑ3 ∈ o1d1fGC black 3 such that: 1. Θ3 = −δϑ3, i.e., Θ3 is exact in o1d1fGC black 3 . 2. Θ3 contains only graphs of the shape C, cf. Figure 8. 3. Each graph of ϑ3 contains at least one red cycle, i.e., ϑ3 /∈ o2d0fGC3. Remark B.2. � The combination of graphs ϑ3 contains 68 black-oriented graphs (48 graphs of shape A and 20 graphs of shape B) while the combination Θ3 contains 288 bi-oriented graphs of shape C. � Although Θ3 is exact in o1d1fGC black 3 , it is crucial to note that Θ3 is not exact in o2d0fGC3, i.e., there is no combination of graphs κ3 ∈ o2d0fGC3 such that Θ3 = −δκ3. Hence Θ3 is a non-trivial cocycle in o2d0fGC3. 21 3 4 6 A 21 3 4 6 B 2 1 34 5 6 C Figure 8. Shape of graphs involved in ϑ3 (A and B) and Θ3 (C). Let E π→ M be a vector bundle. To each Lie bialgebroid structure on (E,E∗) (represented by the Hamiltonian function H ∈ AE Lie), we will associate the functions � ϑ3(H ) := RepE5 (ϑ3)(H ⊗5) ∈ AE Lie-quasi, � Θ3(H ) := RepE6 (Θ3)(H ⊗6) ∈ AE Lie. 97Restricting to Lie-quasi (resp. quasi-Lie) bialgebroids yields ω ≡ 0 (resp. Λ ≡ 0), so that the subalgebra End ( Γ(E) ) acts on the abelian ideal Γ ( ∧2E ) ( resp. Γ ( ∧2E∗)) by rotations. A Note on Multi-Oriented Graph Complexes 35 Note that the condition Θ3 = −δϑ3 ensures that Θ3(H ) ∼ Q[ϑ3(H )] – where the differen- tial Q is defined as Q := { H , · }E Ω – so that Θ3(H ) is a coboundary in the complex (AE Lie-quasi\|Q). However, Θ3(H ) is generically not exact inAE Lie and the obstruction for Θ3(H ) to be a cobound- ary in AE Lie is precisely given by the component of ϑ3(H ) proportional to ζ3. We will denote this obstruction as Ob(H )abc, so that Ob(H )abc = 0 ⇒ ϑ3(H ) ∈ AE Lie and Θ3(H ) is a trivial cocycle in (AE Lie-quasi\|Q). A straightforward computation gives Ob(H )abc = Rd\|µRe\|ν(2ρf λ∂µCe [a\|f∂λνCd bc] − ρd β∂βνR [a\|λ∂µλCe bc] − ρd λ∂µCe [a\|f∂λνCf bc] − ∂µR f \|λfef [a∂νλCd bc] − 2fdf [a∂µCe b\|g∂νCg c]f + 2ffg [a∂µCe b\|f∂νCd c]g ) +Rd\|µ∂µR e\|ν(fdf [aCe fg∂νCg bc] − 2fef [aCd fg∂νCg bc] + ρd λ∂λCe [a\|f∂νCf bc] − 2ρe λ∂λCd [a\|f∂νCf bc] − ρe λ∂λCf [ab∂νCd c]f ) + ρd β∂βR e\|νRd\|µ∂µνR [a\|λ∂λCe bc] + 2ρe λRd\|µ∂λµR [a\|νCd ef∂νCf bc] + ρg νRd\|µCd ef∂νCe [ab∂µCf c]g. The latter encodes the first obstruction for H to define a “quantizable Lie bialgebroid”. Although the obstruction does not vanish for a generic Lie bialgebroid, the following proposition displays two important examples: Proposition B.3. The obstruction vanishes for � Lie bialgebras, � coboundary Lie bialgebroids. Proof. Setting Ra\|µ ≡ 0 yields Ob(H )abc = 0 hence the obstruction vanishes for Lie bialgebras. More generally, it can be checked that each graph appearing in the combinations ϑ3 and Θ3 contains at least one arrow of the type i j so that both ϑ3(H ) and Θ3(H ) vanish identically on the graded Poisson subalgebra98 Ag Lie ⊂ AE Lie. For coboundary Lie bialgebroids, we perform the replacement Ra\|µ = ρb µΛba and Cc ab = −ρcµ∂µΛab − 2Λd[afdc b], with Λab = Λ[ab] a bivector, see Appendix A. Under this replacement, it can be checked that Ob(H )abc identically vanishes modulo the defining conditions C1 ≡ 0, C2 ≡ 0 ensuring that the maps (ρ, f) define a Lie algebroid. ■ Acknowledgements The author is grateful to T. Basile, D. Lejay, H. Y. Liao and P. Xu for discussions as well as to Y. Kosmann–Schwarzbach, S. Merkulov and T. Willwacher for correspondence. The author would also like to thank J. H. Park for support. 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institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-17T12:02:09Z
publishDate	2022
publisher	Інститут математики НАН України
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spelling	Morand, Kevin 2026-01-05T12:24:58Z 2022 A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids. Kevin Morand. SIGMA 18 (2022), 020, 38 pages 1815-0659 2020 Mathematics Subject Classification: 53D55; 18G85; 17B62 arXiv:2102.07593 https://nasplib.isofts.kiev.ua/handle/123456789/211525 https://doi.org/10.3842/SIGMA.2022.020 Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichmüller group via one-colored directed and oriented graphs, respectively. In this note, we study the action of multi-oriented graph complexes on Lie bialgebroids and their ''quasi'' generalisations. Using results due to T. Willwacher and M. Zivković on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck-Teichmüller group on Lie bialgebras and quasi-Lie bialgebras can be generalised to quasi-Lie bialgebroids via graphs with two colors, one of them being oriented. However, this action generically fails to preserve the subspace of Lie bialgebroids. By resorting to graphs with two oriented colors, we instead show the existence of an obstruction to the quantization of a generic Lie bialgebroid in the guise of a new Lie∞-algebra structure non-trivially deforming the ''big bracket'' for Lie bialgebroids. This exotic Lie∞-structure can be interpreted as the equivalent in = 3 of the Kontsevich-Shoikhet obstruction to the quantization of infinite-dimensional Poisson manifolds (in = 2). We discuss the implications of these results with respect to a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids. The author is grateful to T. Basile, D. Lejay, H. Y. Liao, and P. Xu for discussions as well as to Y. Kosmann–Schwarzbach, S. Merkulov, and T. Willwacher for correspondence. The author would also like to thank J. H. Park for support. Finally, the author is grateful to anonymous referees whose suggestions greatly helped to improve the quality of the present paper. This work was supported by the Brain Pool Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science and ICT (2018H1D3A1A01030137), and by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R1A6A1A03047877). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids Article published earlier
spellingShingle	A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids Morand, Kevin
title	A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
title_full	A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
title_fullStr	A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
title_full_unstemmed	A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
title_short	A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
title_sort	note on multi-oriented graph complexes and deformation quantization of lie bialgebroids
url	https://nasplib.isofts.kiev.ua/handle/123456789/211525
work_keys_str_mv	AT morandkevin anoteonmultiorientedgraphcomplexesanddeformationquantizationofliebialgebroids AT morandkevin noteonmultiorientedgraphcomplexesanddeformationquantizationofliebialgebroids

A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids

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