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-Teichm\"uller 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. Zivkovi\'c on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck-Teichm\"uller 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 $\mathsf{Lie}_\infty$-algebra structure non-trivially deforming the"big bracket"for Lie bialgebroids. This exotic $\mathsf{Lie}_\infty$-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 implications of these results with respect to a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids.


Introduction
Graph complexes play an essential rôle in the understanding of the deformation quantization of various algebraic and geometric structures, the paradigmatic example thereof being the Kontsevich graph complex and its relation to the deformation quantization problem for (finitedimensional) 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 GRT 1 -introduced by V. Drinfeld 1 [17] in the context of the absolute Galois group Gal Q /Q and the theory of quasi-Hopf algebras -so that the space of Kontsevich maps 2 is a GRT 1 -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 E 2 -operad, etc.
In the present context of graph complexes, another incarnation of the Grothendieck-Teichmüller 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 bialgebras 4 and furthermore provides a rationale for the classifying rôle played by the Grothendieck-Teichmüller group on the space of quantization maps for Lie and Lie-quasi bialgebrasà la Etingof-Kazhdan [20,61]. The oriented graph complex also plays a crucial rôle regarding the obstruction theory to the existence of a universal quantization 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] deforming 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). Deformation 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 quantizations.
• A middle way class comprising the deformation quantization problems for Courant algebroids: 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 bialgebroids 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 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 bialgebroids 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 quantization 5 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 as 7 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 introduced 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) bialgebras in Section 4.2 and then move on to the Lie bialgebroid case in Section 4.3. Using the 5 Note that the case d = 1 is somehow special among Table 1 as there is no associated quantization problem. The relevant cohomological class H 1 (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). 6 The graph complex oidjfGC d 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. 7 In the remaining of the text, we use the prefix dg to refer to differential graded objects.
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 geometric 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 := g k+n . We will denote s : g[n] → g the corresponding suspension map of intrinsic degree n.

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 generalised 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. Roytenberg [58,59] that dg symplectic manifolds (or NPQ-manifolds) of degree 1 (resp. of degree 2) are 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 generalisations 8 ) 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.

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 coordinates 9 ξ a 1 , ζ a 1 , with a ∈ {1, . . . , dim g}. The latter is a graded symplectic manifold with symplectic 2-form Ω = dξ a ∧ dζ a of degree 2. The associated Poisson bracket of degree −2 acts on homogeneous functions in C ∞ ((g ⊕ g * )[1]) as 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 NPQmanifold) 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., L Q Ω = 0). This last compatibility relation ensures 11 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 as 12 , ϕ abc = ϕ [abc] and ψ abc = 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}: 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 8 As reviewed in Appendix A. 9 The subscript denotes the corresponding degree. 10 Or equivalently, endowing the graded Poisson algebra of functions C ∞ ((g ⊕ g * ) [1]), ·, {·, ·} g Ω with a compatible differential. 11 Via Cartan's homotopy formula, cf. [59, Lemma 2.2]. 12 The signs and coefficients are chosen for later convenience. 13 Here and in the following, round (resp. square) brackets of indices will denote (skew)symmetrisation.
controlled by the dg Lie algebra 14 C ∞ ((g ⊕ g * ) [1]), Q, {·, ·} g Ω . Proto-Lie bialgebras thus constitute 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 subalgebras of C ∞ ((g ⊕ g * ) [1]) whose Hamiltonian functions will encode various sub-classes of proto-Lie bialgebras. Let us start by defining the subspace A g In plain words, the subspace A g Lie-quasi is obtained by discarding all functions of the form ψ a 1 ···am ξ a 1 · · · ξ am , for arbitrary values of m ≥ 0. It can be easily checked that A g Lie-quasi is preserved by both the pointwise product of functions 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 A g 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 reproduce 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 A g quasi-Lie as the subspace obtained by discarding all functions of the form ϕ a 1 ···an ζ a 1 · · · ζ an , for all n ≥ 0, i.e., ). The most general Hamiltonian function of A g 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 defined 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 A g Lie is defined as the intersection A g Lie := A g Lie-quasi ∩ A g quasi-Lie between the two previous Poisson subalgebras. The latter subspace can again be checked to be a graded Poisson subalgebra of C ∞ ((g ⊕ g * ) [1]) obtained by discarding all functions of the form ψ a 1 ···am ξ a 1 · · · ξ am and ϕ a 1 ···an ζ a 1 · · · ζ an for all m, n ≥ 0. In particular, the most general Hamiltonian function of A g 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 * ). • Lie-quasi 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.
14 Note 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]. 15 Remark 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.
In the next section, we turn to the generalisation of this graded geometric interpretation to the larger class of Lie bialgebroids and variations thereof.

Lie bialgebroids
Letting E π → M be a vector bundle over the smooth (finite-dimensional) manifold 16 The associated Poisson bracket of degree −2 acts as follows on homogeneous functions f, g ∈ C ∞ (E ): 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 geometric 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 where {ρ, f, R, C, ϕ, ψ} are functions on the base space M , with symmetries f ab , ϕ abc = ϕ [abc] and ψ abc = 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 14) 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 algebra C ∞ (E ) admits several subalgebras defining in turn various sub-classes of proto-Lie bialgebroids. A convenient way to characterise these Poisson subalgebras is as vanishing ideals of 16 The 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. 17 Here 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. 18 Correspondingly, all formulae appearing in this note will be local. 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 functions vanishing on them. We start by defining the Poisson subalgebra A E Lie-quasi ⊂ C ∞ (E ) as the vanishing ideal of E [1]. In plain words, the subalgebra A E Lie-quasi is obtained by discarding all functions of the form ψ a 1 ···am (x)ξ a 1 · · · ξ am , for all m ≥ 0. The most general Hamiltonian function of A E 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 A E quasi-Lie ⊂ C ∞ (E ) as the vanishing ideal of E * [1], i.e., as the subspace obtained by discarding all functions of the form ϕ a 1 ···an (x)ζ a 1 · · · ζ an with n ≥ 0. The most general Hamiltonian function of A E 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 A E Lie := A E Lie-quasi ∩ A E 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 ψ a 1 ···am (x) ξ a 1 · · · ξ am or ϕ a 1 ···an (x)ζ a 1 · · · ζ an , for all m, n ≥ 0. In particular, the most general Hamiltonian function of A E 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 characterised 21 as Manin triples 22  As usual, gauge transformations for the Hamiltonian function (2.9) are generated by functions 23 of degree 2 in C ∞ (E ) reading explicitly: 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 dim N . 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 . 20 A 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]. 21 We refer to Appendix A for analogues of these statements within the framework of Courant algebroids. 22 A Manin triple is a dg symplectic manifold supplemented with two transverse dg Lagrangian submanifolds C, D ⊂ N such that H ∈ IC and H ∈ ID. 23 Gauge 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. so that the gauge algebra for proto-Lie bialgebroids is isomorphic to Γ(T M ) ⊕ End Γ(E) ⊕ Γ ∧ 2 E ⊕ Γ ∧ 2 E * 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 interpreted 24 in terms of infinitesimal morphisms of proto-Lie bialgebroids as follows: • λ ∈ End Γ(E) : rotation of the fibers of (E, E * ), 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 twisting 25 by an arbitrary bivector Λ ∈ Γ ∧ 2 E . Explicitly, the twisted proto-Lie bialgebroid H Λ is obtained by performing a canonical transformation generated by the flow of the Hamiltonian vector field X Λ := −Λ, · E Ω where Λ := 1 2 Λ ab ζ a ζ b is a function of degree 2 in C ∞ (E ). Explicitly, a twist by Λ corresponds to the canonical transformation 26 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 / ∈ A E quasi-Lie so that Γ ∧ 2 E 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: M be a vector bundle. The following correspondences hold: • Hamiltonians in C ∞ (E ) are in bijective correspondence with proto-Lie bialgebroid structures on (E, E * ).
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. 24 We refer to Appendix A for more details on this interpretation. 25 The twisting procedure was introduced in [16] for quasi-Hopf algebras, in [35] for quasi-Lie bialgebras and in [60] for quasi-Lie bialgebroids. 26 Dually, one can consider twisting by a 2-form field ω ∈ Γ ∧ 2 E * corresponding to the canonical transformation generated by the flow of the Hamiltonian vector field Xω := −ω, · , where ω := 1 2 ω ab ξ a ξ b : which amounts to a shift of the components of H as in equation (A.8). 27 Dually, 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 / ∈ A E Lie-quasi so that Γ ∧ 2 E * is not part of the gauge algebra for quasi-Lie bialgebroids.

(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).

Directed, oriented and sourced graphs
We will denote gra N,k (resp. dgra N,k ) the set of multi(di)graphs 29 with N vertices and k directed edges.
The set d c gra N,k of multi-directed graphs with c colors is defined as the set of ordered pairs (γ, c) where: • γ ∈ dgra N,k is a multidigraph. We will denote V γ (resp. E γ ) the set of vertices (resp. edges) of γ.
• c stands for a map c : A pictorial representation of a multi-directed graph in d c gra N,k can be given by decorating each directed edge of the underlying multidigraph in dgra N,k with c − 1 additional arrows of different colors (cf. Figure 1 for an example). 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 S N (resp. S k ) on elements of d c gra N,k by permutation of the labeling of vertices (resp. edges).
Operads. For all d ∈ N * , we define the collection where sgn k denotes the 1-dimensional sign representation of S k . The subscript stands for taking coinvariants with respect to the diagonal right action of S k and the term between brackets denotes degree suspension. In plain words, this means that edges carry an intrinsic degree 28 Or equivalently, as convolution Lie algebras constructed from suitable graph operads. 29 Recall that multi(di)graphs are undirected (resp. directed) graphs allowed to contain both loops and multiple edges. 30 That is, including the underlying (black) arrow of the multidigraph.
1 − d and are bosonic for d odd and fermionic for d even. The S-module {d c Gra d (N )} N ≥1 can further be given the structure of an operad 31 by endowing it with the usual equivariant partial composition operations Figure 2 for an example and, e.g., of [53, Section 4] for more details.
There is a natural sequence of embeddings of operads 32 given by mapping each graph in d c Gra d to a sum of graphs in d c+1 Gra d where the summation runs over all the possible ways to orient the arrow of the additional direction. We call such mapping the orientation morphism O r : Figure 3 for an example). 33 there is an operad morphism (2) to the graph where the graph 1 2 ∈ d c Gra d (2) is obtained by decorating 1 2 with c − 1 additional colors and summing over all the possible orientations. 34 A multidigraph in dgra N,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 ogra N,k ⊂ dgra N,k . This definition can 31 The identity element id ∈ dcGra d (1) is defined as the graph id := 1 . 32 Here, Gra d ≡ d0Gra d (resp. dGra d ≡ d1Gra d ) stands for the operad of one-colored undirected (resp. directed) graphs. 33 Recall 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. 34 In other words, 1 2 := (O r) c−1 1 2 . For example, 1 2 := 1 2 + 1 2 in d2Gra d . 35 Recall that a cycle (or wheel) is a (non-trivial) directed path from a vertex to itself. be extended to multi-directed graphs by defining the subset 36 o i d j gra N,k ⊆ d c gra N,k of multidirected 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- It is easy to check that the latter is closed under partial compositions and hence defines a suboperad o i d j Gra d ⊆ d c Gra d of multi-oriented graphs. Note that the graph 1 2 is (trivially) multi-oriented and hence defines a morphism of operads 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 s k d j Gra d ⊆ d c Gra d 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 s i,−i d j Gra d , 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 Graph complexes. Given the multi-oriented graph operad o i d j Gra d , one defines the dg Lie algebra of multi-oriented graphs o i d j fGC d as the deformation complex where the terms between brackets denote degree suspension while the superscript stands for taking invariants with respect to the right action of S N , with sgn N the 1-dimensional signature representation of S N . 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 element 39 γ ∈ o i d j fGC d with N vertices and k edges is given by The graded Lie bracket [·, ·] of degree 0 on o i d j fGC d is defined as usual in terms of the partial composition operations (cf., e.g., [53,Section 4.2] for details). The differential δ := [Υ S , ·] is 36 By convention, we identify o0d0gra N,k ≡ gra N,k , o0d1gra N,k ≡ dgra N,k and o1d0gra N,k ≡ ogra N,k . 37 As an example, the graph of Figure 1 belongs to o2d2gra 4,5 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 o2d0gra 2,2 while the second and third graphs belong to o1d1gra 2,2 . 38 We refer to references [41,49] for details. 39 Note that the graph degree is insensitive to the number i of oriented directions.
defined by taking the adjoint action with respect to the Maurer-Cartan element 40 Note to conclude that the dg Lie algebra o |k| d j fGC d of multi-oriented graphs is a sub-dg Lie algebra of the dg Lie algebra of multi-sourced/sinked graphs s k d j fGC d defined for all k ∈ Z and d, j ≥ 0 by 41 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) Multi-oriented graphs with c = i + j colors and i oriented directions Multi-sourced/sinked graphs with c = |k| + j colors and |k| sourced/sinked directions Multi-sourced/sinked graphs with c = i + j colors and i directions being both sourced and sinked

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 43 This is justified by the fact that the cohomology of the graph complex o i d j Gra d is captured by its connected part, 44 as follows from where O(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 40 For example, the graph Υ S := 2 3 = 1 2 + 1 2 + (−1) d 2 1 + 2 1 is a Maurer-Cartan element in oidjfGC d with i, j, d ≥ 0 and i + j = 2. 41 Equivalently, one can define s k djfGC d : Section 5] and [48,Section 7] for reviews. 43 Similarly, one can introduce the suboperads of connected multi-directed graphs djcGra d and connected multisourced graphs sidjcGra d as well as their corresponding sub-dg Lie algebras djfcGC d and sidjfcGC d . 44 We refer to [71] and [72] for the cases i = 0, 1 respectively and to [47] for a general statement.
cohomology of the various complexes previously introduced can be related to one another. This is embodied by the following important theorem, due to T. Willwacher 45 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: 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 sequence 47 of injective quasi-isomorphisms of complexes 48 Hence, adding extra colored direction does not affect the cohomology. The first item of Theorem 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.
for the first item (cf. also [13]), [72] for the second and [73] for the third. 46 The first Betti number is defined as b := k − N + 1 for a connected graph with N vertices and k edges. 47 Here, the dg Lie algebra fcGC d ≡ d0fcGC d (resp. dfcGC d ≡ d1fcGC d ) stands for the usual Kontsevich graph complex of connected undirected (resp. directed) graphs. 48 See [13,71] for the first arrow and [47] for a general statement. 49 The cocycle L3 stands for the triangle loop 2 1 3 . Regarding the Θ-cocycle and the Grothendieck-Teichmüller algebra grt 1 , see below.
Climbing the dimension ladder. Informally, the second item of Theorem 3.1 allows to map familiar structures in low dimensions 50 to novel incarnations thereof in higher dimensions. 51 The most striking example of such hierarchy of structures stems from another important theorem of T. Willwacher [71] showing the existence of an isomorphism of Lie algebras H 0 (fcGC 2 ) grt 1 where grt 1 denotes the infinite-dimensional Grothendieck-Teichmüller algebra. Combined with the second item of Theorem 3.1, we obtain a sequence H 0 (fcGC 2 ) 52 Different incarnations yield different actions of the Grothendieck-Teichmüller group 53 on various algebro-geometric structures. In its original incarnation via directed cocycles in H 0 (d 1 fGC d ), the Grothendieck-Teichmüller group naturally acts via Lie ∞ -automorphisms on the Schouten Lie algebra of polyvector fields T poly 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 GRT 1 via oriented cocycles in H 0 (o 1 d 0 fcGC 3 ) on the deformation complex of Lie bialgebras 54 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 ∈ dgra 2,3 spanning the first cohomology class of fcGC 1 , i.e., H 1 (fcGC 1 ) K Θ . Again, applying the second item of Theorem 3.1 results in a sequence of isomorphisms In its original incarnation as the cocycle spanning H 1 (fcGC 1 ), the Θ-graph can be recursively extended 56 to a non-trivial Maurer-Cartan element Υ Θ := • • + + · · · in the graded Lie algebra fcGC 1 , [·, ·] , see, e.g., [30]. This Maurer-Cartan element is mapped to the Moyal starcommutator [26,54] via the natural action of fcGC 1 on the algebra of functions of a (bosonic) symplectic manifold. Considering the case d = 2 yields another important incarnation of the Θgraph as the oriented Kontsevich-Shoikhet cocycle -denoted Θ 2 in the following, cf. Figure 5 and spanning H 1 (o 1 d • fcGC 2 ). 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 3 . 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 Υ Θ 2 induces an exotic (and essentially unique) universal Lie ∞ -structure on polyvector fields deforming non-trivially 57 the 50 See footnote 3 for the interpretation of the dimension d. 51 Since 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 ∈ dgra N,k in dimension d is mapped to a cocycle γ ∈ dgra N ,k in dimension d + 1 with N = k + 1 and k = 2k − N + 1. More generally, the incarnation of a cocycle γ ∈ dgra N, with b the first Betti number. 52 For example, the tetrahedron graph t2 ∈ H 0 (fcGC2) is mapped to an oriented cocycle t3 ∈ H 0 (o1d•fcGC3) with N = 7 vertices and k = 9 edges (see footnote 51). 53 Recall that the Grothendieck-Teichmüller group is defined by exponentiation of the pro-nilpotent Grothendieck-Teichmüller algebra, i.e., GRT1 ≡ exp(grt 1 ), see [69] for a review. 54 As reviewed in Section 4.2. 55 The fact that quantization of Lie bialgebras involves oriented graphs was already recognised in [18]. 56 Potential obstructions in promoting the Θ-graph to a full Maurer-Cartan element lie in H 2 (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. 57 For 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.
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. Of special interest for our purpose is the incarnation of the Θ-cocycle in dimension 3, dubbed Θ 3 in the following. The latter is a combination of bi-oriented graphs with N = 6 and k = 7 58 (see Appendix B) that will be argued to provide an obstruction to the universal quantization of Lie bialgebroids in Section 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.

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 differential). 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 homotopy 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 canonical dg Lie algebra structure (H(g), 0, [·, ·] H(g) ), i.e., l 1 = 0, l 2 = [·, ·] H(g) and the higher order brackets l >2 are transferred from the dg Lie algebra structure on g. In other words, any quasiisomorphism of complexes Φ : (H(g), 0) δ) can be upgraded to a quasi-isomorphism of Lie ∞ -algebras U : (H(g), l) ∼ −→ (g, δ, [·, ·] g ) with U 1 = 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 formality (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 deformation theory is controlled by the Chevalley-Eilenberg dg Lie algebra CE H(g) endowed with 58 More generally, the Maurer-Cartan element ΥΘ d associated with the incarnation Θ d of the Θ-graph in dimension d is a sum ΥΘ d = 1 2 + Θ d + · · · = p≥0 Υ p Θ d where the graph Υ p Θ d has N = 2p (d − 1) + 2 vertices and k = 2pd + 1 edges. the Nijenhuis-Richardson bracket [·, ·] NR and the differential δ S := [·, ·] H(g) , · NR . Since we are interested in formality maps given by universal formulae, our aim is to introduce -for each deformation 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 Φ HKR : T poly ∼ −→ D poly between: • T poly : the Schouten graded Lie algebra of polyvector fields on the affine space R m , • D poly : the Hochschild dg Lie algebra of multidifferential operators on R m .
According to the previous reasoning, the existence of a formality map U : T poly ∼ −→ D poly can be probed by studying the deformation theory of the Schouten algebra (T poly , 0, [·, ·] S ), controlled by the Chevalley-Eilenberg dg Lie algebra CE(T poly ). In the formulation of his Formality conjecture [32], M. Kontsevich introduced a dg Lie algebra of graphs -denoted fGC 2 -together with a morphism of dg Lie algebras fGC 2 → CE(T poly ) given by explicit local formulae. 60 The dg Lie algebra fGC 2 can therefore be interpreted as a universal model for CE(T poly ) allowing to reduce important questions related to formality theory to the cohomology of fGC 2 : • Existence: Obstructions to the existence of universal formality maps live in H 1 (fGC 2 ).
• Classification: The space of universal formality maps is classified by H 0 (fGC 2 ).
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.

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 ξ a 1 , ζ a 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 dGra 3 -algebra, with dGra 3 the operad of 1-directed graphs in dimension 3. Explicitly, we define a morphism of operads 61 Rep g : dGra 3 → End C ∞ ((g⊕g * ) [1]) via the following sequence of morphisms of graded 62 vector spaces, for N ≥ 1: where 60 The explicit formulae defining the morphism fGC2 → CE(T poly ) 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. 61 Here EndV stands for the endomorphism operad associated with the (graded) vector space V . 62 Recall that the grading of a graph in oidjGra d is given by |γ| = k(1 − d) with k the number of edges.
• The symbol µ N denotes the multiplication map on N elements: • The product is performed over the set E γ of edges of the graph γ ∈ dGra 3 (N ).
• For each edge e ∈ E γ connecting vertices labeled by integers i and j, the operator ∆ e is defined as 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 dGra 3 , 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 Rep g is well-defined and satisfies the axioms of a morphism of operads. The representation 2 − 2 1 . Although this action of dGra 3 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
• A fortiori, the action of 1 2 fails to preserve A g This defect can be cured by carefully restricting the space of graphs, as embodied in the following lemma. • The graded algebra of functions C ∞ ((g ⊕ g * ) [1]) is endowed with a structure of dGra 3algebra via the action of 1-directed graphs.
• The graded subalgebra A g Lie-quasi is endowed with a structure of s 1 d 0 Gra 3 -algebra via the action of sourced 1-directed graphs.
• The graded subalgebra A g quasi-Lie is endowed with a structure of s −1 d 0 Gra 3 -algebra via the action of sinked 1-directed graphs. 63 More generally, the action of the 3-Gerstenhaber operad Ger3 on the algebra of functions factors through , see footnote 64 for more details.
• The graded subalgebra A g Lie is endowed with a structure of s 1,−1 d 0 Gra 3 -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 Rep g . Then the differential operator acting on f is of the form ∂ ∂ξ · · · ∂ ∂ξ f . Assuming that f belongs to the subalgebra A g Lie-quasi ensures that f is at least linear in ζ (by definition of A g Lie-quasi , cf. Section 2.1). Hence the function obtained as the outcome of the action of a sourced graph on A g Lie-quasi either vanishes or is at least linear in ζ, so that it belongs to A g Lie-quasi . The subalgebra A g Lie-quasi is therefore closed under the action of sourced graphs. A similar reasoning shows that A g quasi-Lie is closed under the action of sinked graphs. It follows that the intersection A g Lie := A g Lie-quasi ∩ A g 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 A g sub the three subalgebras A g Lie-quasi , A g quasi-Lie and A g Lie ⊂ C ∞ ((g ⊕ g * ) [1]), the previous lemma thus ensures that A g sub is endowed with a structure of o 1 d 0 Gra 3 -algebra via the action of oriented 1-directed graphs, i.e., the representation Rep g induces morphisms of operads Rep g : o 1 d 0 Gra 3 → End A g 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 Rep g yields the following proposition: where A g sub collectively denotes the subalgebras Proof . The proposition follows straightforwardly from Lemma 4.3 together with the equivariance of Rep g .
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 δ S := {·, ·} g Ω , · 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 d 1 fcGC 3 provides a universal model for the deformation theory of proto-Lie bialgebras while o 1 d 0 fcGC 3 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: Proof . Going to the zeroth cohomology in (4.2) yields a map [2])|δ S . Hence any non-trivial zero degree cocycle γ in o 1 d 0 fGC 3 is mapped to a Lie ∞ -derivation Rep g (γ) of the dg Lie algebras A g sub , {·, ·} g Ω , thus ensuring that exp Rep g (γ) is a Lie ∞ -automorphism thereof. Therefore the pro-unipotent group exp H 0 o 1 d 0 fGC 3 acts on the deformation complexes A g sub , {·, ·} g Ω via Lie ∞ -automorphisms. Focusing on connected graphs, the second item of Theorem 3.1 ensures that H 0 (o 1 d 0 fcGC 3 ) H 0 (fcGC 2 ) grt 1 , where the last equivalence is T. Willwacher's theorem [71]. Exponentienting thus yields an action of the pro-unipotent Grothendieck-Teichmüller group GRT 1 ≡ exp(grt 1 ) on the deformation complexes A g sub , {·, ·} g Ω of Lie-quasi, quasi-Lie and Lie bialgebras via Lie ∞ -automorphisms.
This result is consistent with the known action of GRT 1 (via the GRT 1 -torsor of Drinfeld associators) on quantization maps for Lie bialgebras [20,25,28,50,51,66] and Lie-quasi bialgebras [19,61]. In comparison, there is no such action of GRT 1 on the deformation complex of proto-Lie bialgebras as the above Lie ∞ -automorphisms become trivial when acting on 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.

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. The algebra of functions on E carries a natural action Rep E : d 2 Gra 3 → End C ∞ (E ) of the operad d 2 Gra 3 of bi-directed graphs containing both black and red directions. The action Rep E is defined similarly as the action (4.1) where for each edge e ∈ E γ connecting vertices labeled by integers i and j, the operator ∆ e is defined as 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 dGra 3 , and that ∆ e ∆ e = ∆ e ∆ e as is consistent with the fact that edges are bosonic for d odd.
The representation Rep E maps the graph 2 3 = 1 2 + 1 2 − 2 1 + 2 1 towards the graded Poisson bracket (2.8) and furthermore yields a sequence 64 of morphisms of 64 Similarly 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 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. operads Although the action of d 2 Gra 3 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 A fortiori, the previous graph fails to preserve the intersection Hence A E Lie-quasi , A E quasi-Lie and A E Lie generically fail to be d 2 Gra 3 -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: • The graded algebra of functions C ∞ (E ) is endowed with a structure of d 2 Gra 3 -algebra via the action of bi-directed graphs.
• The graded subalgebra A E Lie-quasi is endowed with a structure of s 1 d 1 Gra black 3 -algebra via the action of bi-directed graphs with a black source.
• The graded subalgebra A E quasi-Lie is endowed with a structure of s 1 d 1 Gra red 3 -algebra via the action of bi-directed graphs with a red source.
• The graded subalgebra A E Lie is endowed with a structure of s 2 d 0 Gra 3 -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 Rep E , the differential operator acting on f is of the form ∂ ∂x · · · ∂ ∂x ∂ ∂ξ · · · ∂ ∂ξ f . Assuming that f belongs to the subalgebra A E Lie-quasi ensures that f is at least linear either in p or ζ (by definition of A E Lie-quasi , cf. Section 2.2). Hence the function obtained as the outcome of the action of a black sourced graph on A E Lie-quasi cannot be of the form ξ · · · ξ so that it belongs to A E Lie-quasi . The subalgebra A E Lie-quasi is therefore closed under the action of black sourced graphs. A similar reasoning shows that A E quasi-Lie is closed under the action of red sourced graphs. 65 It follows that the intersection A E Lie := A E Lie-quasi ∩ A E quasi-Lie is closed under the action of graphs possessing both a black and red source.
As in the Lie bialgebra case, one can make use of the inclusion o • d • Gra 3 ⊂ s • d • Gra 3 (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 A E Lie-quasi (resp. A E quasi-Lie ) is acted upon by the operad of bi-directed graphs with oriented black (resp. red) arrows, denoted o 1 d 1 Gra black 3 (resp. o 1 d 1 Gra red 3 ) in the following. 66

Crucially, preserving the intersection A E
Lie := A E Lie-quasi ∩ A E quasi-Lie requires to orient both directions, thus yielding an action of o 2 d 0 Gra 3 on A E 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 Rep E : o • d • Gra 3 → End A E yields the following proposition: • The morphism of operads Rep E : d 2 Gra 3 → End C ∞ (E ) induces a morphism of dg Lie algebras • The morphisms of operads induce morphisms of dg Lie algebras

Lie induces a morphism of dg Lie algebras
Proof . The proposition is a direct consequence of Lemma 4.7 and the equivariance of the morphism Rep E .
The conventions used are as in the bialgebra case and δ S := {·, ·} E Ω , · NR stands for the Chevalley-Eilenberg differential associated with the graded Poisson bracket (2.8). Going into cohomology and using Theorem 3.1 allows to compute the relevant cohomology groups, as summed up in Table 4.
The morphism (4.3) on the deformation complex of proto-Lie bialgebroids can be seen as a particular subcase of the action of fGC 3 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 • (fcGC 3 ) is located in degree −3. The corresponding cohomology space H −3 (fcGC 3 ) 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 (fcGC 3 ) yields a cocycle function Ω γ ∈ C ∞ (M ) i.e., such that H , Ω γ 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 extension of the action of the Grothendieck-Teichmüller group exp H 0 (o 1 d 1 fcGC 3 ) GRT 1 on the deformation complexes of Lie-quasi and quasi-Lie bialgebras to the "bialgebroid" case. This is the essence of Theorem 1.2. 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 GRT 1 is through graphs with one oriented color (either black or red) and as such generically fails to preserve the sub-deformation complex A E Lie for Lie bialgebroids. This is in contradistinction with the Lie bialgebra case whose deformation complex A g Lie does carry a representation of GRT 1 . Rather the action of o 2 d 0 fcGC 3 endows A E 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: The non-vanishing brackets of the (essentially unique 68 ) exotic Lie ∞ -structure of Theorem 1.1 -denoted θ in the following -take the form 69 . . . 67 Here by split Courant algebroids we mean Courant algebroids whose underlying vector bundle is a Whitney sum E ⊕ E * . 68 Up to gauge transformations and rescalings. 69 The intrinsic degree carried by each bracket is given by |θ4p+2| = −12p − 2. Pulling back the brackets along the suspension map s : A E Lie [2] → A E Lie of degree 2 yields a series of bracketsθ4p+2 on A E Lie [2] with the usual degree −4p.
The minimal 70 Lie ∞ -structure (A E 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 θ 6 = Rep E 6 (Θ 3 ) vanishes identically on the graded Poisson subalgebra 71  referred to simply as formal Lie bialgebroids. Formal Lie bialgebroids being linear in are just Lie bialgebroids and accordingly, we will refer to formal "quantizable Lie bialgebroids" linear in 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 θ 6 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.

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 case 72 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 GRT 1 on the set of inequivalent universal quantization maps. The latter can be traced back to the action of H 0 (o 1 d 0 fcGC 3 ) grt 1 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 finitedimensional 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 U R (E) associated with the Lie algebroid structure on E [52] -with R ≡ C ∞ (M ) -whose semi-classicalisation reproduces the original Lie bialgebroid 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 particular Lie bialgebroids have been exhibited in the literature. Apart from the above mentioned 70 Recall that minimal Lie∞-algebras are characterised by a vanishing differential θ1 ≡ 0. 71  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 bialgebroids 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 methodsà 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: Although Conjecture 4.10 might still hold true in the most general setting, we would like to argue for the non-existence of universal 74 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 deformations (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 infinitedimensional 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: 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 (O E , O E the equivalent of the Gerstenhaber-Schack complex for the standard commutative co-commutative bialgebroid structure on the symmetric algebra O E associated to the vector bundle E, the former should be 73 More precisely, the Kontsevich star product * quantizing the Poisson bivector π provides a Drinfeld twistor J * ∈ UR(E) ⊗R UR(E) [[ ]] for the standard bialgebroid UR(E), where R ≡ C ∞ (M ) and E ≡ T M . Twisting UR(E) by J * then provides a quantization of the Lie bialgebroid associated with π on (T M , T * M ), see [76]. 74 Recall that a universal quantization admits formulae given by expansions in terms of graphs with universal coefficients. 75 While 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 evading such (pr)operadic formulation. The underlying reason is that the relevant category to deal with bialgebroids is the one of R e , R e -bimodules -where R is a ring and R e denotes its enveloping ring R e := R ⊗ R op -which is not symmetric monoidal, as usually required to work with (pr)operads. Rather, the category of R e , R e -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. 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 (O E , O E ), µ 1 should be isomorphic as a graded Lie algebra to the deformation complex A E 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 (O E , O E , 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 , µ 1 and showing that the latter coincides with the exotic Lie ∞ -structure θ on A E 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 , µ , yielding in turn a quantization map for (formal) "quantizable Lie bialgebroids" (the Maurer- , cf. Remark 4.9). Finally, the fact that θ is not Lie ∞ -isomorphic to the big bracket in A E Lie (cf. Theorem 1.1) would prevent the existence of a formality morphism for Lie bialgebroids. We sum up these (non)-formality conjectures for Lie bialgebroids in Figure 6. Note that the situation is markedly different in the Lie-quasi (and quasi-Lie) bialgebroid case. 77 Firstly, recall that the Maurer-Cartan element Υ Θ 3 is not gauge-related to the Maurer-Cartan element Υ S in o 2 d 0 fcGC 3 -since Θ 3 is a non-trivial cocycle in o 2 d 0 fcGC 3 -hence θ is indeed a non-trivial deformation of the big bracket in A E Lie . However, the cocycle Θ 3 is a coboundary in o 1 d 1 fcGC 3 (cf. Appendix B) so that there exists a combination of graphs ϑ 3 ∈ o 1 d 1 fcGC 3 such that Θ 3 = −δϑ 3 ∈ o 2 d 0 fcGC 3 . In order for Υ Θ 3 and Υ S to be gaugerelated in o 1 d 1 fcGC 3 , one needs to find a degree 0 element 78 ϑ = ϑ 3 + ϑ 2 3 + · · · + ϑ p 3 of o 1 d 1 fcGC 3 such that Υ Θ 3 = e ad ϑ Υ S . Contrarily to the problem of prolongating the cocycle Θ 3 to the Maurer-Cartan element Υ Θ 3 -which can be solved by a trivial induction -to display an explicit gauge map ϑ is a highly non-trivial task 79 as the higher obstructions 80 live in H 1 (o 1 d 1 fcGC 3 ) H 1 (fcGC 2 ), i.e., the recipient of the obstructions to the universal quantization of finite-dimensional Poisson manifolds. Although this cohomological space conjecturally vanishes 81 (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 H 0 (fcGC 2 ) grt 1 ). Up to the Drinfeld-Kontsevich conjecture, it is nevertheless expected that, given a Drinfeld associator, Repeating the argument laid down in the Lie bialgebroid case, one needs to find a (trivial) , µ , where the right-hand side stands for the deformation complex of the quantum object associated to Lie-quasi bialgebroids. The relevant category 76 This is only possible thanks to the fact that the exotic Lie∞-structure θ on A E Lie of Theorem 1.1 is minimal and hence is a potential candidate for being the cohomology of another Lie∞-structure. 77 For definiteness, we will focus on the Lie-quasi case, keeping in mind that the arguments apply similarly to the dual case. 78 The graph ϑ p 3 has N = 4p + 1 vertices and k = 6p edges so that to have degree 0 in d = 3. 79 We refer to [50] for an explicit construction in d = 2. 80 The first obstruction vanishes since Θ3 is exact in o1d1fcGC3. The second obstruction Υ 2 Θ 3 − 1 2 [ϑ3, Θ3] can be checked to live in H 1 (o1d1fcGC3). 81 See Table 3.
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 U R (E) as a quasi-bialgebroid. Composing with the (non-trivial) 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.
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 Conjecture 4.12 to Lie bialgebroids ensures that every Lie bialgebroid admits a quantization as a quasiquantum 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.
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 d E * is a derivation of {·, ·} E , where we denoted {·, ·} E the Gerstenhaber bracket on Γ(∧ • E) induced by (ρ, [·, ·] E ) and d E * 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. 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 bracket 84 defined as The exterior algebra Γ(∧ • E), ∧ of a Lie bialgebroid is endowed with both a structure of Gerstenhaber bracket {·, ·} E and of a differential d E * being a derivation for both the graded commutative product ∧ and the Gerstenhaber bracket. Such a quadruplet Γ(∧ • E), ∧, d E * , {·, ·} 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 ∧ 2 E 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 d E * = Λ, · E . The defining condition on Λ is necessary and sufficient to ensure that the inner derivation d E * 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 characterisation of Lie bialgebroids as dg Poisson structures on Γ(∧ • E) [1], ∧ calls for several natural generalisations, as summarised in the following table 85 (see, e.g., [2,22,37,60]): • Lie bialgebroids on (E, E * ) ⇔ dg Poisson algebras on Γ(∧ • E) [1], 83 Note 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. 84 More generally, for a proto-Lie bialgebroid, the Jacobi identity for the bivector is deformed as 85 As 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.
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 ∧ 3 E and ψ ∈ Γ ∧ 3 E * which play the rôle of various obstructions to the usual Lie bialgebroid identities. Letting E * [1] be the shifted dual bundle coordinatised by {x µ , ζ a } 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 form 87 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 • 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 := {l p } 0≤p≤3 , 86 Recall 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. 87 The 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. 88 The 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 ∈ g 1 be an arbitrary element of degree 1, one defines the twisted brackets l m n : g ∧n → g of degree 2 − n for n ≥ 0 as l m n (a1, . . . , an) := k≥0 1 k! l k+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 l m 0 ∈ g 2 vanishes, the element m ∈ g 1 is called a Maurer-Cartan element and (g, l m ) is then a flat Lie∞-algebra.
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 ∧ 2 E 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: 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 l 1 (Λ) + 1 2 l 2 (Λ, Λ) + 1 6 l 3 (Λ, Λ, Λ) = 0 in Γ(∧ • E) [1] preserve the subspace of quasi-Lie bialgebroids 89 (for which ϕ ≡ 0), see [40] for the Lie bialgebroid case and [60] for quasi-Lie bialgebroids.
The resulting proto-Lie bialgebroid is again a quasi-Lie bialgebroid if and onlyφ ≡ 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 ∧ 2 E * which amounts to the following shift of the components of the proto-Lie bialgebroid 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 bundle 91 E, ·, · E is a pair ρ E , [·, ·] E where ρ E : Γ(E) → Γ(T M ) is a C ∞ (M )-linear map called the 89 Equivalently, 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. 90 Recall that twisted Poisson bivectors [31,64] are bivectors satisfying the twisted Jacobi identity [π, π] S = 1 3 H(π, π, π). 91 We 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 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: for all e 1 , e 2 ∈ 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. Courant [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 e 1 , e 2 E := α(Y ) + β(X), where e 1 := X ⊕ α and e 2 := Y ⊕ β for all X, Y ∈ Γ(E) and α, β ∈ Γ(E * ). One can furthermore endow E with the anchor ρ E (e 1 ) := ρ(X) ⊕ R(α) while the Dorfman bracket is defined through the following explicit expression where L E X stands for the unique derivative operator extending the action of [X, ·] E to the tensor algebra of E and similarly for L E * α . The axioms of a proto-Lie bialgebroid thus ensure that the pair ρ E , [·, ·] E defines a Courant algebroid structure on E, ·, · E , cf. [ 92 More general notions of morphisms of Courant algebroids over different base manifolds can be defined, see [68] for a thorough treatment. 93 A morphism of vector bundles F ∈ Hom(E, E ) satisfying e1, e2 E = F(e1), F(e2) E will be called a morphism of pseudo-vector bundles. 94 The vector bundle morphism ρ * E : T * M → E is defined via ρ * E |α, e E = α, ρE |e for all α ∈ Ω 1 (M ) and e ∈ Γ(E), and satisfies ρE • ρ * E = 0.
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 2. Θ 3 contains only graphs of the shape C, cf. Figure 8. 3. Each graph of ϑ 3 contains at least one red cycle, i.e., ϑ 3 / ∈ o 2 d 0 fGC 3 .
• The combination of graphs ϑ 3 contains 68 black-oriented graphs (48 graphs of shape A and 20 graphs of shape B) while the combination Θ 3 contains 288 bi-oriented graphs of shape C.

However, Θ 3 (H ) is generically not exact in A E
Lie and the obstruction for Θ 3 (H ) to be a coboundary in A E Lie is precisely given by the component of ϑ 3 (H ) proportional to ζ 3 . We will denote this obstruction as Ob(H ) abc , so that Ob(H ) abc = 0 ⇒ ϑ 3 (H ) ∈ A E Lie and Θ 3 (H ) is a trivial cocycle in (A E Lie-quasi |Q). A straightforward computation gives 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 R a|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 ϑ 3 and Θ 3 contains at least one arrow of the type i j so that both ϑ 3 (H ) and Θ 3 (H ) vanish identically on the graded Poisson subalgebra 98 A g Lie ⊂ A E Lie . For coboundary Lie bialgebroids, we perform the replacement R a|µ = ρ b µ Λ ba and C c ab = −ρ c µ ∂ µ Λ ab − 2Λ d[a f dc b] , with Λ ab = Λ [ab] a bivector, see Appendix A. Under this replacement, it can be checked that Ob(H ) abc identically vanishes modulo the defining conditions C 1 ≡ 0, C 2 ≡ 0 ensuring that the maps (ρ, f ) define a Lie algebroid.