A note on multi-oriented graph complexes and deformation quantization of Lie bialgebroids
AA note on multi-oriented graph complexesand deformation quantization of Lie bialgebroids
Kevin MorandDepartment of Physics, Sogang University, Seoul 04107, South KoreaCenter for Quantum Spacetime, Sogang University, Seoul 04107, South Korea [email protected]
Abstract
Universal solutions to deformation quantization problems can be conveniently classified by the cohomologyof suitable graph complexes. In particular, the deformation quantization of (finite dimensional) Poissonmanifolds 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 graphcomplexes on Lie bialgebroids and their “quasi” generalisations. Using results due to T. Willwacher andM. Živković on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck–Teichmüller group on Lie bialgebras and Lie-quasi bialgebras can be generalised to Lie-quasi bialgebroidsvia graphs with two colors, one of them being oriented. However, this action generically fails to preservethe subspace of Lie bialgebroids. By resorting to graphs with two oriented colors, we instead show theexistence 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 ∞ -structurecan be interpreted as the equivalent in d = 3 of the Kontsevich–Shoikhet obstruction to the quantization ofinfinite dimensional Poisson manifolds (in d = 2 ). We discuss the implications of these results with respectto a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids. Contents
A Geometry of Lie bialgebroids 19B Incarnation of the Θ -graph in d = 3 a r X i v : . [ m a t h . QA ] F e b Introduction
Graph complexes play an essential rôle in the understanding of the deformation quantization of various algebraicand geometric structures, the paradigmatic example thereof being the Kontsevich graph complex and its relationto the deformation quantization problem for (finite dimensional) Poisson manifolds [25]. In particular, the space ofKontsevich quantization maps is acted upon by the pro-unipotent group exponentiating the zeroth cohomology ofthe Kontsevich graph complex of directed graphs [9]. As shown by T. Willwacher [56], the latter is isomorphic tothe Grothendieck–Teichmüller group
GRT – introduced by V. Drinfel’d [13] in the context of the absolute Galoisgroup Gal ( ¯ Q / Q ) and the theory of quasi-Hopf algebras – so that the space of Kontsevich maps is a GRT -torsor[27]. Since its inception, the Grothendieck–Teichmüller group appeared in a variety of mathematical contexts suchas the Kashiwara–Vergne conjecture, multiple zeta values, rational homotopy of the E -operad, etc. In the present context of graph complexes, another incarnation of the Grothendieck–Teichmüller group can be foundin relation to the deformation quantization problem for Lie bialgebras via the action of the graph complex of oriented graphs [57] in dimension d = 3 . The latter action generalises to Lie-quasi bialgebras and furthermore provides arationale for the classifying rôle played by the Grothendieck–Teichmüller group on the space of quantization mapsfor Lie and Lie-quasi bialgebras à la Etingof–Kazhdan [16, 52]. The oriented graph complex also plays a crucialrôle regarding the obstruction theory to the existence of a universal quantization of infinite dimensional Poissonmanifolds [57]. 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 representedon the space of (infinite dimensional) polyvector fields, the latter yields an exotic Lie ∞ -structure [53] deformingnon-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 infinitedimensional Poisson manifolds. The deformation quantization problem for infinite dimensional Poisson manifoldsthus differs essentially from the finite dimensional case and this discrepancy can be traced back to the fact thattheir respective deformation theory is acted upon by a different graph complex (directed vs oriented).In the present note, we add two threads to this on-going story by introducing some new universal models forthe deformation theory of Lie bialgebroids and their “quasi” versions. Lie bialgebroids have been introduced byMackenzie–Xu in [34] and constitute a common generalisation of the notions of Poisson manifolds and Lie bialgebras.The corresponding quantization problem was spelled out by P. Xu in [61, 62] and remains open to this day. Themain result of this note consists in exhibiting an obstruction to the existence of a universal quantization map for Liebialgebroids in the guise of an exotic Lie ∞ -structure on the deformation complex of Lie bialgebroids ( cf. Theorem4.10). The latter is a non-trivial deformation of the so-called “big bracket” for Lie bialgebroids and can be consideredas an avatar in d = 3 of the Kontsevich–Shoikhet obstruction to the quantization of infinite dimensional Poissonmanifolds. Therefore, the deformation quantization problem for Lie bialgebroids differs essentially from its Liebialgebra counterpart and is in fact more akin to the one for infinite dimensional Poisson manifolds, both exhibitingthe following features: 1) no classification rôle played by the Grothendieck–Teichmüller group; 2) existence of ageneric obstruction to quantization. The origin of this obstruction can be traced back to an action of the graphcomplex of bi-oriented graphs ( i.e. graphs with two oriented colors) on the deformation theory of Lie bialgebroids.Relaxing the orientation on 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 bialgebroidsgeneralising the one on Lie-(quasi) bialgebras ( cf. Proposition 4.9). Our results are summed up in Table 1. Based on a suggestion due to A. Grothendieck in his
Esquisse d’un Programme [21] who proposed to study the combinatorialproperties of the absolute Galois group Gal (¯ Q / Q ) via its natural action on the tower of Teichmüller groupoids. More precisely, the space of homotopy classes [8] of stable [9] formality morphisms is a
GRT -torsor. Recall that the parameter d corresponds to the dimension of the source manifold of the relevant AKSZ σ -model [1]. The latteris related to the degree n of the corresponding target manifold via d = n + 1 and is therefore independent of the dimension ofthe associated algebro-geometric structure (Poisson manifold, Lie bialgebra, etc. ). Consistently, it relates to the dimension of thecompactified configuration spaces of points of the associated de Rham field theories [37, 38]. Therefore, any graph complex relatedto Poisson manifolds has dimension d = 2 while the ones related to Lie bialgebras and generalisations thereof have dimension d = 3 . As well as their dual, referred to as quasi-Lie bialgebras in the following, see footnote 11 for terminology. ( GC ) (cid:39) grt H ( GC ) (cid:39) H ( GC ) (cid:39) K H ( GC ) ? (cid:39) H ( GC ) (cid:39) K H ( GC ) (cid:39) d = 2 Poisson ( dim < ∞ ) Poisson ( dim = ∞ ) d = 3 Lie bialgebras Proto-Lie bialgebrasLie-quasi bialgebrasQuasi-Lie bialgebrasLie-quasi bialgebroids Lie bialgebroids Proto-Lie bialgebroidsQuasi-Lie bialgebroids Courant algebroidsTable 1: Classification of deformation quantization problems via graph cohomology
Organisation of this paper
The original universal model introduced by M. Kontsevich [25] takes advantage ofthe graded geometric interpretation of Poisson manifolds as 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 thedeformation theory of Lie bialgebroids and generalisations thereof will be instrumental in formulating associateduniversal models in Section 4. The class of universal models introduced in this note involves multi-oriented graphs,as introduced in [63] and studied in [39] in the context of multi-oriented props and their representations on homotopyalgebras with branes. The main definitions and results regarding the cohomology of multi-oriented graph complexesare reviewed in Section 3. Following these two review sections, we introduce our main results in Section 4. We startby reviewing the known action of the (one-colored) oriented graph complex on Lie-(quasi) bialgebras in Section4.2 and then move on to the Lie bialgebroid case in Section 4.3. Using the cohomological results reviewed inSection 3, we formulate our main result regarding the existence of an exotic Lie ∞ -structure for Lie bialgebroids ( cf. Theorem 4.10) and the action of the Grothendieck–Teichmüller group on Lie-quasi bialgebroids ( cf.
Proposition4.9). In view of the results of Section 4.3, we formulate two conjectures in Section 4.4: a no-go (Conjecture 4.13)regarding the existence of (universal) quantizations for Lie bialgebroids and a yes-go (Conjecture 4.14) regardingthe one of Lie-quasi bialgebroids (and their dual). Two appendices conclude this note. Appendix A reviewsthe (ungraded) geometric formulation of Lie bialgebroids and related notions, to be compared with the gradedgeometric interpretation of Section 2. Appendix B contains explicit formulas and additional results regarding thegraph 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 introducedin the text live in the category of (graded) vector spaces over K . Given a graded K -vector space g := (cid:76) 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 . 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. Parkfor support. This work was supported by Brain Pool Program through the National Research Foundation ofKorea (NRF) funded by the Ministry of Science and ICT (2018H1D3A1A01030137) and by Basic Science ResearchProgram through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R1A6A1A03047877). In the remaining of the text, we use the prefix dg to refer to differential graded objects. Graded geometry
It is a well-known result that a Lie algebra structure on a vector space g yields a differential structure on theexterior algebra ∧ • g ∗ in the guise of the Chevalley–Eilenberg differential. The exterior algebra can equivalentlybe recast as the algebra of functions on the shifted vector space g [1] seen as a graded manifold of degree 1 onwhich the Chevalley–Eilenberg differential defines a homological vector field. Such a supergeometric formulation ofLie algebras was generalised by A. Y. Vaintrob [55] 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 [49, 51]that dg symplectic manifolds (or NPQ -manifolds) of degree 1 (resp. of degree 2) are in bijective correspondencewith Poisson manifolds (resp. Courant algebroids). Our aim in this section is to review how Lie bialgebra andLie bialgebroid structures (and generalisations ) can be naturally recast as Hamiltonian functions for a specificgraded Poisson algebra of functions on a graded manifold (as pioneered in [32], cf. [49, 30] for details and relatedconstructions). 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 thesenotions in Section 2.2. Lie bialgebra structures (and generalisations thereof) on a vector space g can be conveniently encoded into particularHamiltonian functions on the graded manifold T ∗ ( g [1]) (cid:39) ( g ⊕ g ∗ )[1] with homogeneous coordinates (cid:8) ξ a , ζ a (cid:9) , with a ∈ { , . . . , dim g } . The latter is a graded symplectic manifold with symplectic 2-form ω = dξ a ∧ dζ a of degree .The associated Poisson bracket of degree − acts on homogeneous functions f, g ∈ C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) as: (cid:8) f, g (cid:9) g ω = ( − | f | (cid:16) ∂f∂ξ a ∂g∂ζ a + ∂f∂ζ a ∂g∂ξ a (cid:17) . (2.1)The graded Poisson bracket (2.1) can be seen as the graded geometric formulation of the “big bracket” (introducedby Y. Kosmann–Schwarzbach in [28]) acting on ∧ • ( g ⊕ g ∗ ) (cid:39) C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) . Upgrading the graded symplecticmanifold ( g ⊕ g ∗ )[1] to a dg symplectic manifold (or NPQ -manifold) 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 that Q is necessarily a Hamiltonian vector field i.e. thereexists a nilpotent ( i.e. (cid:8) H , H (cid:9) g ω = 0 ) function of degree 3 called the Hamiltonian and such that Q := (cid:8) H , · (cid:9) g ω .The most general function of degree 3 on ( g ⊕ g ∗ )[1] reads explicitly as : H = − f abc ξ a ξ b ζ c − C cab ζ a ζ b ξ c + 16 ϕ abc ζ a ζ b ζ c + 16 ψ abc ξ a ξ b ξ c (2.2)where f abc = f [ ab ] c , C cab = C c [ ab ] , ϕ abc = ϕ [ abc ] and ψ abc = ψ [ abc ] .The Hamiltonian condition (cid:8) H , H (cid:9) g ω = 0 translates as a set of 5 constraints on the defining maps { f, C, ϕ, ψ } :• D abcd := − f e [ ad f bc ] e − ψ e [ ab C c ] ed = 0 (2.3)• D dabc := − C de [ a C ebc ] − ϕ e [ ab f edc ] = 0 (2.4)• D abcd := 2 f e [ a [ c C b ] d ] e − f abe C ecd − ψ eab ϕ ecd = 0 (2.5)• D abcd := ϕ e [ ab C ecd ] = 0 (2.6)• D abcd := ψ e [ ab f cd ] e = 0 . (2.7) As reviewed in Appendix A. The subscript denotes the corresponding degree. Or equivalently, endowing the graded Poisson algebra of functions (cid:16) C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) , · , {· , ·} g ω (cid:17) with a compatible differential. Via Cartan’s homotopy formula, cf.
Lemma 2.2 of [51]. The signs and coefficients are chosen for later convenience. { 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 controlled by the dg Lie algebra (cid:16) C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) , Q , {· , ·} g ω (cid:17) . Proto-Lie bialgebras thus constitute the most general notion in the bialgebra realmand other structures (Lie-quasi, quasi-Lie and Lie bialgebras) will be defined as particular cases thereof. Theremainder of this section will therefore introduce several particular graded Poisson subalgebras of C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) whose Hamiltonian functions will encode various sub-classes of proto-Lie bialgebras. Let us start by defining thesubspace A g Lie-quasi ⊂ C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) as A g Lie-quasi := (cid:8) f ∈ C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) (cid:12)(cid:12) f | ζ =0 = 0 (cid:9) . In plain words, thesubspace A g Lie-quasi is obtained by discarding all functions of the form ψ a ··· a m ξ a · · · ξ a m , for arbitrary values of m > . It can be easily checked that A g Lie-quasi is preserved by both the pointwise product of functions andthe graded Poisson bracket (2.1) and thus defines a graded Poisson subalgebra of C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) . The mostgeneral Hamiltonian function of A g Lie-quasi reads as (2.2) with ψ ≡ , where the maps { f, C, ϕ } satisfy (2.3)-(2.6) with ψ ≡ . In particular, imposing ψ ≡ in eq.(2.3) ensures that the map f defines a genuine Liealgebra structure on g (while the structure defined on g ∗ is still “quasi” due to the presence of ϕ ). The resultingequations reproduce the defining conditions of a Lie-quasi bialgebra on ( g , g ∗ ) as introduced by Drinfel’d in[12] as semi-classicalisation of the notion of quasi-bialgebra . Dually to the previous case, one defines the gradedPoisson subalgebra A g quasi-Lie := (cid:8) f ∈ C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) (cid:12)(cid:12) f | ξ =0 = 0 (cid:9) ⊂ C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) as the subspace obtainedby discarding all functions of the form ϕ a ··· a n ζ a · · · ζ a n , for all n > . The most general Hamiltonian function of A g quasi-Lie reads as (2.2) with ϕ ≡ , where the functions { f, C, ψ } satisfy (2.3)-(2.5) and (2.7) with ϕ ≡ . Duallyto the Lie-quasi case, setting ϕ ≡ in eq.(2.4) ensures that the map C defines a genuine Lie algebra structureon g ∗ (while the structure defined on g is only “quasi” Lie due to the presence of ψ ). The resulting equationsreproduce the defining conditions of a quasi-Lie bialgebra on ( g , g ∗ ) as introduced and studied in [28, 3]. Finally,let us define the subspace A g Lie := (cid:8) f ∈ C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) (cid:12)(cid:12) f | ξ =0 = 0 and f | ζ =0 = 0 (cid:9) i.e. A g Lie is defined as theintersection A g Lie := A g Lie-quasi ∩ A g quasi-Lie between the two previous Poisson subalgebras. The latter subspace canagain be checked to be a graded Poisson subalgebra of C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) obtained by discarding all functions ofthe form ψ a ··· a m ξ a · · · ξ a m and ϕ a ··· a n ζ a · · · ζ a n for all m, n > . In particular, the most general Hamiltonianfunction of A g Lie reads as (2.2) with ϕ ≡ and ψ ≡ , where the functions { f, C } satisfy (2.3)-(2.5) with ϕ ≡ and ψ ≡ . In particular, constraint (2.3) (cid:0) resp. (2.4) (cid:1) ensures that the map f (resp. C ) defines a genuine Liealgebra 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 ∗ ) (cid:0) cf. (A.1) (cid:1) . We sum up the previous discussion in the followingproposition: Proposition 2.1.
Let g be a vector space. The following correspondences hold: • Hamiltonians in C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) are in bijective correspondence with proto-Lie bialgebra structures on ( g , g ∗ ) . • “ A g Lie-quasi “ Lie-quasi bialgebra “ . • “ A g quasi-Lie “ quasi-Lie bialgebra “ . • “ A g Lie “ Lie bialgebra “ .
This interpretation of the deformation theory for Lie bialgebras and generalisations as graded Poisson algebras willbe put to use in Section 4 where will be discussed universal models thereof. In the next section, we turn to thegeneralisation of this graded geometric interpretation to the larger class of Lie bialgebroids and variations thereof. Note that the graded Poisson bracket has intrinsic degree -2 on C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) . To recover the usual grading, one needs toconsider the 2-suspension C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) [2] . Remark that Lie-quasi bialgebras were denoted “quasi-Lie bialgebras” in [12]. We follow the terminology used in [30] where theterm Lie-quasi bialgebras is used for Lie algebras which fail to be Lie bialgebras (so that they are only “quasi” bialgebras) whilethe term quasi-Lie was reserved for the dual counterpart (not considered in [12]) where the Jacobi identity for f is only “quasi”satisfied. .2 Lie bialgebroids Letting E π → M be a vector bundle over the smooth (finite dimensional) manifold M , the relevant graded Poissonalgebra is the algebra of functions of the graded symplectic manifold (or NP -manifold) denoted (cid:0) T ∗ [2] E [1] , ω (cid:1) .The latter is of degree 2 and is locally coordinatised by the set of homogeneous coordinates (cid:110) x µ , ξ a , ζ a , p µ (cid:111) sothat the symplectic 2-form of degree can be written as ω = dx µ ∧ dp µ + dξ a ∧ dζ a . The associated Poisson bracketof degree − acts as follows on homogeneous functions f, g ∈ C ∞ (cid:0) T ∗ [2] E [1] (cid:1) : (cid:8) f, g (cid:9) Eω = ∂f∂x µ ∂g∂p µ − ∂f∂p µ ∂g∂x µ + ( − | f | (cid:16) ∂f∂ξ a ∂g∂ζ a + ∂f∂ζ a ∂g∂ξ a (cid:17) . (2.8)The latter is sometimes referred to as the “big bracket” for Lie bialgebroids. Upgrading the graded symplecticmanifold T ∗ [2] E [1] to a dg symplectic manifold (or NPQ -manifold) will allow to define various geometric structures.Following the same path as in the bialgebra case, we introduce a compatible differential through a Hamiltonianfunction of degree 3 nilpotent with respect to (2.8). The most general function of degree 3 on T ∗ [2] E [1] reads: H = ρ aµ ( x ) ξ a p µ − f abc ( x ) ξ a ξ b ζ c + R a | µ ( x ) ζ a p µ − C cab ( x ) ζ a ζ b ξ c + 16 ϕ abc ( x ) ζ a ζ b ζ c + 16 ψ abc ( x ) ξ a ξ b ξ c (2.9)where { ρ, f, R, C, ϕ, ψ } are functions on the base space M , with symmetries f abc = f [ ab ] c , C cab = C c [ ab ] , ϕ abc = ϕ [ abc ] and ψ abc = ψ [ abc ] . Imposing the Hamiltonian constraint (cid:8) H , H (cid:9) Eω = 0 yields a set of 9 conditions on thedefining functions { ρ, f, R, C, ϕ, ψ } that we denote as follows:• C abµ := 2 ρ [ aλ ∂ λ ρ b ] µ − ρ cµ f abc + R c | µ ψ cab = 0 (2.10)• C abcd := ρ [ aλ ∂ λ f bc ] d − f e [ ad f bc ] e + R d | λ ∂ λ ψ abc − ψ e [ ab C c ] ed = 0 (2.11)• C ab | µ := 2 R [ a | λ ∂ λ R b ] | µ − R c | µ C cab + ρ cµ ϕ cab = 0 (2.12)• C dabc := R [ a | λ ∂ λ C dbc ] − C de [ a C ebc ] + ρ dλ ∂ λ ϕ abc − ϕ e [ ab f edc ] = 0 (2.13)• C µν := R a ( µ ρ aν ) = 0 (2.14)• C ab | µ := ρ aλ ∂ λ R b | µ − R b | λ ∂ λ ρ aµ − ρ cµ C abc − R c | µ f cab = 0 (2.15)• C abcd := ρ [ aλ ∂ λ C b ] cd + R [ c | λ ∂ λ f abd ] + 2 f e [ a [ c C b ] d ] e − f abe C ecd − ψ eab ϕ ecd = 0 (2.16)• C abcd := R [ d | λ ∂ λ ϕ abc ] + ϕ e [ ab C ecd ] = 0 (2.17)• C abcd := ρ [ dλ ∂ λ ψ abc ] + ψ e [ ab f cd ] e = 0 . (2.18)The latter constraints identify with the component expressions of the defining conditions of a proto-Lie bial-gebroid on ( E, E ∗ ) (cid:0) compare with (A.3)-(A.7) (cid:1) . The graded Poisson algebra C ∞ (cid:0) T ∗ [2] E [1] (cid:1) admits severalsubalgebras defining in turn various sub-classes of proto-Lie bialgebroids. We start by considering the subspace A E Lie-quasi ⊂ C ∞ (cid:0) T ∗ [2] E [1] (cid:1) as A E Lie-quasi := (cid:8) f ∈ C ∞ (cid:0) T ∗ [2] E [1] (cid:1) (cid:12)(cid:12) f | p =0 , ζ =0 = 0 (cid:9) . In plain words, the subspace A E Lie-quasi is obtained by discarding all functions of the form ψ a ··· a m ( x ) ξ a · · · ξ a m , for all m > . It can beeasily checked that A E Lie-quasi is preserved by both the pointwise product of functions and the graded Poissonbracket (2.8) and thus defines a graded Poisson subalgebra of C ∞ (cid:0) T ∗ [2] E [1] (cid:1) . The most general Hamiltonianfunction of A E Lie-quasi reads as (2.9) with ψ ≡ , where the functions { ρ, f, R, C, ϕ } satisfy (2.10)-(2.17) with ψ ≡ . In particular, setting ψ ≡ in eqs.(2.10)-(2.11) ensures that the pair { ρ, f } defines a genuine Lie al-gebroid structure on E . The resulting equations reproduce the defining conditions of a Lie-quasi bialgebroid on ( E, E ∗ ) . Dually to the previous case, we define the graded Poisson subalgebra A E quasi-Lie ⊂ C ∞ (cid:0) T ∗ [2] E [1] (cid:1) as A E quasi-Lie := (cid:8) f ∈ C ∞ (cid:0) T ∗ [2] E [1] (cid:1) (cid:12)(cid:12) f | p =0 , ξ =0 = 0 (cid:9) i.e. as the subspace obtained by discarding all func-tions of the form ϕ a ··· a n ( x ) ζ a · · · ζ a n with n > . The most general Hamiltonian function of A E quasi-Lie readsas (2.9) with ϕ ≡ , where the functions { ρ, f, R, C, ψ } satisfy (2.10)-(2.16) and (2.18) with ϕ ≡ . Duallyto the Lie-quasi case, setting ϕ ≡ in eqs.(2.12)-(2.13) ensures that the pair { R, C } defines a genuine Lie The restriction from the “algebroid” case to the “algebra” case can be done by assuming that M is the one-point manifold sothat E (cid:39) g becomes a K -vector space. Where A [ n ] denotes the shifting of the grading of the fiber of the vector bundle A by n . Correspondingly, all formulas appearing in this note will be local. 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 [30]. We conclude by defining the subspace A E Lie := A E Lie-quasi ∩ A E quasi-Lie which explicitlyreads as A E Lie := (cid:8) f ∈ C ∞ (cid:0) T ∗ [2] E [1] (cid:1) (cid:12)(cid:12) f | p =0 , ξ =0 = 0 and f | p =0 , ζ =0 = 0 (cid:9) . The latter subspace can again bechecked to be a graded Poisson subalgebra of C ∞ (cid:0) T ∗ [2] E [1] (cid:1) obtained by discarding all functions of the form ψ a ··· a m ( x ) ξ a · · · ξ a m or ϕ a ··· a n ( x ) ζ a · · · ζ a n , for all m, n > . In particular, the most general Hamiltonian func-tion of A E Lie reads as (2.9) with ϕ ≡ and ψ ≡ , where the functions { ρ, f, R, C } satisfy (2.10)-(2.16) with ϕ ≡ and ψ ≡ . In particular, constraints (2.10)-(2.11) (cid:0) resp. (2.12)-(2.13) (cid:1) ensure that the pair { ρ, f } (cid:0) resp. { R, C } (cid:1) defines a genuine Lie algebroid structure on E (resp. E ∗ ). These two Lie algebroids are furthermore compatiblewith each other due to (2.13)-(2.16) and hence define a Lie bialgebroid structure on ( E, E ∗ ) (cid:0) cf. Appendix A (cid:1) .We sum up the previous discussion in the following proposition, generalising Proposition 2.1 to the bialgebroidcase:
Proposition 2.2.
Let E π → M be a vector bundle. The following correspondences hold: • Hamiltonians in C ∞ (cid:0) T ∗ [2] E [1] (cid:1) are in bijective correspondence with proto-Lie bialgebroid structures on ( E, E ∗ ) . • “ A E Lie-quasi “ Lie-quasi bialgebroid “ . • “ A E quasi-Lie “ quasi-Lie bialgebroid “ . • “ A E Lie “ Lie bialgebroid “ .
As noted previously, the graded geometric interpretation of algebro-geometric structures is instrumental for theconstruction of corresponding universal models, formulated in terms of graphs complexes. The next section willintroduce the relevant graph complexes which will be shown to act on the various (sub)-algebras previously intro-duced in Section 4.
The aim of this section is to review the definition and main results regarding multi-oriented graph complexes andtheir cohomology, as introduced and studied in [63, 64, 39] ( cf. also [40] for a review). Graph complexes are mostclearly defined as deformation complexes of a suitable morphism of operads [41]. We start by introducing therelevant graph operads of multi-directed and multi-oriented graphs from a combinatorial point of view (Section3.1) before moving to the definition of the associated graph complexes. We conclude by discussing known re-sults regarding the cohomology of multi-oriented graph complexes (Section 3.2) by putting the emphasis on someparticular classes relevant for our purpose ( cf.
Section 4).
We will denote gra
N,k the set of multidigraphs with N vertices and k directed edges. The set gra c +1 N,k of multi-directed graphs is defined as the set of ordered pairs ( γ, c ) where:• γ ∈ gra 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 : E γ × [ c ] → { + , −} where c ∈ N stands for the number of decorating colors and [ c ] := { , , . . . , c } .A pictorial representation of a multi-directed graph in gra c +1 N,k can be given by decorating each directed edge of theunderlying multidigraph in gra
N,k with c arrows of different colors ( cf. Figure 1 for an example). Or equivalently, as convolution Lie algebras constructed from suitable graph operads. Recall that multidigraphs (or quivers) are directed graphs which are allowed to contain both loops and multiple edges. By convention, we set gra N,k := gra N,k i.e. directed blacks arrows are achromatic. iiiiii ivv
Figure 1: Example of a multi-directed graph in gra , The direction of the arrow of color i on the edge e is the same as the black arrow if c ( e, i ) = + and opposite toit if c ( e, i ) = − . There is a natural right-action of the permutation group S N (resp. S k ) on elements of gra c +1 N,k bypermutation of the labeling of vertices (resp. edges). For all d ∈ N ∗ , we define the collection (cid:8) Gra c +1 d ( N ) (cid:9) N ≥ of S N -modules: Gra c +1 d ( N ) := (cid:77) k ≥ (cid:16) K (cid:68) gra c +1 N,k (cid:69) ⊗ S k sgn ⊗| d − | k (cid:17) [ k ( d − (3.1)where sgn k denotes the -dimensional sign representation of S k . The subscript stands for taking coinvariants withrespect to the diagonal right action of S k and the term between brackets denotes degree suspension. In plainwords, this means that edges carry an intrinsic degree − d and are bosonic for d odd and fermionic for d even.The S -module (cid:8) Gra c +1 d ( N ) (cid:9) N ≥ can further be given the structure of an operad by endowing it with the usualequivariant partial composition operations ◦ i : Gra c +1 d ( M ) ⊗ Gra c +1 d ( N ) → Gra c +1 d ( M + N − , cf. Figure 2 for anexample and e.g.
Section 4 of [45] for more details.1 23 i iiiii ◦ i = i iviii ii = iiiiii iv + iiiiii iv + i iiiii iv + i iiiii iv Figure 2: Example of partial composition ◦ : Gra d (3) ⊗ Gra d (2) → Gra d (4) There is a natural sequence of embeddings of operads : Gra d O (cid:126) r (cid:44) −→ Gra d O (cid:126) r (cid:44) −→ Gra d O (cid:126) r (cid:44) −→ Gra d O (cid:126) r (cid:44) −→ · · · (3.2)given by mapping each graph in Gra c +1 d to a sum of graphs in Gra c +2 d where the summation runs over all thepossible ways to orient the edges of the new direction c + 2 . We call such mapping the orientation morphism O (cid:126) r : Gra c +1 d (cid:44) −→ Gra c +2 d ( cf. Figure 3 for an example). Denoting
Lie { − d } the (1 − d ) -suspended Lie operad , O (cid:126) r (cid:0) iii (cid:1) = iii + iii + iii + iii Figure 3: The orientation morphism O (cid:126) r : Gra d (cid:44) −→ Gra d The identity element id ∈ Gra c +1 d (1) is defined as the graph id := Here,
Gra d (resp. Gra d ) stands for the operad of one-colored undirected (resp. directed) graphs. γ : Lie { − d } → Gra c +1 d sending the generator ∈ Lie { − d } (2) to the graph1 2 := + ( − d ∈ Gra c +1 d (2) is obtained by decorating 1 2with c colors and summing over all the possible orientations .A multidigraph in gra N,k will be said oriented (or acyclic) if it does not contain cycles . Contrariwise, it will besaid non-oriented (or cyclic) if it contains at least one cycle, cf. Figure 4. The subset of oriented multidigraphs21 3 , 21 3Figure 4: Example of oriented (left) and cyclic (right) graph in gra , will be denoted gra or N,k ⊂ gra N,k . This definition can be extended to multi-directed graphs by defining the subset gra c +1 | oN,k ⊆ gra c +1 N,k , with ≤ o ≤ c + 1 , of multi-directed graphs for which there exists a subset of o directions –black and/or colored – such that there are no cycles made of the corresponding arrows . Substituting gra c +1 | oN,k in place of gra c +1 N,k in (3.1) allows to define the collection (cid:110)
Gra c +1 | od ( N ) (cid:111) N ≥ of S N -modules. It is easy to checkthat the latter is closed under partial compositions and hence defines a suboperad Gra c +1 | od ⊆ Gra c +1 d of multi-oriented graphs. Note that the graph 1 2 is (trivially) multi-oriented and hence defines a morphism ofoperads γ : Lie { − d } → Gra c +1 | od for all ≤ o ≤ c + 1 . Multi-oriented graph complex
Given the multi-oriented graph operad
Gra c +1 | od , one defines the dg Lie algebraof multi-oriented graphs fGC c +1 | od as the deformation complex fGC c +1 | od := Def (cid:16)
Lie { − d } γ → Gra c +1 | od (cid:17) of themorphism of operads γ . As a graded vector space, fGC c +1 | od is defined as:• d even: fGC c +1 | od := (cid:77) N ≥ (cid:0) Gra c +1 | od ( N )[ d (1 − N )] (cid:1) S N (3.3)• d odd: fGC c +1 | od := (cid:77) N ≥ (cid:0) Gra c +1 | od ( N ) ⊗ sgn N [ d (1 − N )] (cid:1) S N (3.4)where the terms between brackets denote degree suspension while the superscript stands for taking invariants withrespect 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 γ ∈ fGC c +1 | od with N vertices and k edges is given by | γ | = d ( N −
1) + k (1 − d ) . Thegraded Lie bracket [ · , · ] of degree 0 on fGC c +1 | od is defined as usual in terms of the partial composition operations( cf. e.g. Section 4.2 of [45] for details). The differential δ := [Υ S , · ] is defined by taking the adjoint action withrespect to the Maurer–Cartan element Υ S := = + ( − d Note to conclude that the multi-oriented dg Lie algebra fGC c +1 | od is a sub-dg Lie algebra of the dg Lie algebra ofmulti-directed graphs fGC c +1 d := fGC c +1 | d defined by substituting Gra c +1 in place of Gra c +1 | o in (3.3)-(3.4). Recall that representations of
Lie { − 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 − d . In other words, 1 2 := O (cid:126) r c (cid:0) (cid:1) . For example, 1 2 := + Gra d . Recall that a cycle (or wheel) is a (non-trivial) directed path from a vertex to itself. By convention, we identify gra
N,k ≡ gra | N,K and gra or N,k ≡ gra | N,K . As an example, the graph of Figure 1 belongs to gra | , as it does not contain cycles for black and yellow arrows (although itdoes so for red and blue arrows). In the right-hand side of Figure 3, the first and fourth graphs belong to gra | , while the secondand third graphs belong to gra | , . See [33, 41]. Note that the graph degree is insensitive to the number c (resp. o ) of colored (resp. oriented) directions. For example, the graph Υ S := = + +( − d (cid:0) + (cid:1) is a Maurer–Cartan elementin fGC |• d . Equivalently, one can define fGC c +1 d := Def (cid:16)
Lie { − d } γ → Gra c +1 d (cid:17) . .2 Cohomology Having introduced the graph complex of multi-directed graphs as well as its sub-complex of multi-oriented graphs– from now on collectively denoted as fGC c +1 | od where o = 0 corresponds to the multi-directed case and o > tothe multi-oriented case – we conclude this review section by collecting some known facts about their respectivecohomologies . We start by introducing the suboperad cGra c +1 | od ⊂ Gra c +1 | od spanned by connected graphs. Thissuboperad yields a sub-dg Lie algebra fcGC c +1 | od ⊂ fGC c +1 | od . As far as cohomology is concerned, one can restrictthe analysis to the connected part since H • ( fGC c +1 | od ) = (cid:98) O (cid:0) H • ( fcGC c +1 | od )[ − d ] (cid:1) [ d ] . We start by noting thatthe sequence of embeddings of operads (3.2) induces a sequence of injective quasi-isomorphisms of complexes : fcGC d ∼ (cid:44) −→ O (cid:126) r fcGC d ∼ (cid:44) −→ O (cid:126) r fcGC d ∼ (cid:44) −→ O (cid:126) r fcGC d ∼ (cid:44) −→ O (cid:126) r · · · (3.5)More generally, the injection O (cid:126) r : fcGC c +1 | od ∼ (cid:44) −→ fcGC c +2 | od is a quasi-isomorphism for all ≤ o ≤ c + 1 . Hence,adding extra colored direction does not change the cohomology. The situation gets more interesting when orientingextra directions. The following important theorem is due to T. Willwacher for o = 0 [57] and has been generalisedto arbitrary o by M. Živković [63]: Theorem 3.1.
For all ≤ o ≤ c , there is an isomorphism of graded Lie algebras H • ( fcGC c +1 | od ) (cid:39) H • ( fcGC c +1 | o +1 d +1 ) . (3.6) Furthermore, this identification preserves the additional grading provided by the first Betti number . For later use, we summarise in Table 2 the cohomology in degrees 0, 1 and 2 of the (undirected) graph complex indimension d = 1 , , . H ( fcGC d ) H ( fcGC d ) H ( fcGC d ) d = 1 K (cid:104) Θ (cid:105) K (cid:104) L (cid:105) d = 2 grt ? ? d = 3 K (cid:104) L (cid:105) Table 2: Cohomology in low dimension and degree
Climbing the dimension ladder
Informally, Theorem 3.1 allows to map familiar structures in low dimensions to novel incarnations thereof in higher dimensions. The most striking example of such hierarchy of structuresstems from another important theorem of T. Willwacher [56] showing the existence of an isomorphism of Lie algebras H ( fcGC ) (cid:39) grt where grt denotes the infinite dimensional Grothendieck–Teichmüller algebra. Combined withTheorem 3.1, we obtain a sequence H ( fcGC ) (cid:39) H ( fcGC | ) (cid:39) H ( fcGC | ) (cid:39) · · · (cid:39) grt of incarnations of grt See Section 5 of [39] and Section 7 of [40] for reviews. Where (cid:98) O ( g ) denotes the completed symmetric algebra associated with the graded vector space g . See [56] and [57] for the cases o = 0 , respectively and [39] for a general statement. Here, fcGC d stands for the usual Kontsevich graph complex of connected undirected graphs. See [56, 10] for the first arrow and [39] for a general statement. The first Betti number is defined as b := k − N + 1 for a connected graph with N vertices and k edges. The cocycle L stands for the triangle loop 21 3 . Regarding the Θ -cocycle and the Grothendieck–Teichmüller algebra grt ,see below. See footnote 3 for the interpretation of the dimension d . Since the isomorphism (3.6) preserves both the graph degree and the first Betti number, a connected cocycle γ ∈ gra N,k indimension d is mapped to a cocycle γ (cid:48) ∈ gra N (cid:48) ,k (cid:48) in dimension d + 1 with N (cid:48) = k + 1 and k (cid:48) = 2 k − N + 1 . More generally, theincarnation of a cocycle γ ∈ gra N,k of H • ( fcGC c +1 | od ) in dimension d (cid:48) > d is a graph γ (cid:48) ∈ gra N (cid:48) ,k (cid:48) with N (cid:48) = N + ( d (cid:48) − d ) b and k (cid:48) = k + ( d (cid:48) − d ) b with b the first Betti number.
10n arbitrary dimensions d ≥ . Different incarnations yield different actions of the Grothendieck–Teichmüllergroup on various algebro-geometric structures. In its original incarnation via directed cocycles in H ( fcGC ) , theGrothendieck–Teichmüller group naturally acts via Lie ∞ -automorphisms on the Schouten Lie algebra of polyvectorfields T poly on (finite dimensional) manifolds. This yields an action on the space of universal formality maps henceon the space of universal quantization maps for finite dimensional Poisson manifolds [25, 26, 37, 56, 9, 23]. Indimension 3, there is a natural action of GRT via oriented cocycles in H ( fcGC | ) on the deformation complexof Lie bialgebras hence on the space of universal formality maps related to the deformation quantization ofLie bialgebras [57, 42, 43]. Another important result for our story concerns the manifold incarnations of the Θ -cocycle ∈ gra , spanning the first cohomology class of fcGC i.e. H ( fcGC ) (cid:39) K (cid:104) Θ (cid:105) . Again, applyingTheorem 3.1 results in a sequence of isomorphisms H ( fcGC ) (cid:39) H ( fcGC | ) (cid:39) H ( fcGC | ) (cid:39) · · · (cid:39) K . In itsoriginal incarnation as the cocycle spanning H ( fcGC ) , the Θ -graph can be recursively extended to a non-trivialMaurer–Cartan element Υ Θ := • • + + · · · in the graded Lie algebra (cid:0) fcGC , [ · , · ] (cid:1) , see e.g. [24]. ThisMaurer–Cartan element is mapped to the Moyal star-commutator [20, 46] via the natural action of fcGC on thealgebra of functions of a (bosonic) symplectic manifold. Considering the case d = 2 yields another importantincarnation of the Θ -graph as the oriented Kontsevich–Shoikhet cocycle – denoted Θ in the following – andspanning H ( fcGC | ) , cf. Figure 5. The latter first appeared implicitly in [48] (see also [7, 58]) as the obstructionto the existence of a cycle-less universal quantization of Poisson manifolds beyond order (cid:126) . It then appearedexplicitly in [53] 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 [57]. The corresponding Maurer–Cartanelement Υ Θ induces an exotic (and essentially unique) universal Lie ∞ -structure on polyvector fields deformingnon-trivially the Schouten algebra on infinite dimensional manifolds [53]. The latter can then be considered asthe avatar in d = 2 of the Moyal star-commutator in d = 1 .21 34 −
21 34 +
21 34Figure 5: Kontsevich–Shoikhet cocycle Θ ∈ H ( fcGC | ) Of special interest for our purpose is the incarnation of the Θ -cocycle in dimension , dubbed Θ in the following.The latter is a combination of bi-oriented graphs with N = 6 and k = 7 , (see Appendix B) that will be arguedto provide an obstruction to the universal quantization of Lie bialgebroids in Section 4. For example, the tetrahedron graph t ∈ H ( fcGC ) is mapped to an oriented cocycle t ∈ H ( fcGC | ) with N = 7 verticesand k = 9 edges (see footnote 38). Recall that the Grothendieck–Teichmüller group is defined by exponentiation of the pro-nilpotent Grothendieck–Teichmülleralgebra i.e.
GRT := exp( grt ) , see [59] for a review. As reviewed in Section 4.2. The fact that quantization of Lie bialgebras involves oriented graphs was already recognised in [14]. Potential obstructions in promoting the Θ -graph to a full Maurer–Cartan element lie in H ( fcGC ) (cid:39) K (cid:104) L (cid:105) , cf. Table 2.Since the obstruction to the prolongation of the Θ -graph at order k ≥ has Betti number k + 2 , it never hits the loop graph L (of Betti number ) so that the prolongation is unobstructed at all orders and can be performed recursively. The argument carriesidentically for incarnations of the Θ -graph in higher dimensions. 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. [43] for explicit transcendental formulas. More generally, the Maurer–Cartan element Υ Θ d associated with the incarnation Θ d of the Θ -graph in dimension d is a sum Υ Θ d = + Θ d + · · · = (cid:88) p ≥ Υ p Θ d where the graph Υ p Θ d has N = 2 p ( d −
1) + 2 vertices and k = 2 pd + 1 edges.For d = 1 , we recover the sum of graphs Υ Θ := (cid:88) p ≥ p + 1)! p + 1 edges Universal models
The aim of the present section is to introduce universal models of multi-oriented graphs ( cf.
Section 3) for Liebialgebroids – and variations thereof – using the graded geometric picture reviewed in Section 2. We start byproviding an abstract characterisation of universal models and emphasise their relevance to address questionsrelated to formality theory and deformation quantization. We then review universal models for Lie bialgebras (andtheir “quasi” versions) before moving on to the Lie bialgebroid case. We conclude by discussing the implications ofour results regarding the deformation quantization problem for Lie bialgebroids.
Let ( g , δ, [ · , · ] g ) be a dg Lie algebra and denote ( H ( g ) , , [ · , · ] H ( g ) ) the associated cohomology endowed with thecanonical dg Lie algebra structure inherited from g (with trivial differential). Let furthermore Φ : ( H ( g ) , ∼ −→ ( 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 ) ) (cid:54) = [Φ( x ) , Φ( y )] g with x, y ∈ H ( g ) – so that Φ is not a morphism of dg Lie algebras. Accordingto the homotopy transfer theorem , any dg Lie algebra g is quasi-isomorphic (as a Lie ∞ -algebra) to itscohomology H ( g ) endowed with a certain Lie ∞ -structure ( H ( g ) , l ) deforming the canonical dg Lie algebra structure ( H ( g ) , , [ · , · ] H ( g ) ) i.e. l = 0 , l = [ · , · ] H ( g ) and the higher order brackets l > are transferred from the dg Liealgebra structure on g . In other words, any quasi-isomorphism of complexes Φ : ( H ( g ) , ∼ −→ ( g , δ ) can beupgraded to a quasi-isomorphism of Lie ∞ -algebras U : ( H ( g ) , l ) ∼ −→ ( g , δ, [ · , · ] g ) with U = Φ . If the higherbrackets l > vanish, then ( g , δ, [ · , · ] g ) and ( H ( g ) , , [ · , · ] H ( g ) ) are quasi-isomorphic as Lie ∞ -algebras and g is said tobe formal . The homotopy transfer theorem thus allows to reduce questions regarding formality (such as existenceof formality maps and their classification) to the study of the space of Lie ∞ -structures deforming the canonical dgLie structure ( H ( g ) , , [ · , · ] H ( g ) ) . The relevant deformation theory is controlled by the Chevalley–Eilenberg dg Liealgebra CE (cid:0) H ( g ) (cid:1) endowed with the Nijenhuis–Richardson bracket [ · , · ] NR and the differential δ S := (cid:104) [ · , · ] H ( g ) , · (cid:105) NR .Since we are interested in formality maps given by universal formulas, our aim is to introduce – for each deformationquantization problem at hand – a universal model for the deformation theory of H ( g ) in the guise of a dg Liealgebra of graphs (collectively denoted GC ) together with a morphism of dg Lie algebras GC → CE (cid:0) H ( g ) (cid:1) . Example 4.1 (Universal model for polyvector fields) . The paradigmatic example of the above construction is dueto M. Kontsevich [25] 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 bystudying the deformation theory of the Schouten algebra ( T poly , , [ · , · ] S ) , controlled by the Chevalley–Eilenberg dgLie algebra CE ( T poly ) . In [25], M. Kontsevich introduced a dg Lie algebra of graphs – denoted fGC – togetherwith a morphism of dg Lie algebras fGC → CE ( T poly ) given by explicit local formulas . The dg Lie algebra fGC can therefore be interpreted as a universal model for CE ( T poly ) allowing to reduce important questions related toformality theory to the cohomology of fGC :• Existence : Obstructions to the existence of universal formality maps live in H ( fGC ) .• Classification : The space of universal formality maps is classified by H ( fGC ) . See e.g.
Theorem 10.3.1 of [33] for a statement as well as Chap. 10 for details and history. The explicit formulas defining the morphism fGC → CE ( T poly ) take advantage of the graded geometric formulation of Poissonmanifolds as dg symplectic manifolds of degree 1, see [56] for the affine space case ( cf. also the earlier work [37] as well as [45]for a generalisation to dg symplectic manifolds of arbitrary degree), [23] for a globalisation to any smooth manifolds and [11] for ageneralisation to the sheaf of polyvector fields on any smooth algebraic variety. We start by reviewing some known results regarding universal models on Lie bialgebras, see e.g. [57, 42, 43]. Let uscome back to the graded symplectic manifold T ∗ ( g [1]) (cid:39) ( g ⊕ g ∗ )[1] of Section 2.1 endowed with a set of homogeneouslocal coordinates (cid:8) ξ a , ζ a (cid:9) , with a ∈ { , . . . , dim g } . Using this set of coordinates, one can (locally) endow the gradedalgebra of functions C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) with a natural structure of Gra | -algebra, with Gra | the operad of 1-directedgraphs in dimension 3. Explicitly, we define a morphism of operads Rep g : Gra | → End C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) via thefollowing sequence of morphisms of graded vector spaces: Rep g N : Gra | ( N ) ⊗ C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) ⊗ N → C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) for N ≥ as (4.1) Rep g N ( γ )( f ⊗ · · · ⊗ f N ) = µ N (cid:16)(cid:89) e ∈ E γ ∆ e ( f ⊗ · · · ⊗ f N ) (cid:17) where• The f i ’s are functions on ( g ⊕ g ∗ )[1] .• The symbol µ N denotes the multiplication map on N elements: µ N : C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) ⊗ N → C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) : f ⊗ · · · ⊗ f N (cid:55)→ f · · · f N (4.2)• The product is performed over the set E γ of edges of the graph γ ∈ Gra | ( N ) .For each edge e ∈ E γ connecting vertices labeled by integers i and j , the operator ∆ e is defined as: ∆ i j = ∂∂ξ a ( i ) ∂∂ζ ( j ) a (4.3)where the sub(super)scripts ( i ) or ( j ) indicate that the derivative acts on the i -th or j -th factor in the tensorproduct.The representation Rep g yields a sequence of morphisms of operads Lie {− } γ (cid:44) −→ Gra | Rep g −→ End C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) mapping the generator of Lie {− } to the graded Poisson bracket (2.1) via the graph 1 2 := − Gra | on C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) is well-defined, it fails to preserve the various subalgebrasintroduced in Section 2.1 due to the presence of cycle graphs. Example 4.2.
Consider the following action of a cycle graph
Rep g (cid:0) (cid:1) ( f ⊗ f ) = ∂ f ∂ξ a ∂ζ b ∂ f ∂ζ a ∂ξ b :• f = f ∼ ξξζ ∈ A g Lie-quasi yields
Rep g (cid:0) (cid:1) ( f ⊗ f ) ∼ ξξ / ∈ A g Lie-quasi .• f = f ∼ ξζζ ∈ A g quasi-Lie yields Rep g (cid:0) (cid:1) ( f ⊗ f ) ∼ ζζ / ∈ A g quasi-Lie .• A fortiori , the action of 1 2 fails to preserve A g Lie := A g Lie-quasi ∩ A g quasi-Lie .This defect can be cured by resorting to oriented graphs hence by substituting Gra | in place of Gra | in (4.1). Proposition 4.3.
Let g be a vector space. • The graded algebra of functions C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) is endowed with a structure of Gra | -algebra. Here
End V stands for the endomorphism operad associated with the (graded) vector space V . Recall that the grading of a graph in
Gra c +1 | od is given by | γ | = k (1 − d ) with k the number of edges. More generally, the action of the 3-Gerstenhaber operad
Ger on the algebra of functions factors through Gra | as Ger (cid:44) −→ Gra | Rep g −→ End C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) . The graded subalgebras A g Lie-quasi , A g quasi-Lie and A g Lie ⊂ C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) are endowed with a structure of Gra | -algebra. Note that there is no “middle case” between the proto-Lie bialgebra case and the Lie-quasi, quasi-Lie, Lie bialgebracases since restricting to oriented graphs is both necessary and sufficient to preserve any of the three subalgebras.As we will see, this is in contradistinction with the “bialgebroid” case in which Lie-quasi and quasi-Lie bialgebroidsprovide a true intermediate case between Lie and proto-Lie bialgebroids, cf.
Section 4.3.Applying Def ( Lie {− } → · ) on both sides of the morphism Rep g yields the following proposition: Proposition 4.4. • The morphism of operads
Rep g : Gra | → End C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) induces a morphism of dg Lie algebras (cid:16) fcGC , δ, [ · , · ] (cid:17) → (cid:16) CE (cid:0) C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) [2] (cid:1) , δ S , [ · , · ] NR (cid:17) . (4.4)• The morphisms of operads
Rep g : Gra | → End A g sub induce morphisms of dg Lie algebras (cid:16) fcGC | , δ, [ · , · ] (cid:17) → (cid:16) CE ( A g sub [2]) , δ S , [ · , · ] NR (cid:17) (4.5) where A g sub stands for A g Lie-quasi , A g quasi-Lie and A g Lie ⊂ C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) . We denoted CE ( g ) the Chevalley–Eilenberg cochain space (in the adjoint representation) associated with the vectorspace g while [ · , · ] NR stands for the Nijenhuis–Richardson bracket and δ S := (cid:2) {· , ·} g ω , · (cid:3) NR for the Chevalley–Eilenbergdifferential associated with the graded Poisson bracket (2.1). In plain words, Proposition 4.4 states that the graphcomplex fcGC provides a universal model for the deformation theory of proto-Lie bialgebras while fcGC | is auniversal model for the deformation theory of Lie-quasi, quasi-Lie and Lie bialgebras. Combining Proposition 4.4with Theorem 3.1 yields an action of the pro-unipotent Grothendieck–Teichmüller group exp (cid:0) H ( fcGC | ) (cid:1) (cid:39) GRT on the three subalgebras A g sub via Lie ∞ -automorphisms . This is consistent with the known action of GRT (via the GRT -torsor of Drinfel’d associators) on quantization maps for Lie bialgebras [16, 54, 22, 42, 19, 43] and Lie-quasibialgebras [15, 52]. We now move on to the main result of this note by introducing novel universal models for the deformation complexesof the family of variations on Lie bialgebroids reviewed in Section 2.2 and Appendix A. Let E π → M be a vector bun-dle and consider the graded symplectic manifold (cid:0) T ∗ [2] E [1] , ω (cid:1) coordinatised by the set of homogeneous coordinates (cid:26) x µ , ξ a , ζ a , p µ (cid:27) . The algebra of functions on T ∗ [2] E [1] carries a natural action Rep E : Gra | → End C ∞ (cid:0) T ∗ [2] E [1] (cid:1) of the operad Gra | of bi-directed graphs containing both black and red directions. The action Rep E is definedsimilarly 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: ∆ i j = ∂∂x µ ( i ) ∂∂p ( j ) µ , ∆ i j = ∂∂ξ a ( i ) ∂∂ζ ( j ) a (4.6)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.The representation Rep E maps the graph 2 3 = + − (cid:0) + (cid:1) towards thegraded Poisson bracket (2.8) and furthermore yields a sequence of morphisms of operads Lie {− } γ (cid:44) −→ Gra | Rep E −→ End C ∞ (cid:0) T ∗ [2] E [1] (cid:1) . Although the action of Gra | on C ∞ (cid:0) T ∗ [2] E [1] (cid:1) is well-defined, it fails to preserve the varioussubalgebras introduced in Section 2.2 due to the presence of cycle graphs. In comparison, there is no such action of
GRT on C ∞ (cid:0) ( g ⊕ g ∗ )[1] (cid:1) as H ( fcGC ) (cid:39) K , cf. Table 2. Similarly as the Lie bialgebra case, the action of the 3-Gerstenhaber operad
Ger on the algebra of functions factors through Gra | as Ger (cid:44) −→ Gra | Rep E −→ End C ∞ (cid:0) T ∗ [2] E [1] (cid:1) . xample 4.5. Considering the following action of a cycle graph
Rep E (cid:0) (cid:1) ( f ⊗ f ) = ( − | f | ∂ f ∂x µ ∂ξ a ∂ f ∂p µ ∂ζ a (4.7)on f = f ( x ) ab ξ a ζ b and f = f ( x ) µ | ab p µ ζ a ζ b ∈ A E quasi-Lie yielding: Rep E (cid:0) (cid:1) ( f ⊗ f ) = − f ( x ) µ | c [ a ∂ µ f ( x ) cb ] ζ a ζ b / ∈ A E quasi-Lie . (4.8)More generally, it can be checked that a graph fails to preserve A E quasi-Lie if and only if it contains a red cycle.Dually, graphs with black cycles fail to preserve A E Lie-quasi . This defect can be cured by resorting to oriented graphs hence by substituting
Gra | in place of Gra | in (4.1), thus allowing to preserve A E Lie-quasi (resp. A E quasi-Lie ) byorienting black (resp. red) arrows. In the following, we will distinguish between two incarnations of Gra | , denoted Gra | black (resp. Gra | red ) with oriented black (resp. red) directions .Crucially, preserving the intersection A E Lie := A E Lie-quasi ∩ A E quasi-Lie requires to orient both directions, thus yieldingan action of Gra | on A E Lie . As pointed out in Section 3.2 (see Theorem 3.1), the number of oriented colors (incontradistinction with the number of directed colors) is the relevant factor to compute the respective cohomology.From the previous simple observation will then follow that the Lie bialgebroid case differs essentially from the dualcases of Lie-quasi and quasi-Lie bialgebroids. We sum up the previous discussion by the following proposition:
Proposition 4.6.
Let E π → M be a vector bundle. • The graded algebra of functions C ∞ (cid:0) T ∗ [2] E [1] (cid:1) is endowed with a structure of Gra | -algebra. • The graded subalgebra A E Lie-quasi is endowed with an action of the operad
Gra | black of bi-directed graphs withoriented black arrows. • The graded subalgebra A E quasi-Lie is endowed with an action of the operad Gra | red of bi-directed graphs withoriented red arrows. • The graded subalgebra A E Lie is endowed with an action of the operad
Gra | of bi-oriented graphs. Applying Def ( Lie {− } → · ) on both sides of Rep E : Gra |• → End A E yields the following proposition: Proposition 4.7. • The morphism of operads
Rep E : Gra | → End C ∞ (cid:0) T ∗ [2] E [1] (cid:1) induces a morphism of dg Lie algebras (cid:16) fcGC , δ, [ · , · ] (cid:17) → (cid:16) CE (cid:0) C ∞ (cid:0) T ∗ [2] E [1] (cid:1) [2] (cid:1) , δ S , [ · , · ] NR (cid:17) . (4.9)• The morphisms of operads
Rep E : Gra | → End A E quasi induce morphisms of dg Lie algebras (cid:16) fcGC | , δ, [ · , · ] (cid:17) → (cid:16) CE (cid:0) A E quasi [2] (cid:1) , δ S , [ · , · ] NR (cid:17) (4.10) where A E quasi stands for A E Lie-quasi , A E quasi-Lie ⊂ C ∞ (cid:0) T ∗ [2] E [1] (cid:1) . • The morphism of operads
Rep E : Gra | → End A E Lie induces a morphism of dg Lie algebras (cid:16) fcGC | , δ, [ · , · ] (cid:17) → (cid:16) CE (cid:0) A E Lie [2] (cid:1) , δ S , [ · , · ] NR (cid:17) . (4.11)The conventions used are as in the bialgebra case and δ S := (cid:104) {· , ·} Eω , · (cid:105) NR stands for the Chevalley–Eilenbergdifferential 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 upin Table 3. Alternatively, both actions could be written in terms of
Gra | black (say) by making use of ∆ i j = ∂∂ξ a ( i ) ∂∂ζ ( j ) a for A E Lie-quasi and ∆ (cid:48) i j = ∂∂ζ ( i ) a ∂∂ξ a ( j ) for A E quasi-Lie . H • ( fcGC | ) (cid:39) H • ( fcGC ) Lie-quasi bialgebroids Oriented Unoriented H • ( fcGC | black ) (cid:39) H • ( fcGC ) Quasi-Lie bialgebroids Unoriented Oriented H • ( fcGC | red ) (cid:39) H • ( fcGC ) Lie bialgebroids Oriented Oriented H • ( fcGC | ) (cid:39) H • ( fcGC ) Table 3: Cohomology groups for Lie bialgebroid structures
Remark 4.8.
The morphism (4.9) on the deformation complex of proto-Lie bialgebroids can be seen as a particularsubcase of the action of fcGC on the deformation complex of Courant algebroids [45] when restricted to the splitcase . The latter does not yield interesting structures in degrees 0 and 1 as the dominant level of the relevantcohomology H • ( fcGC ) is located in degree − . The corresponding cohomology space H − ( fcGC ) is a unitalcommutative algebra spanned by trivalent graphs modulo the IHX relation where the rôle of the unit is playedby the Θ -graph (see e.g. [4]). Given a proto-Lie bialgebroid E π → M represented by the Hamiltonian function H (cid:0) see (2.9) (cid:1) , each trivalent graph γ ∈ H − ( fcGC ) yields a cocycle function Ω γ ∈ C ∞ ( M ) (cid:0) i.e. such that (cid:8) H , Ω γ (cid:9) Eω = 0 (cid:1) thus yielding a conformal flow on the space of proto-Lie bialgebroids on E ( cf. [45] for details).Turning to Lie-quasi and quasi-Lie bialgebroids, the morphism (4.10) yields a natural extension of the action ofthe Grothendieck–Teichmüller group exp (cid:0) H ( fcGC | ) (cid:1) (cid:39) GRT on the deformation complexes of Lie-quasi andquasi-Lie bialgebras to the “bialgebroid” case. Proposition 4.9.
The Grothendieck–Teichmüller group acts via
Lie ∞ -automorphisms on the deformation complexof Lie-quasi bialgebroids and quasi-Lie bialgebroids. We refer to Section 4.4 for a discussion of this action in the context of deformation quantization. Explicitly, theaction of
GRT is through graphs with one oriented color (either black or red) and as such generically fails topreserve the sub-deformation complex A E Lie for Lie bialgebroids. This is in contradistinction with the Lie bialgebracase whose deformation complex A g Lie does carry a representation of
GRT . Rather the action of fcGC | endows A E Lie with a new
Lie ∞ -structure: Theorem 4.10.
The deformation complex of Lie bialgebroids is endowed with an exotic
Lie ∞ -structure deformingnon-trivially the big bracket (2.8) . This (essentially unique ) exotic Lie ∞ -structure, denoted θ , is obtained by mapping the Maurer–Cartan element Υ Θ ∈ fcGC | to a multi-differential operator on A E Lie via the representation
Rep E . The non-vanishing bracketstake the form : θ = Rep E (Υ S ) = {· , ·} Eω , θ = Rep E (Θ ) , θ = Rep E (Υ ) , . . . , θ p +2 = Rep E p +2 (Υ p Θ ) , . . . The minimal Lie ∞ -structure ( A E Lie , θ ) can be interpreted as the avatar in dimension d = 3 both of the Moyalstar-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 onlyone orientable direction is available. In fact, one can explicitly check that the first non-trivial deformed bracket θ = Rep E (Θ ) vanishes identically on the graded Poisson subalgebra A g Lie ⊂ A E Lie controlling deformations ofLie bialgebras, cf.
Proposition B.3. Here by split Courant algebroids we mean Courant algebroids whose underlying vector bundle is a Whitney sum E ⊕ E ∗ . Up to gauge transformations and rescalings. See Section 3.2, specifically footnote 45. The intrinsic degree carried by each bracket is given by | θ p +2 | = − p − . Pulling back the brackets along the suspensionmap s : A E Lie [2] → A E Lie of degree 2 yields a series of brackets ˜ θ p +2 on A E Lie [2] with the usual degree − p . Recall that minimal
Lie ∞ -algebras are characterised by a vanishing differential θ ≡ . See footnote 13. emark 4.11. • In the following, we will consider Maurer–Cartan elements in the formal extension A E Lie [[ (cid:126) ]] of A E Lie by a formalparameter (cid:126) . By analogy with the d = 2 case [43], we will refer to Maurer–Cartan elements of ( A E Lie [[ (cid:126) ]] , θ ) as formal “quantizable Lie bialgebroids” , to contrast with the Maurer–Cartan elements of ( A E Lie [[ (cid:126) ]] , {· , ·} Eω ) referred to simply as formal Lie bialgebroids . Formal Lie bialgebroids being linear in (cid:126) are just Liebialgebroids and accordingly, we will refer to formal “quantizable Lie bialgebroids” linear in (cid:126) as “quantizableLie bialgebroids”.• Note that “quantizable Lie bialgebroids” are in particular Lie bialgebroids as they satisfy (cid:8) H , H (cid:9) Eω = 0 ontop of some higher consistency conditions (cid:0) θ ( H ∧ ) = 0 , etc. (cid:1) .The distinction between Lie bialgebroids and “quantizable Lie bialgebroids” will be salient when applied to thequantization problem for Lie bialgebroids in Section 4.4. The quantization problem for Lie-(quasi) bialgebras was formulated by V. Drinfel’d ( cf. [14] Question 1.1 for Liebialgebras and §5 for Lie-quasi bialgebras) and solved in [16] by Etingof–Kazhdan for the Lie bialgebra case and in [15, 52] for the Lie-quasi bialgebra case. In both cases, the solution is universal and involves the use ofa Drinfel’d associator, yielding an action of the Grothendieck–Teichmüller group GRT on the set of inequivalentuniversal quantization maps. The latter can be traced back to the action of H ( fcGC | ) (cid:39) grt on the deformationcomplex of Lie and Lie-quasi bialgebras, as reviewed in Section 4.2. The situation for Lie bialgebras (and theirquasi versions) is therefore much akin to the situation for finite dimensional Poisson manifolds, in that both casesshare the following important features ( cf. 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 [61, 62] 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 [44] – with R ≡ C ∞ ( M ) – whose semi-classicalisation reproducesthe original Lie bialgebroid structure. The quantization problem for Lie bialgebroids then consists in associatingto each Lie bialgebroid a quantum groupoid quantizing it. Although the quantization problem for a generic Liebialgebroid remains open, several explicit examples of quantizations for particular Lie bialgebroids have beenexhibited in the literature. Apart from the above mentioned quantization of Lie bialgebras [16], it was shownin [62] that M. Kontsevich’s solution to the quantization problem for (finite dimensional) Poisson manifolds [26]ensures that Lie bialgebroids associated with Poisson manifolds ( cf. Example A.2) constitute another example ofLie bialgebroids admitting a quantization . This result was shown to extend to regular triangular Lie algebroidsin [62] (see also [47]) using methods à la Fedosov [17] and to generic triangular Lie bialgebroids in [6] using ageneralisation of Kontsevich’s formality theorem for Lie algebroids. In this context, the following natural conjecturewas formulated by P. Xu:
Conjecture 4.12 (Xu [62], Section 6) . Every Lie bialgebroid admits a quantization as a quantum groupoid.
Although Conjecture 4.12 might still hold true in the most general setting, we would like to argue for the non-existence of universal quantizations of Lie bialgebroids, on the basis of the following results from Section 4.3: See also [16, 54, 22, 42, 19, 43]. More precisely, the Kontsevich star product ∗ quantizing the Poisson bivector π provides a Drinfel’d twistor J ∗ ∈ (cid:0) U R ( E ) ⊗ R U R ( E ) (cid:1) [[ (cid:126) ]] for the standard bialgebroid U R ( E ) , where R ≡ C ∞ ( M ) and E ≡ T M . Twisting U R ( E ) by J ∗ then provides aquantization of the Lie bialgebroid associated with π on ( T M , T ∗ M ) , see [62]. Recall that a universal quantization admits formulas given by expansions in terms of graphs with universal coefficients.
17. The Grothendieck–Teichmüller group plays no classifying rôle regarding the universal deformations (and hencequantizations) 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 concludethat the quantization problem for Lie bialgebroids differs essentially from its Lie bialgebra analogue and is in factmore akin to the quantization problem for infinite dimensional manifolds. In view of this analogy, the obstructionappearing in Theorem 4.10 can be understood as the avatar in d = 3 of the Kontsevich–Shoikhet obstruction in d = 2 . As shown in [48, 53, 7, 58], 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 infinitedimensional Poisson manifolds. Pursuing the analogy with the d = 2 case, we conjecture the following: Conjecture 4.13 (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 theoryof (associative) bialgebroids, which goes beyond the ambition of the present note . We nevertheless concludethe present discussion by outlining a strategy of proof for Conjecture 4.13 by mimicking the two-steps procedureof [43] for Poisson manifolds and Lie bialgebras and adapting it to the case at hand. Denoting C • GS ( O E , O E (cid:1) the equivalent of the Gerstenhaber–Schack complex for the standard commutative co-commutative bialgebroidstructure on the symmetric algebra O E associated to the vector bundle E , the former should be endowed with a Lie ∞ -algebra structure µ – generalising the one of [35, 41] for the bialgebra case – whose corresponding Maurer–Cartan elements are quantum groupoids. By analogy with the bialgebra case, the cohomology of (cid:0) C • GS ( O E , O E ) , µ (cid:1) should be isomorphic as a graded Lie algebra to the deformation complex (cid:0) A E Lie , {· , ·} Eω (cid:1) of Lie bialgebroids on E .Although these two Lie ∞ -algebras should coincide in cohomology, we do not expect them to be quasi-isomorphicas Lie ∞ -algebras, i.e. (cid:0) C • GS ( O E , O E (cid:1) , µ (cid:1) is not formal. To show explicitly that the obstruction to formality ishit would require computing the Lie ∞ -algebra structure obtained by transfer of µ on H • (cid:0) C • GS ( O E , O E ) , µ (cid:1) andshowing that the latter coincides with the exotic Lie ∞ -structure θ on A E Lie64 , as is the case in d = 2 [53, 43].This would provide a trivial (in the sense that no Drinfel’d associator is needed) formality Lie ∞ quasi-isomorphism (cid:0) A E Lie , θ (cid:1) ∼ −→ (cid:0) C • GS ( O E , O E ) , µ (cid:1) , yielding in turn a quantization map for (formal) “quantizable Lie bialgebroids”(the Maurer–Cartan elements of θ in A E Lie [[ (cid:126) ]] , cf. Remark 4.11). Finally, the fact that θ is not Lie ∞ -isomorphic tothe big bracket in A E Lie ( cf. Theorem 4.10) would prevent the existence of a formality morphism for Lie bialgebroids.We sum up these (non)-formality conjectures for Lie bialgebroids in Figure 6. (cid:0) A E Lie , {· , ·} Eω (cid:1) (cid:0) A E Lie , θ (cid:1) (cid:0) C • GS ( O E , O E ) , µ (cid:1) Lie bialgebroids “QuantizableLie bialgebroids” Quantumgroupoids × ∼× ∼
Figure 6: Conjectural (non)-formality maps for Lie bialgebroidsNote that the situation is markedly different in the Lie-quasi (and quasi-Lie) bialgebroid case . Firstly, recallthat the Maurer–Cartan element Υ Θ is not gauge-related to the Maurer–Cartan element Υ S in fcGC | – since Θ is a non-trivial cocycle in fcGC | – hence θ is indeed a non-trivial deformation of the big bracket in A E Lie .However, the cocycle Θ is a coboundary in fcGC | ( cf. Appendix B) so that there exists a combination of The deformation theory of bialgebras is well understood using the framework of properads [41]. Like their bialgebras counterpart,bialgebroids involve both a monoid and a co-monoid, but over different (lax) monoidal categories whose interplay is rather subtleand as such cannot be described using the theory of properads (at least in its standard form). We are grateful to T. Basile andD. Lejay for clarifications regarding this fact. This is only possible thanks to the fact that the exotic
Lie ∞ -structure θ on A E Lie of Theorem 4.10 is minimal and hence is apotential candidate for being the cohomology of another
Lie ∞ -structure. For definiteness, we will focus on the Lie-quasi case, keeping in mind that the arguments apply similarly to the dual case. ϑ ∈ fcGC | such that Θ = − δϑ ∈ fcGC | . In order for Υ Θ and Υ S to be gauge-related in fcGC | ,one needs to find a degree element ϑ = ϑ + ϑ + · · · + ϑ p of fcGC | such that Υ Θ = e ad ϑ Υ S . Contrarilyto the problem of prolongating the cocycle Θ to the Maurer–Cartan element Υ Θ – which can be solved by atrivial induction – to display an explicit gauge map ϑ is a highly non-trivial task as the higher obstructions live in H ( fcGC | ) (cid:39) H ( fcGC ) , i.e. the recipient of the obstructions to the universal quantization of finite-dimensional Poisson manifolds. Although this cohomological space conjecturally vanishes (Drinfel’d–Kontsevich),maps allowing to convert cocycles into coboundaries are highly non-trivial and necessarily involve the choice ofa Drinfel’d associator (consistently with the fact that two coboundaries differ by the choice of an element in H ( fcGC ) (cid:39) grt ). Up to the Drinfel’d–Kontsevich conjecture, it is nevertheless expected that, given a Drinfel’dassociator, one can define a Lie ∞ -isomorphism (cid:0) A E Lie-quasi , {· , ·} Eω (cid:1) ∼ −→ (cid:0) A E Lie-quasi , θ (cid:1) . Repeating the argumentlaid down in the Lie bialgebroid case, one needs to find a (trivial)
Lie ∞ quasi-isomorphism (cid:0) A E Lie-quasi , θ (cid:1) ∼ −→ (cid:0) C • quasi − GS ( O E , O E ) , µ (cid:1) , where the right-hand side stands for the deformation complex of the quantum objectassociated to a Lie-quasi bialgebroid. The relevant category here is the one of quasi-bialgebroids , a commongeneralisation 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 [5].Composing with the (non-trivial) Lie ∞ -isomorphism (cid:0) A E Lie-quasi , {· , ·} Eω (cid:1) ∼ −→ (cid:0) A E Lie-quasi , θ (cid:1) would yield a formalitymorphism for Lie-quasi bialgebroids. We sum up the discussion by formulating the following conjecture and recapthe corresponding conjectural formality maps in the Lie-quasi case in Figure 7.
Conjecture 4.14 (Yes-go) . Given a Drinfel’d associator, one can define a universal quantization of Lie-quasibialgebroids as quasi-quantum groupoids. (cid:0) A E Lie-quasi , {· , ·} Eω (cid:1) (cid:0) A E Lie-quasi , θ (cid:1) (cid:0) C • quasi − GS ( O E , O E ) , µ (cid:1) Lie-quasibialgebroids “QuantizableLie-quasi bialgebroids” Quasi-quantumgroupoids ∼ (cid:9) GRT ∼∼ (cid:9) GRT ∼ Figure 7: Conjectural formality maps for Lie-quasi bialgebroidsTo conclude, let us note that particularising the conjectural quantization map of Conjecture 4.14 to Lie bialgebroidsensures that every Lie bialgebroid admits a quantization as a quasi-quantum groupoid (but generically not as aquantum groupoid as stated in Conjecture 4.12). However, for the particular subclass of “quantizable Lie bialge-broids” (which are in particular Lie bialgebroids, see Remark 4.11) – such as Lie bialgebras and coboundary Liebialgebroids ( cf.
Appendix B) – the associated quantization should yield a quantum groupoid. According to theabove picture, the exotic
Lie ∞ -structure θ of Theorem 4.10 can therefore be seen as a concrete means to delineatethe 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 → ∧ g is a cocycle for the Lie algebra ( g , [ · , · ] g ) i.e. ∆ g ([ x, y ]) = ad x ∆ g ( y ) − ad y ∆ g ( x ) where the representation used is the extension ad : g ⊗ ( ∧ g ) → ∧ g of the adjoint action of ( g , [ · , · ] g ) on ∧ g asad x ( y ∧ z ) = [ x, y ] g ∧ z + y ∧ [ x, z ] g . Letting { e a } | a ∈{ ,..., dim g } be a basis of g , one denotes [ e a , e b ] g = f abc e c and The graph ϑ p has N = 4 p + 1 vertices and k = 6 p edges so that to have degree 0 in d = 3 . We refer to [43] for an explicit construction in d = 2 . The first obstruction vanishes since Θ is exact in fcGC | . The second obstruction Υ − [ ϑ , Θ ] can be checked to live in H ( fcGC | ) . See Table 2. g ( e c ) = C cab e a ⊗ e b . The three defining conditions of a Lie bialgebra read: f e [ ad f bc ] e = 0 , C de [ a C ebc ] = 0 , f abe C ecd − f e [ a [ c C b ] d ] e = 0 (A.1)where f abc = f [ ab ] c and C cab = C c [ ab ] . 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 → T M is a morphism of vector bundles called the anchor • [ · , · ] E : Γ ( E ) ⊗ Γ ( E ) → Γ ( E ) is a K -bilinear map called the bracket 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 Leibniz rule : [ X, f · Y ] E = ρ X [ f ] · Y + f · [ X, Y ] E 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 ( T M ) defines a morphism of Lie algebras between theLie algebra (Γ ( E ) , [ · , · ] E ) and the Lie algebra of vector fields on M , i.e. ρ [ X,Y ] E = [ ρ X , ρ Y ] for all X, Y ∈ Γ ( E ) . Proposition A.1.
Let ( E, ρ, [ · , · ] E ) be a Lie algebroid. The following statements hold:1. Let { x µ } | µ ∈{ ,..., dim M } be a set of coordinates of M and { e a } | a ∈{ ,..., dim E } be a basis of Γ ( E ) . Setting ρ e a [ f ] = ρ aµ ( x ) ∂ µ f and [ e a , e b ] E = f abc ( x ) e c , the defining conditions of a Lie algebroid can be expressed incomponents as: f abc = − f bac , ρ [ aν ∂ ν ρ b ] µ = ρ cµ f abc , ρ [ cν ∂ ν f ab ] d = f e [ cd f ab ] e . (A.2) Acting on generic sections of E , the Lie algebroid bracket reads [ X, Y ] E = (cid:0) ρ X [ Y c ] − ρ Y [ X c ] + f abc X a Y b (cid:1) e c .2. The exterior algebra Γ ( ∧ • E ∗ ) is naturally endowed with a structure of dg commutative algebra with differential d E : Γ ( ∧ • E ∗ ) → Γ (cid:0) ∧ • +1 E ∗ (cid:1) defined by: • ( d E f )( X ) = ρ X [ f ] • ( d E ω )( X, Y ) = ρ X [ ω ( Y )] − ρ Y [ ω ( X )] − ω ([ X, Y ] E ) • d E ( α ∧ β ) = ( d E α ) ∧ β + ( − | α | α ∧ ( d E β ) for all X, Y ∈ Γ ( E ) , f ∈ C ∞ ( M ) , ω ∈ 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 (cid:0) ∧ • + ◦− E (cid:1) defined as follows: (cid:8) f, g (cid:9) E = 0 , (cid:8) X, f (cid:9) E = ρ X [ f ] , (cid:8) X, Y (cid:9) E = [ X, Y ] E (cid:8) P, Q (cid:9) E = − ( − ( | p |− | q |− (cid:8) Q, P (cid:9) E , (cid:8) P, Q ∧ R (cid:9) E = (cid:8) P, Q (cid:9) E ∧ R + ( − | q | ( | p |− Q ∧ (cid:8) P, R (cid:9) 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: (cid:8) (cid:8) P, Q (cid:9) E , R (cid:9) E + ( − ( | P |− | Q | + | R | ) (cid:8) (cid:8) Q, R (cid:9) E , P (cid:9) E + ( − ( | R |− | P | + | Q | ) (cid:8) (cid:8) R, P (cid:9) E , Q (cid:9) E = 0 . Example A.2. • A Lie algebra is a Lie algebroid whose base manifold is a point. We will use the same symbol ρ to denote the induced map of sections ρ : Γ ( E ) → Γ ( T M ) .
20 Given a manifold M , the standard Lie algebroid is defined as the tangent bundle T M together with theidentity map as anchor and the usual Lie bracket of vector fields as bracket. The corresponding differentialon the space of differential forms Ω • ( M ) (cid:39) Γ ( ∧ • T ∗ M ) coincides with the de Rham differential while theinduced Gerstenhaber bracket on Γ ( ∧ • T M ) identifies with the Schouten bracket on polyvector fields.• Let ( M , π ) be a Poisson manifold. The dual tangent bundle T ∗ M is naturally endowed with a Lie algebroidstructure with anchor π (cid:93) : Γ ( T ∗ M ) → Γ ( T M ) : α µ dx µ (cid:55)→ π µν α ν ∂ µ and bracket [ α, β ] π = (cid:0) L π (cid:93) ( α ) β −L π (cid:93) ( β ) α + i π (cid:93) ( α ) dβ − i π (cid:93) ( β ) dα (cid:1) for all α, β ∈ Γ ( T ∗ M ) . Lie bialgebroids
The concept of Lie bialgebroid was introduced by Mackenzie–Xu in [34] as the infinitesimalvariant of a Poisson groupoid. We follow the modern definition of [29] 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 ∗ ) . 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 T M and T ∗ M as defined in ExampleA.2 are compatible in the above sense and hence define a Lie bialgebroid structure on ( T M , T ∗ M ) .Lie bialgebroids thus generalise both Lie bialgebras and Poisson manifolds. In fact, the base manifold M of anyLie bialgebroid is endowed with a canonical Poisson bracket defined as (cid:8) f, g (cid:9) = (cid:104) d E ∗ f, d E g (cid:105) (cid:0) or in componentsas (cid:8) f, g (cid:9) = R a [ µ ρ aν ] ∂ µ f ∂ ν g (cid:1) .The exterior algebra (cid:0) Γ ( ∧ • E ) , ∧ (cid:1) 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 Gerstenhaberbracket. Such a quadruplet (cid:0) Γ ( ∧ • E ) , ∧ , d E ∗ , {· , ·} E (cid:1) is called a strong differential Gerstenhaber algebra. It wasin fact shown in [60] (see also [29]) that Lie bialgebroid structures on a vector bundle E π → M are in one-to-onecorrespondence with strong differential Gerstenhaber structures on (cid:0) Γ ( ∧ • E ) , ∧ (cid:1) (cid:0) or equivalently with dg Poissonstructures on (Γ ( ∧ • E ) [1] , ∧ ) (cid:1) . Coboundary Lie bialgebroids
Letting E be a Lie algebroid, an r -matrix is a section Λ ∈ Γ (cid:0) ∧ E (cid:1) satisfying (cid:8) X, (cid:8) Λ , Λ (cid:9) E (cid:9) E = 0 for all X ∈ Γ ( E ) . An r -matrix endows E with a structure of Lie bialgebroid by defining d E ∗ = (cid:8) Λ , · (cid:9) 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 strongercondition (cid:8) Λ , Λ (cid:9) E = 0 holds, the induced Lie bialgebroid is said to be triangular [34] (cid:0) cf. for example the Liebialgebroid on Poisson manifolds defined in Example A.2 (cid:1) . If furthermore, Λ is of constant rank, the triangularLie bialgebroid is said to be regular . Quasi-Lie, Lie-quasi and proto-Lie bialgebroids
The above mentioned characterisation of Lie bialgebroidsas dg Poisson structures on (cid:0)
Γ ( ∧ • E ) [1] , ∧ (cid:1) calls for several natural generalisations, as summarised in the followingtable (cid:0) see e.g. [30, 50, 18, 2] (cid:1) : Note that the defining condition of a Lie bialgebroid can equivalently be stated as the fact that the differential d E 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 bialgebroidis self-dual. More generally, for a proto-Lie bialgebroid, the Jacobi identity for the bivector is deformed as π ρ [ λ ∂ ρ π µν ] = 13 R a [ λ R b | µ R c | ν ] ψ abc + 13 ρ a [ λ ρ bµ ρ cν ] ϕ abc . As is transparent from the correspondence below, there is a series of inclusions of bialgebroids: Lie ⊂ Quasi-Lie ⊂ Proto-Lie andLie ⊂ Lie-quasi ⊂ Proto-Lie. Note furthermore that the notions of Lie bialgebroids and proto-Lie bialgebroids are self-dual (cid:0) andthus can be defined on both
Γ ( ∧ • E ) [1] and Γ ( ∧ • E ∗ ) [1] (cid:1) while the notions of Lie-quasi bialgebroids and quasi-Lie bialgebroids are
21 Lie bialgebroids on ( E, E ∗ ) ⇔ dg Poisson algebras on Γ ( ∧ • E ) [1] • Quasi-Lie bialgebroids on ( E, E ∗ ) ⇔ Homotopy Poisson algebras 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 (cid:0) two anchors ρ , R and two brackets [ · , · ] E , [ · , · ] E ∗ (cid:1) , a proto-Lie bialgebroid contains two elements ϕ ∈ Γ (cid:0) ∧ E (cid:1) and ψ ∈ Γ (cid:0) ∧ E ∗ (cid:1) which play the rôle of various obstructions to the usual Lie bialgebroid identities. Letting E ∗ [1] be the shifted dualbundle coordinatised by { x µ , ζ a } of respective degree and , the graded commutative algebra (cid:0) C ∞ ( E ∗ [1]) , · (cid:1) isisomorphic to the exterior algebra of sections (cid:0) Γ ( ∧ • E ) , ∧ (cid:1) . The most general curved homotopy Poisson structure l on Γ ( ∧ • E ) [1] thus take the form :• l := ϕ abc ζ a ζ b ζ c • l ( f ) := d E ∗ f = R a | µ ζ a ∂f∂x µ − C cab ζ a ζ b ∂f∂ζ c • l ( f, g ) := (cid:8) f, g (cid:9) E = − ρ aµ (cid:16) ∂f∂ζ a ∂g∂x µ + ( − | f | ∂f∂x µ ∂g∂ζ a (cid:17) + ( − | f | f abc ζ c ∂f∂ζ a ∂g∂ζ b • l ( f, g, h ) := ( − | g | ψ abc ∂f∂ζ a ∂g∂ζ b ∂h∂ζ c .Imposing the defining quadratic condition [ l, l ] NR = 0 of a (curved) Lie ∞ -algebra yields a series of identities whichprecisely reproduce the components conditions (2.10)-(2.18) as:• [ l , l ] NR = 0 ⇔ C = 0 (A.3)• [ l , l ] NR + [ l , l ] NR = 0 ⇔ C = C = 0 (A.4)• [ l , l ] NR + [ l , l ] NR = 0 ⇔ C = C = C = 0 (A.5)• [ l , l ] NR + [ l , l ] NR = 0 ⇔ C = C = 0 (A.6)• [ l , l ] NR = 0 ⇔ C = 0 . (A.7)Imposing ψ ≡ (resp. ϕ ≡ ) yields a Lie-quasi (resp. quasi-Lie) bialgebroid and Lie bialgebroids are recoveredby setting ψ ≡ , ϕ ≡ . Assuming that the base manifold M is the one-point manifold and denoting the vectorspace Γ ( E ) as g allows to define the counterparts of these notions in the “bialgebra” realm, cf. Section 2.1.
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 Theorem4.10 generated by the cocycle class [Θ ] ∈ H ( fcGC | ) . For concreteness, we fix a representative of the class [Θ ] as follows: Proposition B.1.
There is a unique pair of combination of graphs Θ ∈ fcGC | and ϑ ∈ fcGC | black such that:1. Θ = − δϑ i.e. Θ is exact in fcGC | black .2. Θ contains only graphs of the shape C , cf. Figure 8 .3. Each graph of ϑ contains at least one red cycle i.e. ϑ / ∈ fcGC | . Remark B.2. • The combination of graphs ϑ contains 68 black-oriented graphs (48 graphs of shape A and 20 graphs of shape B ) while the combination Θ contains 288 bi-oriented graphs of shape C . dual to each other. 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. [36, 31]. A dg Poisson algebra is thus a homotopyPoisson algebra for which the brackets of arity above 2 vanish. The bracket l p being of degree − p , the fact that each bracket is a multi-derivation for the underlying graded commutativealgebra constrains all brackets of arity higher than 3 to vanish.
22 Although Θ is exact in fcGC | black , it is crucial to note that Θ is not exact in fcGC | i.e. there is nocombination of graphs κ ∈ fcGC | such that Θ = − δκ . Hence Θ is a non-trivial cocycle in fcGC | .21346 A B
21 34 56 C Figure 8: Shape of graphs involved in ϑ ( A and B ) and Θ ( C )Let E π → M be a vector bundle. To each Lie bialgebroid structure on ( E, E ∗ ) (represented by the Hamiltonianfunction H ∈ A E Lie ), we will associate the functions• ϑ ( H ) := Rep E ( ϑ )( H ⊗ ) ∈ A E Lie-quasi • Θ ( H ) := Rep E (Θ )( H ⊗ ) ∈ A E Lie .Note that the condition Θ = − δϑ ensures that Θ ( H ) ∼ Q [ ϑ ( H )] – where the differential Q is defined as Q := (cid:8) H , · (cid:9) Eω – so that Θ ( H ) is a coboundary in the complex ( A E Lie-quasi , Q ) . However, Θ ( H ) is generically notexact in A E Lie and the obstruction for Θ ( H ) to be a coboundary in A E Lie is precisely given by the component of ϑ ( H ) proportional to ζ . We will denote this obstruction as Ob ( H ) abc , so that Ob ( H ) abc = 0 ⇒ ϑ ( H ) ∈ A E Lie and Θ ( H ) is a trivial cocycle in ( A E Lie-quasi , Q ) . A straightforward computation gives:Ob ( H ) abc = R d | µ R e | ν (cid:0) ρ f λ ∂ µ C e [ a | f ∂ λν C dbc ] − ρ dβ ∂ βν R [ a | λ ∂ µλ C ebc ] − ρ dλ ∂ µ C e [ a | f ∂ λν C f bc ] − ∂ µ R f | λ f ef [ a ∂ νλ C dbc ] − f df [ a ∂ µ C eb | g ∂ ν C gc ] f + 2 f fg [ a ∂ µ C eb | f ∂ ν C dc ] g (cid:1) + R d | µ ∂ µ R e | ν (cid:0) f df [ a C efg ∂ ν C gbc ] − f ef [ a C dfg ∂ ν C gbc ] + ρ dλ ∂ λ C e [ a | f ∂ ν C f bc ] − ρ eλ ∂ λ C d [ a | f ∂ ν C f bc ] − ρ eλ ∂ λ C f [ ab ∂ ν C dc ] f (cid:1) + ρ dβ ∂ β R e | ν R d | µ ∂ µν R [ a | λ ∂ λ C ebc ] +2 ρ eλ R d | µ ∂ λµ R [ a | ν C def ∂ ν C f bc ] + ρ gν R d | µ C def ∂ ν C e [ ab ∂ µ C f c ] g . The latter encodes the first obstruction for H to define a “quantizable Lie bialgebroid”. Although the obstructiondoes 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 ≡ yields Ob ( H ) abc = 0 hence the obstruction vanishes for Lie bialgebras. More generally, it can bechecked that each graph appearing in the combinations ϑ and Θ contains at least one arrow of the type i j so that both ϑ ( H ) and Θ ( H ) vanish identically on the graded Poisson subalgebra A g Lie ⊂ A E Lie .For coboundary Lie bialgebroids, we perform the replacement R a | µ = ρ bµ Λ ba and C cab = − ρ cµ ∂ µ Λ ab − d [ a f dcb ] ,with Λ ab = Λ [ ab ] a bivector, see Appendix A. Under this replacement, it can be checked that Ob ( H ) abc identicallyvanishes modulo the defining conditions C ≡ , C ≡ ensuring that the maps ( ρ, f ) define a Lie algebroid. See footnote 13. eferences [1] M. Alexandrov, A. Schwarz, O. Zaboronsky and M. Kontsevich “ The Geometry of the masterequation and topological quantum field theory ” Int. J. Mod. Phys. A arXiv:hep-th/9502010 [2] P. Antunes, J. M. Nunes da Costa “ Split Courant algebroids as L ∞ -structures ” Journal of Geometryand Physics arXiv:1912.09791 [3] M. Bangoura, Y. Kosmann–Schwarzbach “ The double of a Jacobian quasi-bialgebra ” Letters in Math-ematical Physics D. Bar-Natan “ On the Vassiliev knot invariants ” Topology T. Basile, D. Lejay, K. Morand “ Quasi-bialgebroids ” (In preparation)[6]
D. Calaque “ Formality for Lie Algebroids ” Communications in Mathematical Physics arXiv:math/0404265 [7]
G. Dito “ The necessity of wheels in universal quantization formulas ” Communications in MathematicalPhysics arXiv:1308.4386 [8]
V. A. Dolgushev “ Erratum to: “A Proof of Tsygan’s Formality Conjecture for an Arbitrary SmoothManifold” ” arXiv:math/0703113 [9] V. A. Dolgushev “ Stable Formality Quasi-isomorphisms for Hochschild Cochains ” arXiv:1109.6031 [10] V. A. Dolgushev, C. L. Rogers “ The cohomology of the full directed graph complex ” Algebras andRepresentation Theory 1-45 (2019) arXiv:1711.04701 [11]
V. A. Dolgushev, C. L. Rogers and T. Willwacher “ Kontsevich’s graph complex, GRT, and thedeformation complex of the sheaf of polyvector fields ” Annals of Mathematics Second Series arXiv:1211.4230 [12]
V. G. Drinfel’d “ Quasi-Hopf algebras ” Algebra i Analiz V. G. Drinfel’d “ On quasitriangular quasi-Hopf algebras and on a group that is closely connected with
Gal( Q / Q ) ” Algebra i Analiz V. G. Drinfel’d “ On some unsolved problems in quantum group theory ” Lecture Notes in Mathematics,Springer Berlin Heidelberg 1-8 (1992)[15]
B. Enriquez, G. Halbout “ Quantization of quasi-Lie bialgebras ” J. Amer. Math. Soc. arXiv:0804.0496 [16] P. Etingof, D. Kazhdan “ Quantization of Lie bialgebras, I ” Selecta Mathematica arXiv:q-alg/9506005 [17] B. Fedosov “ A simple geometrical construction of deformation quantization ” J. Differential Geom. Y. Fregier, M. Zambon “ Simultaneous deformations and Poisson geometry ” Compositio Mathematica arXiv:1202.2896 [19]
G. Ginot, S. Yalin “ Deformation theory of bialgebras, higher Hochschild cohomology and formality ” arXiv:1606.01504 [20] H. J. Groenewold “ On the Principles of elementary quantum mechanics ” Physica A. Grothendieck “ Esquisse d’un Programme ” London Math. Soc. Lect. Note Ser. V. Hinich, D. Lemberg “ Formality theorem and bialgebra deformations ” Annales de la Faculté des sciencesde Toulouse : Mathématiques Série Tome 25 no. 2-3 569-582. (2016) arXiv:1410.2132 [23]
C. Jost “ Globalizing
Lie ∞ automorphisms of the Schouten algebra of polyvector fields ” Differential Geometryand its Applications arXiv:1201.1392 A. Khoroshkin, T. Willwacher and M. Živković “ Differentials on graph complexes ” Advances inMathematics arXiv:1411.2369 [25]
M. Kontsevich “ Formality conjecture ” Deformation Theory and Symplectic Geometry M. Kontsevich “ Deformation quantization of Poisson manifolds. I. ” Lett. Math. Phys. arXiv:q-alg/9709040 [27] M. Kontsevich “ Operads and Motives in Deformation Quantization ” Lett. Math. Phys. arXiv:math/9904055 [28] Y. Kosmann-Schwarzbach “ Jacobian quasi-bialgebras and quasi-Poisson Lie groups ” in M. Gotay,J. E. Marsden and V. Moncrief (eds) Mathematical Aspects of Classical Field Theory Contemporary Math-ematics
American Mathematical Society Providence, RI 459-489 (1992)[29]
Y. Kosmann–Schwarzbach “ Exact Gerstenhaber algebras and Lie bialgebroids ” Acta Applicandae Math-ematica Y. Kosmann-Schwarzbach “ Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory ” inJ. E. Marsden, T. S. Ratiu (eds) The Breadth of Symplectic and Poisson Geometry Progress in Mathematics
H. Lang, Y. Sheng, and X. Xu “ Strong homotopy Lie algebras, homotopy Poisson manifolds and Courantalgebroids ” Letters in Mathematical Physics arXiv:1312.4609 [32]
P. Lecomte, C. Roger “ Modules et cohomologies des bigèbres de Lie ” Comptes rendus de l’Académie dessciences Série 1 Mathématique
13 893-894 (1990)[33]
J. L. Loday, B. Vallette “ Algebraic Operads ” Grundlehren der mathematischen Wissenschaften
K. C. H. Mackenzie, P. Xu “ Lie bialgebroids and Poisson groupoids ” Duke Mathematical Journal M. Markl “ Intrinsic brackets and the L ∞ -deformation theory of bialgebras ” arXiv:math/0411456 [36] R. A. Mehta “ Supergroupoids, double structures, and equivariant cohomology ” arXiv:math/0605356 [37] S. Merkulov “ Exotic automorphisms of the Schouten algebra of polyvector fields ” arXiv:0809.2385 [38] S. Merkulov “ Operads, configuration spaces and quantization ” Bulletin of the Brazilian MathematicalSociety arXiv:1005.3381 [39] S. Merkulov “ Multi-oriented props and homotopy algebras with branes ” Letters in Mathematical Physics arXiv:1712.09268 [40]
S. Merkulov “ Grothendieck-Teichmueller group, operads and graph complexes: a survey ” arXiv:1904.13097 [41] S. Merkulov, B. Vallette “ Deformation theory of representations of prop(erad)s ” Journal für die reineund angewandte Mathematik
634 51-106 (2009) arXiv:0707.0889 [42]
S. Merkulov, T. Willwacher “ Classification of universal formality maps for quantizations of Lie bialge-bras ” arXiv:1605.01282 [43] S. Merkulov, T. Willwacher “ An explicit two step quantization of Poisson structures and Lie bialgebras ”Commun. Math. Phys. arXiv:1612.00368 [44]
I. Moerdijk, J. Mrčun “ On the universal enveloping algebra of a Lie algebroid ” Proc. Amer. Math. Soc. arXiv:0801.3929 [45]
K. Morand “ M. Kontsevich’s graph complexes and universal structures on graded symplectic manifolds I ” arXiv:1908.08253 [46] J. E. Moyal “ Quantum mechanics as a statistical theory ” Mathematical Proceedings of the CambridgePhilosophical Society R. Nest, B. Tsygan “ Formal deformations of symplectic manifolds with boundary ” Journal für die reineund angewandte Mathematik (Crelles Journal)
M. Penkava, P. Vanhaecke “ Deformation Quantization of Polynomial Poisson Algebras ” Journal ofAlgebra arXiv:math/9804022 [49]
D. Roytenberg “ Courant algebroids, derived brackets and even symplectic supermanifolds ” arXiv:math/9910078 [50] D. Roytenberg “ Quasi-Lie bialgebroids and twisted Poisson manifolds ” Letters in Mathematical Physics arXiv:math/0112152 [51] D. Roytenberg “ On the structure of graded symplectic supermanifolds and Courant algebroids ” Contemp.Math.
Amer. Math. Soc. (2002) arXiv:math/0203110 [52]
Š. Sakáloš, P. Ševera “ On quantization of quasi-Lie bialgebras ” Sel. Math. New Ser. arXiv:1304.6382 [53] B. Shoikhet “ An L ∞ algebra structure on polyvector fields ” Selecta Mathematica arXiv:0805.3363 [54] D. E. Tamarkin “ Quantization of lie Bialgebras via the Formality of the operad of Little Disks ” GAFAGeometric And Functional Analysis A. Y. Vaintrob “ Lie algebroids and homological vector fields ” Russian Mathematical Surveys T. Willwacher “ M. Kontsevich’s graph complex and the Grothendieck-Teichmueller Lie algebra ” Invent.math. arXiv:1009.1654 [57]
T. Willwacher “ The Oriented Graph Complexes ” Communications in Mathematical Physics arXiv:1308.4006 [58]
T. Willwacher “ The obstruction to the existence of a loopless star product ” Comptes Rendus Mathematique
11 881-883 (2014) arXiv:1309.7921 [59]
T. Willwacher “ The Grothendieck–Teichmüller Group ” Unpublished notes[60]
P. Xu “ Gerstenhaber algebras and BV-algebras in Poisson geometry ” Communications in MathematicalPhysics arXiv:dg-ga/9703001 [61]
P. Xu “ Quantum groupoids and deformation quantization ” Comptes Rendus de l’Académie des Sciences -Series I - Mathematics issue 3 289-294 (1998) arXiv:q-alg/9708020 [62]
P. Xu “ Quantum groupoids ” Communications in Mathematical Physics arXiv:math/9905192 [63]
M. Živković “ Multi-directed graph complexes and quasi-isomorphisms between them I: oriented graphs ” arXiv:1703.09605 [64] M. Živković “ Multi-directed graph complexes and quasi-isomorphisms between them II: Sourced graphs ” arXiv:1712.01203arXiv:1712.01203