OON HOMOTOPY LIE BIALGEBROIDS
DENIS BASHKIROV AND ALEXANDER A. VORONOV
Abstract.
Lie algebroids appear throughout geometry and mathematical physicsimplementing the idea of a sheaf of Lie algebras acting infinitesimally on asmooth manifold. A well-known result of A. Vaintrob characterizes Lie alge-broids and their morphisms in terms of homological vector fields on super-manifolds, which might be regarded as objects of derived geometry. This leadsnaturally to the notions of an L ∞ -algebroid and an L ∞ -morphism. The situ-ation with Lie bialgebroids and their morphisms is more complicated, as theycombine covariant and contravariant features. We approach Lie bialgebroidsin the way of odd symplectic dg-manifolds, building on D. Roytenberg’s thesis.We extend Lie bialgebroids to the homotopy Lie case and introduce the no-tions of an L ∞ -bialgebroid and an L ∞ -morphism. The case of L ∞ -bialgebroidsover a point coincides with the O. Kravchenko’s notion of an L ∞ -bialgebra,for which the notion of an L ∞ -morphism seems to be new. Contents
1. Introduction 12. Lie algebroids 23. Lie bialgebroids 104. The Hamiltonian approach 135. L ∞ -bialgebroids 16References 191. Introduction
The notion of a Lie bialgebra was introduced in the seminal works of V. Drinfeld[Dri83, Dri87] on algebraic aspects of the quantum inverse scattering method. A Liebialgebra g is a Lie algebra equipped with a one-cocycle δ : g → g ∧ g (a cobracket ),whose dual δ ∗ yields a Lie bracket on g . As a quintessential example, Lie bialgebrasappear as infinitesimal counterparts of Poisson-Lie groups. Geometrization of thisnotion leads to the concept of a Lie bialgebroid that natually arises in the Poisson-geometric context. In particular, there is a canonical Lie bialgebroid associated toany Poisson manifold. The aim of this note is to introduce an extension of thisconcept to the case of graded manifolds and homotopy Lie structures.We survey the basic definitions, motivating examples and results concerning Liealgebroids, L ∞ -algebroids, and Lie bialgebroids in sections 2 and 3. In section 4 wereview the Hamiltonian approach to Lie (bi)algebroids of D. Roytenberg and thengive a Hamiltonian characterization of Lie (bi)algebroid morphisms (Theorem 4.9).In the final section, we introduce the notions of an L ∞ -bialgebroid and an L ∞ -morphism of L ∞ -bialgebroids and list some relevant examples. Date : August 16, 2017. a r X i v : . [ m a t h . QA ] A ug D. BASHKIROV AND A. A. VORONOV
Conventions.
The ground field is R by default. The dual V ∗ of a graded vectorbundle V is understood as the direct sum of the duals of its graded components,graded in such a way that the natural pairing V ∗ ⊗ V → R is grading-preserving. Inparticular, ( V [ n ]) ∗ = V ∗ [ − n ]. By default, the degree on a bigraded vector bundle,such as S ( V ), stands for the total degree. Differentials are assumed to have degree1. We use the exterior algebra ∧ • V and the symmetric algebra S ( V [ − V , whereas thelatter is reserved for graded ones.We assume vector bundles to have finite rank and graded vector bundles tohave locally finite rank. Likewise, all graded manifolds will be assumed to havefinited-dimensional graded components. We will work with smooth graded mani-folds , which we will understand as locally ringed spaces ( M, C ∞ V ) := ( M, S ( V ∗ )),where V → M is a graded vector bundle over a manifold M and S ( V ∗ ) is the gradedsymmetric algebra over C ∞ M on the graded dual to the sheaf V of sections.A morphism V → W of graded manifolds is a morphism of locally ringed spaces( M, S ( V ∗ )) → ( N, S ( W ∗ )). will assume that a differential , i.e ., a degree-one, R -linear derivation d of the structure sheaf satisfying d = 0 is given on a gradedmanifold. Morphisms of dg-manifolds will have to respect differentials. Since wework in the C ∞ category, we will routinely substitute sheaves with spaces of theirglobal sections.A graded manifold V over M comes with a morphism V → M given by theinclusion of C ∞ M into the function sheaf of V . There is also a relative basepoint given by the zero section of the vector bundle V → M , which induces a morphism M → V of graded manifolds over M . A based morphism must respect zero sectionsof the structure vector bundles. The structure differential will also be assumed tobe based , that is to say, the zero section M → V must be a dg-morphism.We will follow a common trend and confuse the notation V for a vector bundle V and the sheaf V of its sections, when it is clear what we mean from the context.For a manifold M with a Poisson tensor π , we will use π to denote the naturalmorphism T ∗ M → T M determined by the condition (cid:104) α ∧ β, π (cid:105) = (cid:104) β, π ( α ) (cid:105) , α, β ∈ T ∗ M. Lie algebroids
Basic definitions and examples.Definition 2.1. A Lie algebroid structure on a vector bundle V → M over asmooth manifold M consists of • an R -bilinear Lie bracket [ , ] : Γ( V ) ⊗ R Γ( V ) → Γ( V ) on the space ofsections; • a morphism of vector bundles ρ : V → T M , called the anchor map ,subject to the Leibniz rule[
X, f Y ] = f [ X, Y ] + ( ρ ( X )( f )) Y, X, Y ∈ Γ( V ) , f ∈ C ∞ ( M ) . It follows, in particular, that the anchor map is a morphism of Lie algebras:(1) ρ ([ X, Y ]) = [ ρ ( X ) , ρ ( Y )] , X, Y ∈ Γ( V ) . Examples 2.2. (1) Any Lie algebra can be regarded as a Lie algebroid over a point.(2) The tangent bundle
T M taken with the standard Lie bracket of vector fieldsand ρ = id : T M → T M is trivially a Lie algebroid.
N HOMOTOPY LIE BIALGEBROIDS 3 (3) More generally, any integrable distribution V ⊂ T M is a Lie algebroid with ρ : V → T M being the inclusion. Thus, a regular foliation on a manifoldgives rise to a Lie algebroid.(4) A family of Lie algebras over a manifold M , i.e ., a vector bundle with a Liebracket bilinear over functions on M , is a Lie algebroid with a zero anchor.(5) Let g be a Lie algebra acting on a manifold M via an infinitesimal actionmap g → Γ( T M ), X (cid:55)→ ξ X . Then g × M → M is a Lie algebroid with( X, m ) (cid:55)→ ξ X ( m ) as the anchor and the bracket defined pointwise by[ X, Y ]( m ) := [ X ( m ) , Y ( m )] g + ξ X Y ( m ) − ξ Y X ( m ) , where a section X of g × M → M is identified with a function X : M → g .(6) Every Lie groupoid gives rise to a Lie algebroid, see Example 3.7(5). Forexample the tangent Lie algebroid of Example (2) above comes from the pairgroupoid of a manifold M . The manifold of objects of the pair groupoid is M itself, the manifold of morphisms is M × M , with morphism compositiongiven by ( x, y ) ◦ ( y, z ) := ( x, z ) , x, y, z ∈ M. (7) If M is a Poisson manifold with the Poisson bivector π ∈ Γ( ∧ T M ), thenthe canonical morphism π : T ∗ M → T M together with the Koszul bracket { α, β } π = L π ( α ) ( β ) − L π ( β ) ( α ) − d ( ι π ( α ∧ β ))determines a Lie algebroid structure on T ∗ M .(8) Given a vector bundle V → M , the space of derivative endomorphisms Der ( V ) is defined as the space of all linear endormorphisms D : Γ( V ) → Γ( V ) such that there exists D M ∈ T M , and D ( f X ) = f D ( X ) + D M ( f ) X for any X ∈ Γ( V ), f ∈ C ∞ ( M ). Then Der ( V ) equipped with the standardcommutator bracket and the mapping ρ : D (cid:55)→ D M as the anchor is a Liealgebroid.(9) For a principal G -bundle P over a manifold M , the quotient T P/G of T P by the induced action of G is known as the Atiyah Lie algebroid of P . Thebracket and the anchor map are naturally inherited from T P .The notion of a morphism of Lie algebroids V → W defined over the same basemanifold M is rather straightforward: it is a vector bundle morphism φ : V → W such that φ ([ X, Y ]) = [ φ ( X ) , φ ( Y )] and ρ W ◦ φ = ρ V . In general, the definition ofsuch a morphism in terms of brackets and anchor maps is more involved due to thefact that a morphism of vector bundles defined over different bases does not inducea morphism of sections.To introduce relevant notation, let φ : V → W be a morphism of vector bundles V → M , W → N over f : M → N and φ ! : V → f ∗ W be a canonical morphismarising from the universal property of the pullback f ∗ W . This induces a mappingof sections Γ( V ) → Γ( f ∗ W ) (cid:39) C ∞ ( M ) ⊗ C ∞ ( N ) Γ( W )that we, by a slight abuse of notation, will keep denoting by φ ! . Definition 2.3.
Then a morphism V → W of Lie algebroids is a morphism V φ −−−−→ W (cid:121) (cid:121) M −−−−→ N of vector bundles subject to the following conditions: D. BASHKIROV AND A. A. VORONOV • ρ W ◦ φ = df ◦ ρ V ; • for X, Y ∈ Γ( V ), write φ ! ( X ) = (cid:80) i f i ⊗ X (cid:48) i , φ ! ( Y ) = (cid:80) j g j ⊗ Y (cid:48) j ; then φ ! ([ X, Y ]) = (cid:88) i,j f i g i ⊗ [ X (cid:48) i , Y (cid:48) j ] + ( ρ V ( X )( g j )) ⊗ Y (cid:48) j − ( ρ V ( Y )( f i )) ⊗ X (cid:48) i . It is an exercise to check that the second condition is independent of the tensor-product expansions.
Examples 2.4. (1) For any smooth map f : M → N , the tangent map df : T M → T N is amorphism of tangent Lie algebroids as defined in Example 2.2(3).(2) Given a Lie algebroid V → M , its anchor map T M → V is a morphism ofLie algebroids.(3) In Example 2.2(8) above, Lie algebroid morphisms T M → Der ( V ) right-inverse to the anchor map ρ : Der ( V ) → T M correspond to flat connectionon V .2.2. The dg-manifold approach.
Given a Lie algebroid V → M , the coboundaryoperator d : Γ( ∧ k V ∗ ) → Γ( ∧ k +1 V ∗ ) defined by dϕ ( X , . . . , X k +1 ) = k +1 (cid:88) i =1 ( − i +1 ρ ( X i ) ϕ ( X , . . . , ˆ X i , . . . , X k +1 )+ (cid:88) i In fact, Γ( ∧ • V ∗ ) admits a slightly richer structure. Namely, the contrac-tion i X : Γ( ∧ k +1 V ∗ ) → Γ( ∧ k V ∗ ) and the Lie derivative L X defined by( L X ϕ )( Y , . . . , Y k ) = ρ ( X )( ϕ ( Y , . . . , Y k )) − k (cid:88) i =1 ϕ ( Y , . . . , [ X, Y i ] , . . . , Y k )satisfy all the standard rules of Cartan calculus.Passage from Lie algebroids to the “Koszul dual” picture encoded by the cor-responding dg-algebras simplifies the matters concerning Lie algebroid morphisms.This is due to the following Theorem 2.6 (A. Vaintrob [Vai97]) . Let V → M be a vector bundle. Then thestructures of(1) a Lie algebroid on V , N HOMOTOPY LIE BIALGEBROIDS 5 (2) a dg-manifold, d V : C ∞ ( V [1]) → C ∞ ( V [1]) , on the graded manifold V [1] are equivalent. Furthermore, there are natural bijections between the following sets:(1) The set of morphisms of Lie algebroids V → M and W → N ;(2) The set of dg-manifold morphisms ( V [1] , d V ) → ( W [1] , d W ) . In the context of graded manifolds the differential d V is commonly referred toas a homological vector field on V [1]. The complex S ( V ∗ [ − , d V ) is often calledthe cohomological Chevalley-Eilenberg complex of the Lie algebroid. Idea of proof. A derivation d V : C ∞ ( V [1]) → C ∞ ( V [1]) of degree one is determinedby its restriction to the subalgebra C ∞ ( M ):(2) C ∞ ( M ) → Γ( V ∗ [ − , which must be a derivation of the algebra C ∞ ( M ) with values in a C ∞ ( M )-module,and by the restriction to the module of generators:(3) Γ( V ∗ [ − → Γ( S ( V ∗ [ − . By the universal property of K¨ahler differentials, (2) is equivalent to a C ∞ ( M )-module morphism Ω ( M ) → Γ( V ∗ [ − , whose dualization gives an anchor. The R -dual of (3) gives a bracket. The differ-ential property ( d V ) = 0 then translates into the Jacobi identity for the bracketand the Lie algebra morphism property for the anchor. The Lebniz rule for d V translates into the Leibniz rule for the bracket. (cid:3) Example 2.7. The tangent algebroid T M → M of a (graded) manifold M corre-sponds to the graded manifold T [1] M , a shifted tangent bundle, whose dg-algebraof smooth functions is the de Rham algebra (Ω • ( M ) , d dR ).2.3. The graded case and L ∞ -algebroids. Treating the notion of a Lie al-gebroid via dg-manifolds leads naturally to its graded version known as an L ∞ -algebroid. The concept of an L ∞ -algebroid was conceived in the works of H. Khu-daverdian and Th. Th. Voronov [KV08], H. Sati, U. Schreiber and J. Stasheff[SSS09], A. J. Bruce [Bru11], and Th. Th. Voronov [Vor10]. At the same time, L ∞ -algebras disguised as formal dg-manifolds or R | -action have been known toV. Drinfeld, V. A. Hinich, M. Kontsevich, D. Quillen, V. Schechtman, D. Sulli-van, and likely some others, since the last three decades of the 20th century; see[Kon94] and [Sta16] and references therein. L ∞ -algebroids have made their wayto physics, as one of the most general models of quantum field theory, the AKSZmodel [ASZK97]; it is a sigma model with an odd symplectic dg-manifold as thetarget space.One may also think of an L ∞ -algebroid as a sheaf of L ∞ -algebras acting in-finitesimally on a smooth manifold, see a remark after Definition 2.8. This notionis essentially K. Costello’s notion of an L ∞ space [Cos11], see [GG15] and anotherremark after Definition 2.8 for details. Formal graded and differential graded manifolds. From now on we will befocusing on formal graded manifolds ( M, (cid:98) C ∞ V ) := ( M, S ( V ) ∗ ), where M is a gradedmaniold, V a graded vector bundle over M and the sheaf of functions S ( V ) ∗ ) is thegraded dual of the symmetric coalgebra with respect to the shuffle coproduct. Onecan think of the algebra S ( V ) ∗ as an algebraic version of completion of the algebra S ( V ∗ ). Any graded manifold M may be regarded as a formal manifold over itselfassociated with the zero vector bundle over M . D. BASHKIROV AND A. A. VORONOV A morphism V → W of formal graded manifolds is a morphism of locally ringedspaces ( M, S ( V ) ∗ ) → ( N, S ( W ) ∗ ) induced by a vector bundle morphism S ( V ) −−−−→ W (cid:121) (cid:121) M −−−−→ N This requirement may be understood as a continuity condition with respect to ouralgebraic completions. A formal dg-manifold is a formal graded manifold endowedwith a differential , i.e ., a degree-one, R -linear derivation d of the structure sheafsatisfying d = 0. Morphisms of ( formal ) dg-manifolds will have to respect differentials. Since wework in the C ∞ category, we will routinely substitute sheaves with spaces of theirglobal sections.A graded manifold V over M of any of the above flavors comes with a morphism V → M given by the inclusion of C ∞ M into the function sheaf of V . There isalso a relative basepoint given by the zero section of the vector bundle V → M ,which induces a morphism M → V of (formal) graded manifolds over M . A basedmorphism must respect zero sections of the structure vector bundles. The structuredifferential in the dg case will also be assumed to be based , that is to say, the zerosection M → V must be a dg-morphism.Let V → M be a graded vector bundle over a (possibly, graded) manifold M .We will think of its total space as a pointed formal graded manifold V [1] , fiberedover M . The term “pointed” is understood in a fiberwise (relative) sense and refersto the fact that the fiber bundle V [1] → M has a canonical zero section, given by astandard augmentation (cid:98) C ∞ ( V [1]) → C ∞ ( M ). A dg structure on the pointed formalgraded manifold V [1] over M is a choice of a square-zero, degree-one derivation d ofthe R -algebra (cid:98) C ∞ ( V [1]) such that the zero section (cid:98) C ∞ ( V [1]) → C ∞ ( M ) respectsthe differentials. Here the differential on C ∞ ( M ) is assumed to be zero. Definition 2.8. (1) An L ∞ -algebroid is a graded vector bundle V → M with the structure of adg-manifold on the pointed formal graded manifold V [1] over M . That is,it is the structure of a dg-algebra over C ∞ ( M ) on the graded commutativealgebra Γ( M, S ( V [1]) ∗ ) such that the differential is compatible with theaugmentation Γ( M, S ( V [1]) ∗ ) → C ∞ ( M ).(2) An L ∞ -morphism of L ∞ -algebroids V → M and W → N is a formalpointed dg-manifold morphism ( V [1] , d V ) → ( W [1] , d W ). Equivalently, itis an augmented dg-algebra morphism Γ( N, S ( W [1]) ∗ ) → Γ( M, S ( V [1]) ∗ )over a graded algebra morphism C ∞ ( N ) → C ∞ ( M ). Example 2.9. When M is a point, an L ∞ -algebroid is nothing but an L ∞ -algebra,and the notion of an L ∞ -morphism of Lie algebroids over a point reproduces thestandard notion of an L ∞ -morphism of L ∞ -algebras. Remark. There is a generalized L ∞ -anchor map associated to an L ∞ -algebroid V → M . Indeed, the composition of its structure differential with the unit map C ∞ ( M ) → Γ( M, S ( V [1]) ∗ ) gives a degree-one derivation with values in Γ( M,S ( V [1]) ∗ ). This gives rise to a C ∞ ( M )-module morphism Ω ( M ) → Γ( M, S ( V [1]) ∗ )by the universal property of K¨ahler differentials. It extends uniquely to a dg-algebramorphism Ω • ( M ) → Γ( M, S ( V [1]) ∗ ). This, in its turn, induces a morphism of for-mal pointed dg-manifolds V [1] → T [1] M , or an L ∞ -morphism of L ∞ -algebroids, N HOMOTOPY LIE BIALGEBROIDS 7 generalizing the anchor. One can also think of it as an L ∞ -action of the L ∞ -algebroid V on the base graded manfiold M .2.4. The Poisson manifold approach. The data of a Lie algebroid on V → M can also be cast in the form of a Poisson structure on the linear dual bundle V ∗ → M generalizing the well-known Kostant-Kirillov (also known as the Lie-Poisson) bracket defined on the linear dual of a Lie algebra. More specifically,identifying smooth functions on V ∗ constant along the fibers with functions on M and identifying functions on V ∗ linear along the fibers with sections of V , we set { f, g } V ∗ = [ f, g ] , f, g ∈ Γ( V ) ρ ( f ) g, f ∈ Γ( V ) , g ∈ C ∞ ( M )0 , f, g ∈ C ∞ ( M )Extending this bracket further to the polynomial and smooth functions via theLeibniz rule and completion endows V ∗ with a well-defined Poisson structure. Notethat the corresponding Poisson tensor will be linear along the fibers of V ∗ → M . Theorem 2.10 (T. J. Courant [Cou90]) . Let V → M be a vector bundle. Thenthe following structures are equivalent:(1) A Lie algebroid structure on V → M ;(2) A Poisson structure on the total space of the vector bundle V ∗ → M suchthat the Poisson structure is linear along the fibers. Two Poisson structures arising in this fashion on the linear duals of Lie algebroids V → M , W → N can be related by means of Lie algebroid comorphisms rather thanmorphisms. Namely, a Lie algebroid comorphism from V to W over f : M → N isa (base-preserving) morphism of vector bundles φ : f ∗ W → V over M such that φ ([ X, Y ]) = [ φ ( X ) , φ ( Y )]for all X, Y ∈ Γ( W ) and df ◦ ρ V ◦ φ = ρ W . Here, φ is the natural compositionΓ( W ) f ∗ → Γ( f ∗ W ) φ ! → Γ( V ). Such a comorphism yields a vector bundle morphism V ∗ → ( f ∗ W ) ∗ ∼ −→ f ∗ W ∗ → W ∗ . Theorem 2.11 (P. Higgins, K. Mackenzie [HM93]) . Lie algebroid comorphismsfrom V → M to W → N are in one-to-one corresponence with vector bundlemorphisms V ∗ → W ∗ that are Poisson. Lie coalgebroids. Keeping in mind our main objective of studying Lie bial-gebroids and their homotopy generalization, we briefly describe the notion of a Liecoalgebroid. Definition 2.12. A Lie coalgebroid structure on a vector bundle V → M over asmooth manifold M consists of • an R -linear mapping δ : Γ( V ) → Γ( V ) ∧ R Γ( V ) (a Lie cobracket ) satisfyingthe co-Jacobi identity (cid:8) ( δ ⊗ id ) δ = 0 , where (cid:8) ( x ⊗ y ⊗ z ) = x ⊗ y ⊗ z + y ⊗ z ⊗ x + z ⊗ x ⊗ y ; • a vector-bundle morphism σ : T ∗ M → V , called the coanchor , subject tothe co-Leibniz rule δ ( f X ) = f δ ( X ) + σ ( df ) ∧ X, X ∈ Γ( V ) , f ∈ C ∞ ( M ) . D. BASHKIROV AND A. A. VORONOV This implies, in particular, that δ ( σ ( ω )) = ( σ ∧ σ )( dω ) , ω ∈ Γ( T ∗ M ) . That is, σ induces a morphism Γ( T ∗ M ) → Γ( V ) of Lie coalgebras. Theorem 2.13. Let V → M be a vector bundle. Then the structures of(1) a Lie coalgebroid on V ,(2) a dg-manifold on the graded manifold V ∗ [1] are equivalent. The proof of this theorem is a rather straightforward exercise on the definitions;it is similar to the proof of Theorem 2.6. Examples 2.14. (1) The cotangent bundle T ∗ M → M is trivially a Lie coalgebroid with thecobracket being the restriction of the de Rham differential d dR onto T ∗ M and the coanchor σ = id T ∗ M .(2) Any Lie algebroid structure on V → M gives rise to a Lie coalgebroidstructure on the linear dual bundle V ∗ → M [BD04, Section 1.4.14]. Inparticular, taking V to be the standard tangent Lie algebroid T M → M recovers the previous example. Coversely, a Lie coalgebroid structure on V → M induces a Lie algebroid structure on V ∗ → M .The last example combined with the Poisson bracket construction outlined inthe previous section implies, in particular, that the structure of a Lie coalgebroidon V → M induces a fiberwise linear Poisson structure on V . More concretely,identifying, as before, fiberwise linear functions on V with the sections of V ∗ andfiberwise constant functions on V with the elements of C ∞ ( M ), we get { α, β } V = ( α ⊗ β ) δ, α, β ∈ Γ( V ∗ ) α ( σ ( dβ )) , α ∈ Γ( V ∗ ) , β ∈ C ∞ ( M )0 , α, β ∈ C ∞ ( M ) . This is to be extended further via the Leibniz rule and completion. As an upshot,it enables us to give a concise definition of a Lie coalgebroid morphism. Definition 2.15. A morphism V → W of Lie coalgebroids is a vector bundlemorphism ( V → M ) → ( W → N ) such that the map of total spaces V → W isPoisson. Example 2.16. The coanchor map σ : T ∗ M → V of a Lie coalgebroid V → M is a Lie coalgebroid morphism, where T ∗ M is given the standard cotangent Liecoalgebroid structure as in Example 2.14(1). In this case, the Poisson structureon T ∗ M corresponding to the Lie coalgebroid structure is the standard Poissonstructure on the cotangent bundle.2.6. The odd Poisson manifold approach. Yet another algebraic structure nat-urally associated with a Lie algebroid V → M is defined on Γ( ∧ • V ). Namely, thereis a canonical extension of the Lie bracket on Γ( V ) to a bracket[ , ] : Γ( ∧ k V ) ⊗ Γ( ∧ l V ) → Γ( ∧ k + l − V ) , k, l ≥ − X ∈ Γ( V ) , f ∈ C ∞ ( M ), we set [ X, f ] = ρ ( X )( f ) . We remind the reader that we assume all vector bundles to be of finite rank. N HOMOTOPY LIE BIALGEBROIDS 9 Altogether, this turns Γ( ∧ • V ) into a Gerstenhaber (or an odd Poisson ) algebra.The converse is also true: Theorem 2.17 (A. Vaintrob [Vai97]) . Let V → M be a vector bundle. The fol-lowing structures are equivalent:(1) A Lie algebroid structure on V → M ;(2) A Gerstenhaber algebra structure on Γ( ∧ • V ) (taken with the standard mul-tiplication);(3) A graded Poisson structure of degree − on V ∗ [1] . This theorem can be regarded as an odd analogue of Courant’s Theorem 2.10.Likewise, the following statement is an odd analogue of Higgins-Mackenzie’s The-orem 2.11. Proposition 2.18. There are natural bijections between the following sets:(1) The set of Lie algebroid comorphisms from V → M to W → N ;(2) The set of Gerstenhaber algebra morphisms Γ( ∧ • W ) → Γ( ∧ • V ) ;(3) The set of graded Poisson manifold morphisms V ∗ [1] → W ∗ [1] . Examples 2.19. (1) For a Lie algebra V , Γ( ∧ • V ) is the underlying space of the homologicalChevalley-Eilenberg complex with trivial coefficients. The odd Poissonbracket on Γ( ∧ • V ) is a derived bracket[KS04] generated by the homologicalChevalley-Eilenberg differential.(2) For a tangent Lie algebroid T M , Γ( ∧ • T M ) is the Schouten-Nijenhuis alge-bra of multivector fields.(3) For a Lie algebroid associated with a Poisson manifold M as in Example2.2(7), Γ( ∧ • T ∗ M ) = Ω • ( M ) is the underlying space of the homologicalPoisson complex. The differential in this case (known as the Brylinski differential) is d = [ i π , d dR ]. Remark. The Poisson manifold V ∗ , the dg-manifold V [1] and the odd Poissonmanifold V ∗ [1] determined by a Lie algebroid V → M are known as P -, Q - and S - manifolds , respectively, associated to V [Vor02]. In that regard, Lie bialgebroids(see Section 3) manifest themselves in the form of QP - or QS -manifolds, comprisinga pair of such structures in a compatible way.We also have analogous statements for Lie coalgebroids. Theorem 2.20. Let V → M be a vector bundle. Then the structures of(1) a Lie coalgebroid on V ,(2) a graded Poisson structure of degree − on V [1] are equivalent. Furthermore, there are natural bijections between the following sets:(1) The set of morphisms of Lie coalgebroids V → M and W → N ;(2) The set of graded Poisson morphisms V [1] → W [1] . Connections and associated BV algebras. This section follows the paper[Xu99] by Ping Xu. For a Lie algebroid V → M , endowing the associated Gersten-haber algebra Γ( ∧ • V ) with some extra data in the form of a differential operatorof order one or two, subject to certain compatibility conditions, determines an ad-ditional structure on V → M . The former case will be addressed in Section 3; tohandle the latter case, we need the following Definition 2.21. Let V → M be a Lie algebroid and E → M be a vector bundle.A linear mapping ∇ : Γ( V ) ⊗ Γ( E ) → Γ( E ) , X ⊗ s (cid:55)→ ∇ X ( s ) is called a V -connection ifi. ∇ fX ( s ) = f ∇ X ( s );ii. ∇ X ( f s ) = ( ρ ( X ) f ) s + f ∇ X ( s )for all f ∈ C ∞ ( M ) , X ∈ Γ( V ) , s ∈ Γ( E ).The curvature of a V -connection ∇ on E → M is an element R ∈ Γ( ∧ V ∗ ) ⊗ End ( E ) defined by R ( X, Y ) = ∇ X ∇ Y − ∇ Y ∇ X − ∇ [ X,Y ] , X, Y ∈ Γ( V ) , and the torsion is T ( X, Y ) = ∇ X Y − ∇ Y X − [ X, Y ] X, Y ∈ Γ( V ) . A V -connection is said to be flat if R ≡ V → M is of rank n as a vector bundle, then any V -connectionof the canonical line bundle E = ∧ n V determines an operator ∆ : Γ( ∧ • V ) → Γ( ∧ •− V ) on the Gerstenhaber algebra Γ( ∧ • V ):∆ ω ( X , . . . , X p +1 ) := (cid:88) i ( − i − ∇ X i ω ( X , . . . , ˆ X i , . . . , X p +1 )+ (cid:88) i Let V be a vector bundle over M such that V → M is both a Lie algebroid and aLie coalgebroid. Denote by d the coboundary operator on Γ( ∧ • V ) = Γ( S ( V [ − Definition 3.1. A vector bundle V → M with the structures of a Lie algebroidand a Lie coalgebroid is a Lie bialgebroid if d is a derivation of bracket:(4) d ([ X, Y ]) = [ dX, Y ] + [ X, dY ] , X, Y ∈ Γ( V ) . Note that in the equation above, dX, dY ∈ Γ( ∧ V ) and the Lie bracket is ex-tended from V to ∧ • V as a biderivation of degree − 1, see Section 2.6. N HOMOTOPY LIE BIALGEBROIDS 11 Remark. In view of the equivalence between Lie coalgebroid structures on V → M and Lie algebroid structures on V ∗ → M , see Example 2.14(2), one may think ofa Lie bialgebroid as a pair ( V, V ∗ ) of Lie algebroids satisfying the compatibilitycondition (4). This is a more common point of view. However, we prefer to thinkof a Lie bialgebroid as two compatible structures on V in this section for functo-riality reasons. We will return to the former viewpoint later, when we discuss theHamiltonian approach. Theorem 3.2. If V is a Lie bialgebroid, then so is V ∗ . Theorem 3.3. Let V → M be a vector bundle. The structures of(1) a Lie bialgebroid on V ,(2) a dg-Gerstenhaber algebra on Γ( ∧ • V ∗ ) ,(3) a dg-Poisson manifold on V [1] with Poisson bracket of degree − are equivalent.Remark. The multiplication on Γ( ∧ • V ∗ ) and the differential are to be related viathe Leibniz rule. Sometimes, dg Gerstenhaber algebras with this property are called strong , [Xu99]. Definition 3.4. A morphism V → W of Lie bialgebroids is a vector bundle mor-phism ( V → M ) → ( W → N ) which is a morphism of Lie algebroids and amorphism of Lie coalgebroids.Taking into account Definition 2.15, the above definition may immediately bereworded as follows. Proposition 3.5. Let V and W be Lie bialgebrodis. A Lie algebroid morphism V → W is a morphism of Lie bialgebroids iff it is a Poisson map with respect tothe Poisson structures on V and W induced by the Lie coalgebroid structures on V and W , respectively, as in Section 2.5. In the vein of Theorem 3.3, one obtains a characterization of Lie bialgebroidmorphisms: Theorem 3.6. There are natural bijections between the following sets:(1) The set of Lie bialgebroid morphisms from V to W ;(2) The set of dg-Gerstenhaber algebra morphisms Γ( ∧ • W ∗ ) → Γ( ∧ • V ∗ ) ;(3) The set of dg-Poisson manifold morphisms V [1] → W [1] . Examples 3.7. (1) If M is a point, a Lie bialgebroid over M is a Lie bialgebra in the sense ofDrinfeld.(2) Let M be a Poisson manifold with a Poisson bivector π , T ∗ M → M bethe cotangent Lie algebroid associated to π as in Example 2.2(7) with thecanonical Lie coalgebroid structure as in Example 2.14(1). The Lichnerow-icz differential d π = [ π, − ] is a derivation of the Schouten-Nijenhuis bracketon Γ( ∧ • T M ), thus giving the cotangent bundle T ∗ M a Lie bialgebroidstructure.Conversely, let V be a Lie bialgebroid over M . Then π V := ρ ◦ σ : T ∗ M → T M , where ρ , σ are the anchor and the coanchor maps of V → M ,respectively, defines a Poisson structure on M .(3) Suppose V → M is a Lie algebroid with an anchor ρ and a structuredifferential d V on Γ( ∧ • V ∗ ). Let r ∈ Γ( ∧ V ) be such that [ r, r ] = 0. Denote by r the associated bundle map V ∗ → V . One can show that ρ ∗ := ρ ◦ r : V ∗ → T M and[ φ, ψ ] = L r ( φ ) ( ψ ) − L r ( ψ ) ( φ ) − d V (( φ ∧ ψ ) r ) , φ, ψ ∈ Γ( V ∗ ) , where L X := [ d V , ι X ] , X ∈ Γ( V ) , determine a Lie algebroid structure on V ∗ → M . Furthermore, a sim-ple check confirms that the pair ( V, V ∗ ) is actually a Lie bialgebroid. Inparticular, taking V = T M recovers the previous example.(4) A Nijenhuis structure on a smooth manifold M is a vector bundle endo-morphism N : T M → T M such that its Nijenhuis torsion [ N ( X ) , N ( Y )] − N ([ N ( X ) , Y ] + [ X, N ( Y )]) + N ([ X, Y ])vanishes for any X, Y ∈ Γ( T M ). A prototypical example of a Nijenhuisstructure arises in the form of a recursion operator of an integrable bi-Hamiltonian system [Mag78, Olv90]. Namely, given a pair π , π of Poissontensors on a manifold M such that any linear combination λπ + µπ isPoisson as well and π is symplectic, then N := π ◦ ( π ) − is a Nijenhuisstructure on M .A Nijenhuis structure on M induces [KS96] a Lie algebroid structure on T M with the bracket[ X, Y ] N := [ N ( X ) , Y ] + [ X, N ( Y )] − N ([ X, Y ])and N : T M → T M being an anchor map. Now, if M is a Poisson manifold,then T ∗ M can be given the Lie algebroid structure of Example 2.2(7). Thisproduces a Lie coalgebroid structure on T M by Example 2.14(2). It turnsout [KS96] that these two structures on T M are compatible, thereby making T M a Lie bialgebroid, provided N ◦ π = π ◦ N ∗ , and { α, β } Nπ = { N ∗ ( α ) , β } π + { α, N ∗ ( β ) } π − N ∗ ( { α, β } π )for all α, β ∈ Γ( T ∗ M ).(5) Recall that a Lie groupoid is a (small) groupoid s, t : G ⇒ M such thatits set of objects M and the set morphisms G are smooth manifolds, andthe source and the target maps s, t along with the composition G × G → G ,the unit M → G and the inverse map G → G are smooth. The source andtarget maps are also assumed to be submersions.Given a Lie groupoid G ⇒ M , we define the associated Lie algebroid V → M as follows. As a vector bundle, V = ker( ds ) | M , where the restrictionis taken along the unit map M → G , x (cid:55)→ x . A Lie bracket on Γ( V ) isobtained by identifying the sections of V with the right-invariant vectorfields on G and the anchor map is dt : T G → T M restricted onto V ⊂ T G .A Poisson groupoid is a Lie groupoid G ⇒ M with a Poisson structure π such that the graph of the composition m : G × G → G is a coisotropicsubmanifold of G × G × ¯ G , where ¯ G denotes G with the opposite Poissontensor − π . One can show [MX94] that a Poisson structure on G inducesa Lie algebroid structure on V ∗ . Furthermore, it is compatible with theLie algebroid structure on V → M , giving rise to a Lie bialgebroid over M . This generalizes the well-known construction of Lie bialgebras arisingas infinitesimal counterparts of Poisson-Lie groups. N HOMOTOPY LIE BIALGEBROIDS 13 The Hamiltonian approach Let Q be a (graded) vector field on a graded manifold V . The cotangent (or Hamiltonian ) lift Γ( V, T V ) → C ∞ ( T ∗ V ) ,Q (cid:55)→ µ Q , is defined by setting(5) µ Q ( x, p ) = p ( Q x ) , p ∈ T ∗ x V. In local Darboux coordinates ( x ∗ i , x i ) on T ∗ V , if Q = (cid:80) i Q i ( x ) ∂/∂x i , then µ Q = (cid:80) i Q i ( x ) x ∗ i , a function linear along the fibers of π : T ∗ V → V . Remark. This construction has been rediscovered in the case where M is a pointby C. Braun and A. Lazarev [BL15] under the name of doubling , see also [Kra07]. Proposition 4.1. Let Q , Q , Q be vector fields on V . Then(1) { µ Q , µ Q } = µ [ Q ,Q ] (2) π ∗ ( { µ Q , −} ) = Q . Note that if [ Q, Q ] = 0, the proposition implies { µ Q , µ Q } = 0.The case we are interested in corresponds to the graded manifold V [1] associatedwith a vector bundle V → M . By Theorem 2.6, a vector field Q of degree +1 andsuch that [ Q, Q ] = 0 determines a Lie algebroid structure on V → M . This leadsto the following chain of correspondences: Lie algebroidstructureson V → M (cid:33) Homologicalvector fields Q of degree +1on V [1] (cid:33) “Integrable oddHamiltonians”: { µ Q , µ Q } = 0 Theorem 4.2 (D. Roytenberg [Roy99]) . Lie algebroid structures on V → M arein one-to-one correspondence with functions µ on T ∗ V [1] which are linear along thefibers of T ∗ V [1] → V [1] , of total degree one in the natural Z -grading on C ∞ ( T ∗ V [1]) and such that { µ, µ } = 0 . In terms of local Darboux coordinates ( x i , ξ a , x ∗ i , ξ ∗ a ), where • { x i } are coordinates on U ⊂ M , • { e a } is a basis of sections of V over U , • { ξ a } are the corresponding generators of Γ( U, ∧ • V ∗ ), • { x ∗ i } are coordinates along T ∗ U → U , • { ξ ∗ a } are coordinates along T ∗ V [1] | U → V [1] | U ,the above correspondence takes the following form: (cid:26) ρ ( e a ) = A ia ( x ) ∂∂x i [ e a , e b ] = C cab ( x ) e c (cid:27) (cid:33) (cid:110) d = ξ a A ia ( x ) ∂∂x i + C cab ( x ) ξ a ξ b ∂∂ξ c (cid:111) (cid:33) (cid:8) µ = ξ a A ia ( x ) x ∗ i + C cab ( x ) ξ a ξ b ξ ∗ c (cid:9) Examples 4.3. (1) The Hamiltonian of a tangent Lie algebroid T M → M is µ = ξ i x ∗ i .(2) A Hamiltonian µ = ξ a A ia x ∗ i + C cab ξ a ξ b ξ ∗ c with coordinate-independentstructure coefficients C cab corresponds to an action Lie algebroid, cf . Ex-ample 2.2(5). Namely, in that case, C cab are the structure constants of theLie algebra g acting on M and A ia ( x ) are the coefficients of the anchor map ρ ( e a ) = A ia ∂∂x i . (3) For a Poisson manifold M with a Poisson bivector π = π ij ∂∂x i ∧ ∂∂x j , theHamiltonian of the corresponding Lie algebroid T ∗ M → M is µ = (cid:88) i,j ξ j π ij x ∗ i + (cid:88) i,j,k ∂∂x k ( π ij ) ξ i ξ j ξ ∗ k . We would like to give a Hamiltonian characterization of Lie algebroid morphisms.To this end, consider Lie algebroids V → M , W → N . We let F : f ∗ ( T ∗ W [1]) → T ∗ W [1] be the pullback of f : V [1] → W [1] along the projection T ∗ W [1] → W [1],and Φ : f ∗ ( T ∗ W [1]) → T ∗ V [1] be the morphism of graded vector bundles over V [1]given as the dual of the bundle map T V [1] → f ∗ ( T W [1]) induced by the differential df : T V [1] → T W [1]. Proposition 4.4. Let µ ∈ C ∞ ( T ∗ V [1]) , ν ∈ C ∞ ( T ∗ W [1]) be the Hamiltonianscorresponding to the Lie algebroid structures on V → M , W → N . Then Lie al-gebroid morphisms V → W are in one-to-one correspondence with graded manifoldmorphisms f : V [1] → W [1] such that F ∗ ( ν ) = Φ ∗ ( µ ) . Proof. We have to show that, given Hamiltonians µ, ν , the condition above is equiv-alent to the homological vector fields Q V = dp ( { µ, −} ) and Q W = dp ( { ν, −} ) being f -related. That is, df ( Q Vv ) = Q Wf ( v ) for any v ∈ V [1].Indeed, for any point ( v, ζ ) in f ∗ ( T ∗ W [1]), we have( F ∗ ( ν ))( v, ζ ) = ( ν ◦ F )( v, ζ ) = ν ( f ( v ) , ζ ) = ζ ( Q Wf ( v ) ) . On the other hand,(Φ ∗ ( µ ))( v, ζ ) = ( µ ◦ Φ)( v, ζ ) = µ ( v, f ∗ ( ζ )) = ( f ∗ ( ζ ))( Q Vv ) = ζ ( df ( Q Vv )) . Thus, the equation F ∗ ( ν ) = Φ ∗ ( µ ) is equivalent to ζ ( Q Wf ( v ) ) = ζ ( df ( Q Vv )) for all v and ζ . (cid:3) Given a pair ( V, V ∗ ) of Lie algebroids over M , we get the corresponding pair ofHamiltonians µ ∈ C ∞ ( T ∗ V [1]) and µ ∗ ∈ C ∞ ( T ∗ V ∗ [1])of degree one in the Z -grading. We can bring them together in C ∞ ( T ∗ V [1]) bymeans of a canonical symplectomorphism L : T ∗ [2] V [1] → T ∗ [2] V ∗ [1] , known as the Legendre transform . In local coordinates, it reads( x, ξ, x ∗ , ξ ∗ ) (cid:55)→ ( x, ξ ∗ , x ∗ , ξ )in our Z -graded setting, see [Roy99, Section 3.4] for the Z / Z -graded case. Notethat V ∗ [1] = ( V [1]) ∗ [2] and the shift along the cotangent directions is needed tomake sure that L respects grading. After the shift, the Poisson bracket { , } acquiresdegree − 2, and both µ and µ ∗ become elements of degree 3 in C ∞ ( T ∗ [2] V [1]) and C ∞ ( T ∗ [2] V ∗ [1]), respectively. Lemma 4.5. Let µ ∗ ∈ C ∞ ( T ∗ [2] V ∗ [1]) be a Hamiltonian corresponding to a Liealgebroid structure on V ∗ . Then L ∗ µ ∗ ( v, ζ ) = π V ∗ ,v ( ζ, ζ ) , ( v, ζ ) ∈ T ∗ [2] V [1] where π V ∗ is the graded Poisson tensor induced by the Lie algebroid structure on V ∗ and π V ∗ ,v is its value at v ∈ V [1] . N HOMOTOPY LIE BIALGEBROIDS 15 Proof. In local coordinates we may write ( v, ζ ) = ( x i , ξ a , x ∗ i , ξ ∗ a ), and L ( v, ζ ) =( x i , ξ ∗ a , x ∗ i , ξ a ). Then( L ∗ µ ∗ )( v, ζ ) = µ ∗ ( L ( v, ζ )) = ξ ∗ a A ia ( x ) x ∗ i + 12 C abc ( x ) ξ ∗ a ξ ∗ b ξ c , where A ia ( x ) and C abc ( x ) are the structure functions of µ ∗ .On the other hand, π V ∗ ,v ( ζ, ζ ) = (cid:88) a,b π V ∗ ,v ( dξ a , dξ b ) + (cid:88) i,j π V ∗ ,v ( dx i , dx j ) + (cid:88) a,i π V ∗ ,v ( dξ a , dx i ) , where the first and the last summands contribute C abc ( x ) ξ ∗ a ξ ∗ b ξ c and ξ ∗ a A ia ( x ) x ∗ i ,respectively, and the second one is identically zero. (cid:3) Theorem 4.6 (D. Roytenberg [Roy99]) . A pair ( V, V ∗ ) of Lie algebroids is a Liebialgebroid if and only if { µ + L ∗ µ ∗ , µ + L ∗ µ ∗ } = 0 . Corollary 4.7. A structure of a Lie bialgebroid on a vector bundle V → M is equiv-alent to a Hamiltonian χ on T ∗ [2] V [1] , which is linear-quadratic along the fibers of T ∗ [2] V [1] → V [1] and is of degree three in the natural Z -grading on functions on T ∗ [2] V [1] , and such that { χ, χ } = 0 . Example 4.8. For a Lie bialgebroid ( T ∗ M, T M ) associated to a Poisson manifold M as in Example 3.7(2), the Hamiltonian χ on T ∗ [2] T ∗ M [1] is given by χ = (cid:88) i ξ ∗ i x ∗ i + (cid:88) i,j ξ j π ij x ∗ i + (cid:88) i,j,k ∂∂x k ( π ij ) ξ i ξ j ξ ∗ k . Theorem 4.9. Lie bialgebroid morphisms V → W are in one-to-one correspon-dence with formal graded manifold morphisms f : V [1] → W [1] such that F ∗ ( ψ ) = Φ ∗ ( χ ) , where and χ, ψ are the Hamiltonians on T ∗ [2] V [1] and T ∗ [2] W [1] corresponding, re-spectively, to the given Lie bialgebroid structures and F, Φ are as in Proposition . .Proof. As in the proof of Proposition 4.4, at any point ( v, ζ ) in f ∗ ( T ∗ [2] W [1]), thevalue ( F ∗ ( ψ ))( v, ζ ) = ψ ( f ( v ) , ζ ) is given by(1) evaluating the linear in ζ ∈ T ∗ f ( v ) [2] W [1] part ν of ψ = ν + L ∗ ε given by thevector field Q Wf ( v ) on W [1] as a linear functional on the cotangent bundle T ∗ [2] W [1], see Equation (5),(2) evaluating the quadratic part L ∗ ε , which by Lemma 4.5 is given by thePoisson tensor on W [1] corresponding to the Lie algebroid structure on W ∗ , and(3) adding the results together.Likewise, the value (Φ ∗ ( χ ))( v, ζ ) = χ ( v, f ∗ ( ζ )) at ( v, ζ ) is the sum of the valuesat f ∗ ( ζ ) of the linear part µ of χ = µ + L ∗ µ ∗ given by the vector field Q Vv and thequadratic part L ∗ µ ∗ given by the Poisson tensor on V [1]. Since equality of polyno-mial functions is equivalent to equality of their homogeneous parts, the agreementof the functions F ∗ ( ψ ) and Φ ∗ ( χ ) implies that f is a morphism of Lie algebroidsrespecting the graded Poisson structures. (cid:3) L ∞ -bialgebroids Corollary 4.7 and Theorem 4.9 motivate the following L ∞ generalizations of thenotions of a Lie bialgebroid and a Lie-bialgebroid morphism. Definition 5.1. An L ∞ - bialgebroid over a (graded) manifold M is a graded vectorbundle V → M along with a degree-three function χ on the pointed formal gradedmanifold T ∗ [2] V [1] such that • { χ, χ } = 0, i.e ., χ is an integrable Hamiltonian ; • χ vanishes on the zero section V [1] ⊂ T ∗ [2] V [1] of the vector bundle T ∗ [2] V [1] → V [1] as well as on the restriction T ∗ [2] V [1] | M of this bun-dle to the zero section M ⊂ V [1] of the vector bundle V [1] → M .Removing the second condition leads to an L ∞ generalization of the notion of a quasi-Lie bialgebroid , [Roy99], also known as a curved Lie bialgebroid , [GG15]. Remark. A seemingly natural attempt to define the notion of an L ∞ -bialgebroidin a way similar to Definition 3.1, as a pair of L ∞ -algebroids and L ∞ -coalgebroidstructures on V subject to some compatibility conditions, would be too restrictive,as such a structure would fail to comprise higher L ∞ operations with multiple inputsand multiple outputs, cf. Examples 5.2 and 5.5. However, defining the notion ofan L ∞ -bialgebroid via Manin L ∞ -triples , as an L ∞ -algebroid structure on V ⊕ V ∗ under some finite-rank conditions, should be possible, see [Kra07], where this isdone for L ∞ -bialgebras, i.e ., when M is a point. Example 5.2. This example generalizes triangular Lie bialgebras in the sense ofDrinfeld [Dri87].A generalized (or higher ) Poisson structure on a graded manifold M is a (total)degree-two multivector field P ∈ Γ( S ( T [ − M )) such that [ P, P ] = 0, where thebracket is the standard Schouten bracket. As shown by H. Khudaverdian andTh. Th. Voronov [KV08], such a structure induces L ∞ brackets on the algebraof smooth functions C ∞ ( M ) and on the de Rham complex Γ( S ( T ∗ [ − M )) of M .These higher brackets are known as the higher Poisson and higher Koszul brackets,respectively. The former generalizes the standard Poisson bracket construction,while the latter generalizes Example 2.2(7), see also [Bru11].Pursuing these ideas in the direction of Example 3.7(3), we start with a gradedmanifold M and a graded Lie algebroid V → M , determined by a Hamiltonian µ .Let r ∈ Γ( S ( V [ − ⊂ C ∞ ( V ∗ [1]) be a degree-two element such that [ r, r ] = 0,where the bracket is the degree-( − 1) Poisson bracket on V ∗ [1] induced by the Liealgebroid structure on V , as described in Section 2.6. Then the following sequenceof maps takes place: α : C ∞ ( V ∗ [1]) → Γ( V ∗ [1] , T V ∗ [1]) → C ∞ ( T ∗ V ∗ [1]) . Here, the first mapping associates the Hamiltonian vector field [ f, − ] to a function f , using the odd Poisson bracket on V ∗ [1], and the second one is the cotangent lift.Each of these morphisms respects the brackets, thus letting r pass to a degree-oneelement α ( r ) such that { α ( r ) , α ( r ) } = 0 on T ∗ V ∗ [1]. After the degree shift to T ∗ [2] V ∗ [1], the element α ( r ) acquires degree 3. Altogether, as a Hamiltonian, thesum µ + L ∗ ( α ( r )) determines an L ∞ -bialgebroid structure on V .Compatibility of the coalgebroid component L ∗ ( α ( r )) with µ becomes more ap-parent upon recognizing that L ∗ ( α ( r )) = { µ, L ∗ ( r ) } , where { , } is the canonicalbracket on T ∗ [2] V [1]; see [KSR10]. The L ∞ -algebroid part on V is a just a gradedLie algebroid, and there are no higher mixed operations. A construction givingnontrivial higher brackets, higher cobrackets and higher mixed operations alongthese lines in the case of M being a point can be found in [BV15]. N HOMOTOPY LIE BIALGEBROIDS 17 Definition 5.3. A semistrict L ∞ -morphism V → W of L ∞ -bialgebroids V → M and W → N is a morphism f : V [1] → W [1] of pointed formal graded manifoldsrelating the Hamiltonians on the shifted cotangent bundles T ∗ [2] V [1] and T ∗ [2] W [1]in the sense of Proposition 4.4. The “pointed” condition means that f maps thezero section of V [1] → M to the zero section of W [1] → N .Even though this definition looks like a direct generalization of Theorem 4.9,which gives a Hamiltonian charaterization of morphisms of Lie bialgebroids, wehave chosen to use the word semistrict , because the morphism V → W of L ∞ -coalgebroids under this definition is strict, while the morphism V → W of L ∞ -algebroids may have “higher,” i.e ., L ∞ , components. See also Example 5.5 below.The definition of a full-fledged L ∞ -morphism beautifully overlaps with the ideaof deformation quantization, whence we need to introduce a formal quantizationparameter (cid:126) , to which it would be convenient to assign a degree 2 in our gradedcontext. It is common in deformation quantization to consider functions on thecotangent bundle polynomial in the momenta, see e.g . [BNW98], but the presenceof a formal parameter will allow us to consider formal series in the momenta.Let V be a graded vector bundle over a graded manifold M and χ ∈ (cid:98) C ∞ ( T ∗ [2] V [1])an integrable Hamiltonian defining an L ∞ -bialgebroid structure on V . We want todefine an action of χ on functions g ∈ (cid:98) C ∞ ( V [1]) on V [1] by differential operators.The differential dg of g defines a section of the cotangent bundle T ∗ ( V [1]). Thedifferential d vert χ along the vertical directions of the vector bundle T ∗ [2] V [1] → V [1]is a section of the relative cotangent bundle T ∗ ( T ∗ [2] V [1] /V [1]) → T ∗ [2] V [1], whoserestriction to the zero section V [1] ⊂ T ∗ [2] V [1] may be identified with the shiftedtangent bundle T [ − V [1]. (Formally speaking, there should be a double dual,but given our assumption of finite dimensionality of each graded component, thecorresponding tangent bundle will be reflexive.) Using the natural paring betweenthe tangent and the cotangent bundles of V [1], we can pair ( d vert χ ) | V [1] and dg ,producing a function on V [1], denoted χ ( g ) := (cid:104) ( d vert χ ) | V [1] , dg (cid:105) . Note that this pairing will shift the degree by − 2, given that ( d vert χ ) | V [1] is asection of T [ − V [1].Extend this pairing to pairings of degree − k between iterated differentials( d k vert χ ) (cid:12)(cid:12) V [1] and d k g , viewed as sections of the k th symmetric powers S k ( T [ − V [1])and S k T ∗ V [1], respectively, for all k ≥ χ ( g ) k := (cid:104) ( d k vert χ ) | V [1] , d k g (cid:105) . Note that because of the vanishing condition χ | V [1] = 0, the k = 0 term χ ( g ) willautomatically be zero. Finally, define a differential operator C ∞ ( V [1]) χ −→ C ∞ ( V [1])[[ (cid:126) ]] ,g (cid:55)→ χ ( g ) := ∞ (cid:88) k =1 (cid:126) k − χ ( g ) k . (6)Since the degree of χ as a function on T ∗ [2] V [1] is three and | (cid:126) | = 2, the degree of χ as an operator on functions on V [1] is one, i.e ., | χ ( g ) | = | g | + 1 . Note that the vanishing condition χ | T ∗ [2] V [1] | M = 0 on the Hamiltonian implies that χ ( g ) | M = 0, whereas the integrability, { χ, χ } = 0, yields χ ( χ ( g )) = 0 or χ ◦ χ = 0.This action has a more familiar form in coordinates. Let q i ’s denote coordinateson V [1] and p i ’s denote the conjugate momenta, resulting in Darboux coordinates( p i , q i ) on T ∗ [2] V [1]. Under stronger assumptions of finite dimensionality, such as the finiteness of the total rank of V and the total dimension of M , so as the index i takes finitely many values, we would have χ ( g ) = ∞ (cid:88) k =1 (cid:126) k − k ! (cid:88) i ,...,i k ± ∂ k χ∂p i . . . ∂p i k (cid:12)(cid:12)(cid:12)(cid:12) V [1] ∂ k g∂q i . . . ∂q i k , where ± is a suitable Koszul sign.Compare this with the star product of standard type: χ ∗ g := ∞ (cid:88) k =0 (cid:126) k k ! (cid:88) i ,...,i k ± ∂ k χ∂p i . . . ∂p i k ∂ k g∂q i . . . ∂q i k , which in our case, when g is a function on the base V [1] of the cotangent bundle,coincides with the star product of Moyal-Weyl type, see [BNW98]. Since (cid:126) hasdegree 2, the above star product is homogeneous.We will be considering smooth maps f : V [1] → S ( W [1]) , with which we would like to associate certain linear maps C ∞ ( W [1]) → C ∞ ( V [1]) . A map f : V [1] → S ( W [1]) induces a morphism f ∗ : C ∞ ( S ( W [1])) → C ∞ ( V [1])of algebras of smooth functions. On the other hand, starting from a smooth function g ∈ C ∞ ( W [1]), we can build a linear (and thereby, smooth) function on S ( W [1])by taking a formal Taylor series T ( g ) := ∞ (cid:88) k =0 k ! d k vert g (cid:12)(cid:12) M , d k vert g (cid:12)(cid:12) M ∈ Γ( M, S k ( W [1]) ∗ ) . This produces a linear map T : C ∞ ( W [1]) → Γ( N, S ( W [1]) ∗ ) ⊂ C ∞ ( S ( W [1])) . Definition 5.4. An L ∞ -morphism V → W of L ∞ -bialgebroids V → M and W → N is a morphism f : V [1] → S ( W [1])of pointed formal graded manifolds relating the Hamiltonians χ V and χ W on theshifted cotangent bundles T ∗ [2] V [1] and T ∗ [2] W [1], respectively, as follows: χ V ◦ f ∗ ◦ T = f ∗ ◦ T ◦ χ W , where the structure Hamiltonians χ V and χ W are regarded as operators on functionsvia the action (6) and the condition above is understood as an equation on linearoperators C ∞ ( W [1]) → C ∞ ( V [1])[[ (cid:126) ]] . Example 5.5. If M is a point, T ∗ [2] V [1] can be identified, as a graded manifold,with V ∗ [1] ⊕ V [1] equipped with a Poisson bracket of degree − bigbracket , [KS04, Kra07]. The algebra of smooth functions on V ∗ [1] ⊕ V [1] may bethought of as a completion of S ( V [ − ⊕ V ∗ [ − V . Schematically, the big bracket of two homogeneous tensors f and g , interpreted as linear maps between graded symmetric powers of V [ ± N HOMOTOPY LIE BIALGEBROIDS 19 Here, the summation is done over all possible ways to form an input-output pairfor f and g ; the relevant signs are suppressed.Now, an integrable Hamiltonian on T ∗ [2] V [1] is a degree-three function χ on V ∗ [1] ⊕ V [1] satisfying { χ, χ } = 0. The condition that χ vanishes on V [1] and V ∗ [1]implies that χ belongs to a functional completion of S > ( V [ − ⊗ S > ( V ∗ [ − L ∞ -bialgebra structure on V , equivalent, up tocompletion, to Kravchenko’s notion [Kra07] of an L ∞ -bialgebra. We will adopt thealgebraic version of [BV15] and assume χ ∈ (cid:81) m,n ≥ Hom( S m ( V [1]) , S n ( V [ − { χ, χ } = 0.The notion of an L ∞ -morphism for two L ∞ -bialgebroids, see Definition 5.4, leadsto the following version of the notion of an L ∞ -morphism of two L ∞ -bialgebras V and W , regarded as L ∞ -algebroids over a point. An L ∞ - morphism V → W of L ∞ -bialgebras is a linear map Φ : S ( V [1]) → S ( W [1]) commuting with the solutions χ V ∈ (cid:81) m,n ≥ Hom( S m ( V [1]) , S n ( V [ − χ W ∈ (cid:81) m,n ≥ Hom( S m ( W [1]) , S n ( W [ − { χ, χ } = 0 defining the L ∞ -bialgebra structures:Φ ◦ χ V = χ W ◦ Φ . Here, as in the definition of an L ∞ -morphism of L ∞ -algebroids, χ V and χ W are un-derstood as operators. 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