Atiyah classes of strongly homotopy Lie pairs
aa r X i v : . [ m a t h . QA ] M a y ATIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS
ZHUO CHEN, HONGLEI LANG, AND MAOSONG XIANGA
BSTRACT . The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L ∞ -algebras. Weextract an intrinsic character, the Atiyah class, which measures the nontriviality of an SH Lie algebra A when itis extended to L . In fact, given such an SH Lie pair ( L, A ) , and any A -module E , there associates a canonicalcohomology class, the Atiyah class [ α E ] , which generalizes earlier known Atiyah classes out of Lie algebrapairs. We show that the Atiyah class [ α L/A ] induces a graded Lie algebra structure on H • CE ( A, L/A [ − , andthe Atiyah class [ α E ] of any A -module E induces a Lie algebra module structure on H • CE ( A, E ) . Moreover,Atiyah classes are invariant under gauge equivalent A -compatible infinitesimal deformations of L . Keywords : Homotopical algebra, L ∞ -algebra, Atiyah class. MSC : Primary 16E45, 18G55. Secondary 58C50. C ONTENTS
Introduction 21. Preliminaries 31.1. Graded linear algebra 31.2. Strongly homotopy Lie algebras 51.3. Connections, curvatures and modules of SH Lie algebras 72. Atiyah classes of SH Lie pairs 112.1. SH Lie pairs 112.2. Construction of Atiyah classes 122.3. An equivalent description of Atiyah classes 172.4. Vanishing of Atiyah classes 193. Atiyah classes as functors 203.1. Atiyah operators 203.2. Atiyah classes as Lie structures 213.3. Atiyah functors 244. Invariance of Atiyah classes under infinitesimal deformations 244.1. Compatible infinitesimal deformations 254.2. Gauge invariance of Atiyah classes 265. Appendix: Morphisms of SH Lie algebras 27Acknowledgments 28References 28
Research partially supported by NSFC 11471179. I NTRODUCTION
This work is motivated by two sources: Atiyah classes and strongly homotopy Lie algebras. Originally,the Atiyah class [3] of a holomorphic vector bundle U over a complex manifold constitutes the obstructionto the existence of a holomorphic connection on U . Molino [26, 27] defined the Atiyah-Molino class ofa foliation of a manifold to capture the existence of a locally projectable connection. The Atiyah classof a Lie algebra pair was studied by Wang [36], Nguyen-van [28] and Bordemann [6], to characterize theexistence of invariant connections on a homogeneous space. Atiyah classes have enjoyed renewed vigor dueto Kontsevich’s seminal work on deformation quantization [16, 17]. They are also related to the Rozansky-Witten theory [15, 30]. Calaque and Van den Bergh [10] considered the Atiyah class of a DG module overa DG-algebra. They also inferred that, given a Lie algebra pair ( d , g ) , the Atiyah class of the quotient d / g coincides with the class capturing the obstruction to the “PBW problem” studied earlier by Calaque–C˘ald˘araru–Tu [9] (see also [8, 14]).The notion of strongly homotopy (SH) Lie algebras, also called L ∞ [1] -algebras (see Definition 1.5), wasintroduced by Lada and Stasheff [19, 20]. The investigation of SH Lie algebras from various perspectivesstarted a while ago. Attention on this subject in the past ten years is largely due to its role in mathematicalphysics and supergeometry. For example, Kontsevich and Soibelman [18] approached this notion via thelanguage of formal geometry. Meanwhile, Bashkirov and Voronov [5] used the Batalin-Vilkovisky formal-ism to treat an SH Lie algebra as a special pointed BV ∞ -manifold. The interested reader is referred to arecent talk by Stasheff [31] in which many related topics are reviewed.Motivated by the various constructions of Atiyah classes, we will study SH Lie algebra pairs, show thatanalogous Atiyah classes exist, and how they play the role of sending homotopical objects to Lie objects.It is usually nontrivial to construct examples of L ∞ [1] -algebras. One “trivial” way is the semi-direct productof an L ∞ [1] -algebra A with its module B (see Proposition 1.28 (4)). We are particularly interested ingrasping the information when a smaller L ∞ [1] -algebra, say A , “non-trivially” extends to a bigger one,say L . Hence we introduce the SH Lie pair ( L, A ) , where L is an L ∞ [1] -algebra and A ⊂ L is a sub-algebra. What we discovered, is the so-called Atiyah class [ α L/A ] of ( L, A ) , which generalizes previousconstructions of Lie algebra pairs. It measures how “nontrivial” it is when the sub-algebra A is extendedto L , while the A -module structure on L/A is maintained. Moreover, it refines the homotopical data of ( L, A ) , to a canonical graded Lie algebra H • CE ( A, L/A [ − (see Theorem 3.5). One can even involve anexternal object — an A -module E , and use the Atiyah class [ α E ] of E to test the nontrivial information of A being extended to L . Moreover, [ α E ] gives rise to a Lie algebra module structure on H • CE ( A, E ) , over theaforesaid Lie algebra object (see Theorem 3.5).The following is a summary of this paper:After a review of Z -graded linear algebra and SH Lie algebras in Section 1, we focus on the constructionof Atiyah classes in Section 2. Given an SH Lie pair ( L, A ) and an A -module E , one is able to extend the A -module structure on E to an L -connection ∇ on E . The curvature R ∇ measures the failure of E beingan L -module. From R ∇ , we extract a particular element α E ∇ := ( J ⊗ R ∇ ) ∈ O ( A ) ⊗ A ⊥ ⊗ End( E ) , which is a degree cocycle. Here O ( A ) is the graded algebra of formal power series on A . We call α E ∇ theAtiyah cocycle of the SH Lie pair ( L, A ) with respect to the A -module E and the L -connection ∇ extending ( A, E ) . Theorem (A) . The cohomology class, called Atiyah class, [ α E ∇ ] ∈ H ( A, A ⊥ ⊗ End( E )) is canonical,i.e., independent of the choice of ∇ . In particular, for the canonical A -module L/A , there associates acanonical Atiyah class [ α L/A ] ∈ H ( A, Hom(
L/A ⊗ L/A, L/A )) . TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 3
In Section 3, we introduce the Atiyah operator and functor, which manifest the nature of Atiyah classes fromdifferent perspectives. The Atiyah operator is an O ( A ) -linear map which arises from the construction ofAtiyah cocycles: α E : O ( A ) ⊗ E −→ O ( A ) ⊗ A ⊥ ⊗ E ; Or, α E : ( O ( A ) ⊗ L/A ) × ( O ( A ) ⊗ E ) −→ O ( A ) ⊗ E. We prove
Theorem (B) . The graded vector space H • CE ( A, ( L/A )[ − with the binary operation induced by theAtiyah operator α L/A is a Lie algebra. Furthermore, if E is an A -module, then H • CE ( A, E ) is a Lie algebramodule over H • CE ( A, ( L/A )[ − , with the action induced by the Atiyah operator α E . This certainly generalizes previous results in [7, 12, 15]. An alternative point of view is that the processof taking Atiyah classes defines a functor, called Atiyah functor, from the category of A -modules to thecategory of H • CE ( A, ( L/A )[ − -modules.In Section 4, we study a special kind of deformations of the given SH Lie algebra pair ( L, A ) , namely A -compatible infinitesimal deformations of L . Roughly speaking, they are L ∞ [1] -algebra structures on L [ ~ ] = L ⊕ ~ L, where ~ is a formal parameter with ~ = 0 , such that the subspace A [ ~ ] is trivially extended from A , andthe A [ ~ ] -module structure on L [ ~ ] /A [ ~ ] = ( L/A )[ ~ ] is trivially extended from the A -module L/A . A smallperturbation of L [ ~ ] is an isomorphism σ : O ( L )[ ~ ] → O ( L )[ ~ ] of graded algebras.We prove the following invariance property of Atiyah classes under gauge equivalences (See Definition 4.3): Theorem (C) . If two A -compatible infinitesimal deformations of L are gauge equivalent, then the twoassociated Atiyah classes are the same. It is our hope that these results may lead to new insights in homotopical algebras and DG-manifolds. Wewould also like to point out other works that are related to the present paper: Chen, Sti´enon and Xu [12]proposed a notion of the Atiyah class of a Lie algebroid pair ( L, A ) , which encompasses both the originalAtiyah class of holomorphic vector bundles and the Atiyah-Molino class of a foliation as special cases.Shortly after that, an L ∞ -algebra structure on the space Γ( ∧ • A ∨ ⊗ L/A ) was constructed in [21, 22], wherethe Atiyah class determines the -bracket l . A similar theory for Lie groupoid pairs is available in [23]. Wealso mention that Shoikhet [32] studied the Atiyah class of a DG-manifold; Costello [13] defined the Atiyahclass of a DG-vector bundle in his geometric approach to Witten genus; Mehta, Sti´enon and Xu [25] studiedthe Atiyah class of a DG-Lie algebroid with respect to a DG-vector bundle.1. P RELIMINARIES
Graded linear algebra.
Throughout this paper, we fix a base field K of characteristic zero. A Z -graded vector space is a K -vector space V = ⊕ n ∈ Z V n , where each V n = { v ∈ V | | v | = n } is anordinary K -vector space consisting of elements of homogeneous degree n . Henceforth, we will simply call V a graded vector space. And K is considered as concentrated in degree .A degree r morphism from a graded vector space V to a graded vector space W is a linear map from V to W that sends V n to W n + r , where r could be any integer. The set Hom(
V, W ) consisting of suchhomogeneous morphisms is also a graded vector space. Thus the category of graded vector spaces over K ,denoted by GVS K , is a K -linear category.The dual of V , denoted by V ∨ , is the graded vector space whose degree n part is the ordinary dual ( V − n ) ∗ of V − n . If V is of finite dimension, then the dual of V ∨ is isomorphic to V . In this paper, we will alwaysassume that V is finite dimensional if V ∨ is involved. ZHUO CHEN, HONGLEI LANG, AND MAOSONG XIANG
For k ∈ Z , we denote by V [ k ] the graded vector space with k -shifted gradings ( V [ k ]) n = V n + k . Hence ( V [ k ]) ∨ = V ∨ [ − k ] .The category of graded vector spaces is monoidal. The tensor product of two objects V and W is the gradedvector space whose degree n part is ( V ⊗ W ) n = ⊕ i + j = n V i ⊗ W j . We have isomorphisms of graded vector spaces: V ∨ ⊗ W ∼ = W ⊗ V ∨ ∼ = Hom( V, W ) ,ξ ⊗ w ( − | ξ || w | w ⊗ ξ φ ( − ) , where ξ ∈ V ∨ , w ∈ W and φ is the map v ( ξ ⊗ w )( v ) = ( − | v || w | ξ ( v ) w .For any homogeneous element φ ∈ Hom(
V, W ) , its dual φ ∨ ∈ Hom( W ∨ , V ∨ ) , which is also homogeneousof degree | φ | , is defined in the standard manner: h φ ∨ ( α ) , v i = ( − | φ || α | h α, φ ( v ) i , α ∈ W ∨ , v ∈ V. The symmetric algebra of V and its formal completion are, respectively, S • ( V ) = ⊕ n ≥ S n ( V ) , b S • ( V ) = Y n ≥ S n ( V ) . Note that they might be infinite dimensional. The product in S • ( V ) , as well as that in b S • ( V ) , is denoted by ⊙ . The Koszul sign ǫ ( σ ) of a permutation σ of homogeneous vectors v , · · · , v n in V is determined by theequality v ⊙ · · · ⊙ v n = ǫ ( σ ) v σ (1) ⊙ · · · ⊙ v σ ( n ) . Given v ∈ V , there induces two natural contractions, one from left and one from right, denoted respectively ι v and x v , on V ∨ : ι v ξ = ( − | ξ || v | ξ x v = ( − | ξ || v | ξ ( v ) , ∀ ξ ∈ V ∨ . The left contraction ι v is extended to ι v : S • ( V ∨ ) → S •− ( V ∨ ) by the Leibniz rule ι v ( ξ ⊙ η ) = ι v ( ξ ) ⊙ η + ( − | v || ξ | ξ ⊙ ι v ( η ) , ∀ ξ, η ∈ S • ( V ∨ ) . The extension of the right contraction is similar: ( ξ ⊙ η ) x v = ( − | v || η | ( ξ x v ) ⊙ η + ξ ⊙ ( η x v ) , ∀ ξ, η ∈ S • ( V ∨ ) . We define a duality pairing S • ( V ) × S • ( V ∨ ) → K by h v ⊙ · · · ⊙ v p , ξ ⊙ · · · ⊙ ξ q i = ( ι v · · · ι v p ( ξ ⊙ · · · ⊙ ξ p ) , p = q, , otherwise.The pairing between S • ( V ∨ ) and S • ( V ) is similarly defined, and we have h v ⊙ · · · ⊙ v n , ξ ⊙ · · · ⊙ ξ n i = ( − ( P ni =1 | v i | )( P nj =1 | ξ j | ) h ξ ⊙ · · · ⊙ ξ n , v ⊙ · · · ⊙ v n i . Let O ( V ) = b S • ( V ∨ ) be the space of formal power series on the graded vector space V , which is a local algebra with the uniquemaximal ideal O + ( V ) = O ( V ) ⊙ V ∨ = b S n ≥ ( V ∨ ) = Y n ≥ S n ( V ∨ ) . Denote by r + : O ( V ) → O + ( V ) the obvious projection. TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 5
For all k ≥ , consider the product µ Vk +1 : S k ( V ) ⊗ V → S k +1 ( V ) , x ⊗ v x ⊙ v, ∀ x ∈ S k ( V ) , v ∈ V. (1.1)The dual map will be denoted by I Vk +1 : S k +1 ( V ∨ ) → S k ( V ∨ ) ⊗ V ∨ . (1.2)The summation of these I Vk +1 defines an operator I V = X k ≥ I Vk +1 ◦ r + : O ( V ) → O ( V ) ⊗ V ∨ , (1.3)which is in fact the algebraic de Rham operator of the K -algebra O ( V ) . It is clear that I V is an O ( V ) -derivation valued in the O ( V ) -bimodule O ( V ) ⊗ V ∨ , i.e., for all ω, ω ′ ∈ O ( V ) , I V ( ω ⊙ ω ′ ) = ω ⊙ I V ( ω ′ ) + I V ( ω ) ⊙ ω ′ = ω ⊙ I V ( ω ′ ) + ( − | ω || ω ′ | ω ′ ⊙ I V ( ω ) . (1.4)A degree n derivation D of O ( V ) is a degree n K -linear map D : O ( V ) → O ( V ) such that the followingLeibniz rule holds: D ( ξ ⊙ η ) = D ( ξ ) ⊙ η + ( − n | ξ | ξ ⊙ D ( η ) , ∀ ξ, η ∈ O ( V ) . The space
Der( O ( V )) of derivations of O ( V ) , together with the graded commutator [ D , D ] := D ◦ D − ( − | D || D | D ◦ D , ∀ D , D ∈ Der( O ( V )) , is a graded Lie algebra.1.2. Strongly homotopy Lie algebras.Definition 1.5.
An SH Lie algebra (or L ∞ [1] -algebra) is a pair ( L, { λ k } ∞ k =0 ) , simply denoted by ( L, λ • ) ,where L is a graded vector space and λ k : S k ( L ) → L, k ≥ , called the k th-bracket, are degree linearmaps satisfying the following generalized Jacobi identities: X k + l = n X σ ∈ sh( l,k ) ǫ ( σ ) λ k +1 ( λ l ( u σ (1) , · · · , u σ ( l ) ) , · · · , u σ ( n ) ) = 0 , (1.6)for all k, l, n ≥ and homogeneous elements u i ∈ L, ≤ i ≤ n . Here sh( s, k ) is the set of ( s, k ) -unshuffles. Example 1.7.
Let E be a graded vector space. Then End( E ) together with the graded commutator [ − , − ] is a Lie algebra. Thus End( E )[1] with the shifted commutator {− , −} : { ¯ φ, ¯ ψ } := ( − | φ | [ φ, ψ ] = ( − | φ | ( φ ◦ ψ − ( − | φ || ψ | ψ ◦ φ ) , ∀ ¯ φ, ¯ ψ ∈ End( E )[1] , (1.8)is an L ∞ [1] -algebra with only one nontrivial bracket λ = {− , −} . Here φ and ¯ φ are the same element withdifferent degrees: | ¯ φ | = | φ | − . The relation between ¯ ψ and ψ is similar. Remark . Our Definition 1.5 of L ∞ [1] -algebras is not the more commonly known notion of L ∞ -algebras(e.g., see standard texts [19, 20]). In particular, one should notice the different convention of degrees andsigns of L ∞ and L ∞ [1] -algebras. What we adopt is similar to that in [34], where Z grading is used. For apassage connecting our definition to that in [19, 20], we refer to [34, Remark 2.1].SH Lie algebras could also be characterized as Q -manifolds [2]: Definition 1.10.
A homological vector field on L is a degree derivation Q on O ( L ) such that Q = [ Q, Q ] = 0 . Proposition 1.11.
Let L be a graded vector space. Then there is a one-to-one correspondence between L ∞ [1] -algebra structures on L and homological vector fields on L . ZHUO CHEN, HONGLEI LANG, AND MAOSONG XIANG
In fact, on the one hand, if ( L, { λ k } k ≥ ) is an L ∞ [1] -algebra, then we can construct a homological vectorfield Q L as follows:For each k ≥ , the dual of λ k : S k ( L ) → L is a map λ ∨ k : L ∨ → S k ( L ∨ ) , which can be uniquely extendedto a degree derivation O ( L ) → O ( L ) . Then define Q L by Q L = X k ≥ ( − k λ ∨ k : O ( L ) → O ( L ) , i.e., for all ξ ∈ L ∨ , u i ∈ L, ≤ i ≤ k , we have h ξ, λ k ( u , · · · , u k ) i = ( − | ξ | + k h Q L ( ξ ) , u ⊙ · · · ⊙ u k i . (1.12)On the other hand, given a homological vector field Q L on L , we can define a collection of degree linearmaps λ k : S k ( L ) → L, k ≥ by λ k ( u , · · · , u k ) = ι − ([[ · · · [[ Q, ι u ] , ι u ] · · · ] , ι u k ]) ∈ L, ∀ u i ∈ L. (1.13)Here the map ι − : Der( O ( L )) → L is defined by h ι − ( D ) , ξ i = pr ◦ D ( ξ ) ∈ K , ∀ D ∈ Der( O ( L )) , ξ ∈ L ∨ , where pr : O ( L ) = b S • ( L ∨ ) → S ( L ∨ ) = K is the obvious projection. It is clear that the map ι − is the left inverse of the contraction operator ι − : L → Der( O ( L )) in the sense that ι − ( ι u ) = u , for all u ∈ L .The fact that Q L in Equation (1.12) is of square zero and the fact that { λ k } k ≥ in Equation (1.13) definesan L ∞ [1] -algebra structure on L are equivalent. Details can be found in [34, 35].For this reason, an L ∞ [1] -algebra can be denoted by any of the notations ( L, λ • ) , ( L, Q L ) or ( L, λ • ∼ Q L ) .Now we recall morphisms of SH Lie algebras: Definition 1.14.
Let ( L, λ • ∼ Q L ) and ( L ′ , λ ′• ∼ Q L ′ ) be two L ∞ [1] -algebras. An L ∞ [1] -morphism from L to L ′ is a morphism φ : O ( L ′ ) → O ( L ) of K -algebras such that φ ◦ Q L ′ = Q L ◦ φ : O ( L ′ ) → O ( L ) . (1.15)Equivalently, an L ∞ [1] -morphism from L to L ′ is a family of degree zero linear maps f k : S k ( L ) → L ′ , k ≥ satisfying the following two conditions: (1) The element f ∈ ( L ′ ) satisfies X k ≥ k ! λ ′ k ( f , · · · , f ) + λ ′ = f ( λ ) . (1.16) (2) For each n ≥ , the relation X k + l = n X σ ∈ sh( l,k ) ǫ ( σ ) f k +1 ( λ l ( u σ (1) , · · · , u σ ( l ) ) · · · , u σ ( n ) ) (1.17) = X i , ··· ,i r ≥ i + ··· + i r = n X τ ∈ sh( i , ··· ,i r ) X j ≥ r + j )! ǫ ( τ ) λ ′ r + j ( f ⊙ j , f i ( u τ (1) , · · · , u τ ( i ) ) , · · · f i r ( · · · , u τ ( n ) )) holds, where k, l ≥ and u i ∈ L are homogeneous. TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 7
For completeness, we will give a proof on the equivalence of the two definitions of morphisms in Appen-dix 5.1.3.
Connections, curvatures and modules of SH Lie algebras.
This part can be thought of as formaldifferential geometry of L ∞ [1] -algebras. What we shall deal with, namely connections and curvatures, aredefined in the same manner of super-connections in [29]. Some closely related contents can be found in[1, 19, 24, 33].Let us fix an L ∞ [1] -algebra ( L, λ • ∼ Q L ) and a graded vector space E . A degree n operator ∂ : O ( L ) ⊗ E → O ( L ) ⊗ E is called an E -derivation if it is K -linear and there exists a degree n derivation ∂ ∈ Der( O ( L )) , called the symbol of ∂ , such that ∂ ( ω ⊗ e ) = ∂ ( ω ) ⊗ e + ( − n | ω | ω ⊙ ∂ ( e ) , ω ∈ O ( L ) , e ∈ E. Let us denote the space of all E -derivations by Der( O ( L ) ⊗ E ) . There is a Lie bracket on Der( O ( L ) ⊗ E ) defined by its graded commutator [ ∂, ∂ ′ ] = ∂ ◦ ∂ ′ − ( − | ∂ || ∂ ′ | ∂ ′ ◦ ∂, ∀ ∂, ∂ ′ ∈ Der( O ( L ) ⊗ E ) . Lemma 1.18.
The Lie algebra
Der( O ( L ) ⊗ E ) is isomorphic to the semidirect product of Der( O ( L )) and O ( L ) ⊗ End( E ) .Proof. Note that O ( L ) ⊗ End( E ) consists of E -derivations with zero symbol, thus a Lie subalgebra of Der( O ( L ) ⊗ E ) . And Der( O ( L )) is also a Lie subalgebra by the inclusion j : Der( O ( L )) → Der( O ( L ) ⊗ E ) , j ( X )( ω ⊗ e ) = X ( ω ) ⊗ e, ∀ X ∈ Der( O ( L )) , ω ∈ O ( L ) , e ∈ E, which gives rise to a natural splitting of the short exact sequence of Lie algebras → O ( L ) ⊗ End( E ) ֒ → Der( O ( L ) ⊗ E ) → Der( O ( L )) → , where the Lie algebra morphism Der( O ( L ) ⊗ E ) → Der( O ( L )) is taking symbols. It gives rise to asemidirect product Der( O ( L )) ⋉ ( O ( L ) ⊗ End( E )) that is isomorphic to Der( O ( L ) ⊗ E ) . (cid:3) Connections and curvatures.
Definition 1.19. An L -connection on E is a degree E -derivation ∇ ∈ Der( O ( L ) ⊗ E ) whose symbol is Q L , i.e., the following Leibniz rule holds: ∇ ( ω ⊗ e ) = Q L ( ω ) ⊗ e + ( − | ω | ω ⊙ ∇ ( e ) , ∀ ω ∈ O ( L ) , e ∈ E. The degree E -derivation R ∇ := ∇ = 12 [ ∇ , ∇ ] : O ( L ) ⊗ E → O ( L ) ⊗ E is of zero symbol, i.e., O ( L ) -linear, and will be called the curvature of ∇ . An L -connection ∇ is said to beflat if its curvature R ∇ vanishes.The difference of two connections is a degree endomorphism of the O ( L ) -module O ( L ) ⊗ E . Thus theset of all L -connections on E is an affine space over ( O ( L ) ⊗ End( E )) .According to Lemma 1.18, any L -connection ∇ is determined by an element D E ∈ ( O ( L ) ⊗ End( E )) sothat ∇ = Q L + D E . An easy computation shows that the curvature has the form R ∇ = ∇ = ( Q L + D E ) ◦ ( Q L + D E ) = Q L ( D E ) + ( D E ) . Lemma 1.20.
We have the Bianchi identity: Q L ( R ∇ ) + [ D E , R ∇ ] = 0 . (1.21) ZHUO CHEN, HONGLEI LANG, AND MAOSONG XIANG
Proof.
This follows from straightforward computations: [ Q L + D E , R ∇ ] = [ ∇ , ∇ ] = ∇ ◦ ∇ − ( − × ∇ ◦ ∇ = 0 . (cid:3) Given an L -connection ∇ on E , there corresponds a dual L -connection ∇ ∨ : O ( L ) ⊗ E ∨ → O ( L ) ⊗ E ∨ on E ∨ . Explicitly, we have h∇ ∨ ( f ) , g i = Q L h f, g i − ( − | f | h f, ∇ ( g ) i ∈ O ( L ) , ∀ f ∈ O ( L ) ⊗ E, g ∈ O ( L ) ⊗ E ∨ . (1.22)For two graded vector spaces with L -connections ( E, ∇ E ) and ( F, ∇ F ) , the induced L -connection on E ⊗ F is given by ∇ E ⊗ F ( ω ⊗ e ⊗ f ) = Q L ( ω ) ⊗ e ⊗ f + ( − | ω | ω ⊙ ( ∇ E ( e ) ⊗ f ) + ( − | ω | + | e | ( ω ⊗ e ) ⊗ O ( L ) ∇ F ( f ) . (1.23)Here we used the canonical isomorphism ( O ( L ) ⊗ E ) ⊗ O ( L ) ( O ( L ) ⊗ F ) ∼ = −→ O ( L ) ⊗ ( E ⊗ F ) to view the last term ( ω ⊗ e ) ⊗ O ( L ) ∇ F ( f ) as an element in O ( L ) ⊗ ( E ⊗ F ) .In particular, we have the induced L -connection ∇ Hom(
E,F ) on Hom(
E, F ) ∼ = E ∨ ⊗ F . Lemma 1.24.
For each Ψ ∈ O ( L ) ⊗ Hom(
E, F ) ∼ = Hom O ( L ) ( O ( L ) ⊗ E, O ( L ) ⊗ F ) , ∇ Hom(
E,F ) (Ψ) = ∇ F ◦ Ψ − ( − | Ψ | Ψ ◦ ∇ E . (1.25)1.3.2. Modules over SH Lie algebras.
Definition 1.26. An L -module ( E, ∂ EL ) is a graded vector space E together with a flat L -connection ∂ EL : O ( L ) ⊗ E → O ( L ) ⊗ E , which will be called the Chevalley-Eilenberg differential of E . Theassociated cohomology H • ( O ( L ) ⊗ E, ∂ EL ) will be denoted by H • CE ( L, E ) .Recall that as an L -connection, ∂ EL is determined by an element D E ∈ ( O ( L ) ⊗ End( E )) such that ∂ EL = Q L + D E . The flat condition of ∂ EL becomes a Maurer-Cartan equation: Q L ( D E ) + ( D E ) = 0 . (1.27)We have alternative descriptions of L -modules. Proposition 1.28.
Let L = ( L, λ • ∼ Q L ) be an L ∞ [1] -algebra, and E be a graded vector space. Thefollowing data are mutually determined: (1) An L -module structure on E ; (2) An L -module structure on E ∨ ; (3) A degree derivation Q ∈ Der( O ( L ⊕ E )) such that Q | O ( L ) = Q L , Q ( E ∨ ) ⊂ O ( L ) ⊗ E ∨ and Q = 0 ; (4) An abelian extension L ⊕ E of L along E (also called a semi-direct product of L with E ): i.e., L ⊕ E carries an L ∞ [1] -algebra structure { ˜ λ k } k ≥ such that (i) L is an L ∞ [1] subalgebra (in particular, ˜ λ = λ ∈ L ); (ii) E is an ideal: ˜ λ k ( E, · · · ) ⊂ E, k ≥ ; (iii) E is abelian: ˜ λ k ( E, E, · · · ) = 0 , k ≥ ; (5) A family of degree linear maps m Ek : S k − ( L ) ⊗ E → E, (1.29) k = 1 , , · · · , such that X k + l = n X σ ∈ sh( l,k ) ǫ ( σ ) m Ek +2 ( λ l ( u σ (1) , · · · , u σ ( l ) ) , · · · , u σ ( n ) , e ) TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 9 = − X k + l = n X τ ∈ sh( k,l ) ǫ ( τ )( − † τk m Ek +1 ( u τ (1) , · · · , u τ ( k ) , m El +1 ( u τ ( k +1) , · · · , u τ ( n ) , e )) (1.30) hold for all k, l, n ≥ and homogeneous elements u i ∈ L, e ∈ E , where † τk = P ki =1 | u τ ( i ) | ; (6) An L ∞ [1] -morphism f E = { f Ek } from L to End( E )[1] , where f Ek : S k ( L ) → End( E )[1] , k ≥ . (1.31) Proof.
For the equivalence (1) ⇔ (2) , we need to show that an L -connection ∂ EL on E is flat if and only ifits dual L -connection ∂ E ∨ L on E ∨ is flat. In fact, by Equation (1.22), we have h ( ∂ E ∨ L ) ( e ∨ ) , e i = Q L h ∂ E ∨ L ( e ∨ ) , e i − ( − | e ∨ | +1 h ∂ E ∨ L ( e ∨ ) , ∂ EL ( e ) i = Q L h ∂ E ∨ L ( e ∨ ) , e i − ( − | e ∨ | +1 Q L h e ∨ , ∂ EL ( e ) i − h e ∨ , ( ∂ EL ) ( e ) i = −h e ∨ , ( ∂ EL ) ( e ) i . Thus ( ∂ EL ) = 0 is equivalent to ( ∂ E ∨ L ) = 0 .The equivalence (2) ⇔ (3) is obvious.To see the equivalence (3) ⇔ (4) , we note that: Q = 0 ⇔ L ⊕ E is an L ∞ [1] -algebra (by Proposition1.11); Q | O ( L ) = Q L ⇔ (i) in (4); Q ( E ∨ ) ⊂ O ( L ) ⊗ E ∨ ⇔ (ii) and (iii) in (4).To see (4) ⇔ (5) , we note that by setting m Ek ( u , · · · , u k − , e ) = ˜ λ k ( u , · · · , u k − , e ) , Equation (1.30) is equivalent to the generalized Jacobi identity (1.6) of ( L ⊕ E, ˜ λ • ) .Finally, we show that (5) ⇔ (6) . In fact, for all k ≥ , f Ek and m Ek +1 are mutually determined by f Ek ( u , · · · , u k )( e ) = m Ek +1 ( u , · · · , u k , e ) . Now we reformulate Equation (1.30) in terms of f Ek : For n = 0 , it becomes m E ( λ , e ) = − m E ( m E ( e )) , ∀ e ∈ E, which is equivalent to f E ( λ )( e ) = −
12 [ f E , f E ]( e ) = 12 { f E , f E } ( e ) , ∀ e ∈ E. Here [ − , − ] is the graded commutator in End( E ) and {− , −} is the shifted graded commutator in End( E )[1] (see Example 1.7), which is the only nontrivial bracket in End( E )[1] .For n ≥ , Equation (1.30) can be reorganized as X k + l = n X σ ∈ sh( l,k ) ǫ ( σ ) f Ek +1 ( λ l ( u σ (1) , · · · , u σ ( l ) ) , · · · , u σ ( n ) )= 12 { f E , f En ( u , · · · , u n ) } + X k + l = n X τ ∈ sh( l,k ) ǫ ( τ ) 12 { f Ek ( u τ (1) , · · · , u τ ( k ) ) , f El ( u τ ( k +1) , · · · , u τ ( n ) ) } . (1.32)These are exactly Equation (1.17). (cid:3) Remark . According to this theorem, we can write the explicit relation between D E (or ∂ EL ) and m E • : m Ek +1 ( u , · · · , u k , e ) = ( − | e |∗ k + k +1 D E ( e ) x u x · · · x u k , ∀ u , · · · , u k ∈ L, e ∈ E, (1.34)where ∗ k = P ki =1 | u i | . From now on, we will denote an L -module by any of the notations ( E, ∂ EL ) , ( E, D E ) , ( E, m E • ) , ( E, f E • ) or ( E, ∂ EL ∼ m E • ) , etc. The data ∂ EL : O ( L ) ⊗ E → O ( L ) ⊗ E , D E ∈ ( O ( L ) ⊗ End( E )) , { m Ek } k ≥ inEquation (1.29), and { f Ek } in Equation (1.31) are mutually determined by the above theorem. Example 1.35 (Adjoint module) . Let ( L, λ • ) be an L ∞ [1] -algebra. Then the maps m Lk +1 = λ k +1 ◦ µ Lk +1 : S k ( L ) ⊗ L → L, k ≥ (1.36)make L an L -module, where µ Lk +1 are defined by Equation (1.1). In analogy to Lie algebras, we call it theadjoint L -module or the adjoint representation of L .Now we introduce morphisms of L -modules. Definition 1.37 (Morphisms of L -modules) . Let ( E, ∂ EL ∼ m E • ) and ( F, ∂ FL ∼ m F • ) be two L -modules. Amorphism of L -modules from E to F is an O ( L ) -linear map φ : O ( L ) ⊗ E → O ( L ) ⊗ F of degree which is also a chain map, i.e., φ ◦ ∂ EL = ∂ FL ◦ φ. (1.38)Equivalently, an L -module morphism from E to F is an element φ = P k ≥ φ k ∈ O ( L ) ⊗ Hom(
E, F ) ,where φ k ∈ S k ( L ∨ ) ⊗ Hom(
E, F ) , such that, for all n ≥ , X k + l = n X σ ∈ sh( k,l ) ǫ ( σ )( − ∗ n ( † σk +1)+ k φ l ( u σ ( k +1) , · · · , u σ ( n ) )( m Ek +1 ( u σ (1) , · · · , u σ ( k ) , e ))= X k + l = n X τ ∈ sh( k,l ) ǫ ( τ )( − k φ n − k +1 ( λ k ( u τ (1) , · · · , u τ ( k ) ) , · · · , u τ ( n ) )( e )+ X k + l = n X τ ∈ sh( k,l ) ǫ ( τ )( − ( ∗ n +1) † τk + n − k m Fn − k +1 ( u τ ( k +1) , · · · , u τ ( n ) , φ k ( u τ (1) , · · · , u τ ( k ) )( e )) (1.39)holds for all e ∈ E, u i ∈ L, k, l ≥ , where ∗ n = P ni =1 | u i | , † σk = P ki =1 | u σ ( i ) | and † τk = P ki =1 | u τ ( i ) | .The set of such morphisms will be denoted by Hom L ( E, F ) .Let ( E, f E • ) be an L -module. Consider the family of maps φ L,Ek = f Ek +1 ◦ µ Lk +1 : S k ( L ) ⊗ L → End( E )[1] , k ≥ , (1.40)where µ Lk +1 : S k ( L ) ⊗ L → S k +1 ( L ) is defined by Equation (1.1). It turns out to be a canonical morphismof L -modules: Lemma 1.41.
The family of maps φ L,E = { φ L,E • } defines an L -module morphism from the adjoint L -module L to End( E )[1] .Proof. To see that φ L,E is an L -module morphism, it suffices to show that φ L,E satisfies Equation (1.39). Infact, note that m End( E )[1] ( ϕ ) = { m E , ϕ } , ∀ ϕ ∈ End( E )[1] , where {− , −} is the shifted graded commutator (1.8) in End( E )[1] . Then Equation (1.39) follows byreformulating Equation (1.32) via Equations (1.36) and (1.40). (cid:3) TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 11
2. A
TIYAH CLASSES OF
SH L
IE PAIRS
SH Lie pairs.Definition 2.1.
By an SH Lie pair ( L, A ) , we mean an L ∞ [1] -algebra ( L, λ • ∼ Q L ) with a subalgebra A ⊂ L . The structure maps in A are again denoted by { λ k } k ≥ . In particular, λ ∈ A ⊂ L .The homological vector field Q A : O ( A ) → O ( A ) on the subalgebra A is determined by Q L . In fact, theinclusion map j : A → L gives rise to j ∨ : L ∨ → A ∨ and a surjective morphism of commutative algebras j ∨ : O ( L ) → O ( A ) . The condition that A is a subalgebra in L is equivalent to j ∨ ◦ Q L = Q A ◦ j ∨ : O ( L ) → O ( A ) . We fix such an SH Lie pair ( L, A ) . For simplicity, here and in the sequel, we write B = L/A . Lemma 2.2.
The quotient space B is a canonical A -module.Proof. There is an exact sequence of graded vector spaces / / A j / / L p / / B / / . (2.3)The canonical A -module structure on Bm k : S k − ( A ) ⊗ B → B, k ≥ is defined by m k ( a , · · · , a k − , b ) = p ◦ λ k ( a , · · · , a k − , l ) , ∀ a i ∈ A, l ∈ L such that p ( l ) = b. These m k , k ≥ , are well-defined because A is a subalgebra. That { m k } k ≥ satisfies Equation (1.30)follows from the generalized Jacobi identity (1.6). (cid:3) It follows that the dual vector space B ∨ = ( L/A ) ∨ ∼ = A ⊥ = ker( j ∨ : L ∨ → A ∨ ) is also an A -module, which will be denoted by ( A ⊥ , ∂ ⊥ A ) .Define an operator J : O ( L ) → O ( A ) ⊗ L ∨ by the commutative diagram: O ( L ) I L / / J % % ▲▲▲▲▲▲▲▲▲▲ O ( L ) ⊗ L ∨ j ∨ ⊗ (cid:15) (cid:15) O ( A ) ⊗ L ∨ . Here I L : O ( L ) → O ( L ) ⊗ L ∨ is defined by Equation (1.3). It follows immediately that (1 ⊗ j ∨ ) ◦ J = j ∨ ◦ I L = I A ◦ j ∨ : O ( L ) → O ( A ) ⊗ A ∨ . (2.4)It is also easy to see that J is a derivation valued in the O ( L ) -bimodule O ( A ) ⊗ L ∨ , i.e., for all ω, ω ′ ∈ O ( L ) , J ( ω ⊙ ω ′ ) = j ∨ ( ω ) ⊙ J ( ω ′ ) + J ( ω ) ⊙ j ∨ ( ω ′ ) = j ∨ ( ω ) ⊙ J ( ω ′ ) + ( − | ω || ω ′ | j ∨ ( ω ′ ) ⊙ J ( ω ) . (2.5)For any graded vector space E , O ( L ) ⊗ End( E ) has an obvious associative product ◦ . By Equation (2.5),the map J ⊗ O ( L ) ⊗ End( E ) → O ( A ) ⊗ L ∨ ⊗ End( E ) satisfies, for all φ, ψ ∈ O ( L ) ⊗ End( E ) , ( J ⊗ φ ◦ ψ ) = ( j ∨ ⊗ φ ) ◦ ( J ⊗ ψ ) + ( J ⊗ φ ) ◦ ( j ∨ ⊗ ψ ) . (2.6) Proposition 2.7.
Let ∂ ⊥ A = Q A + D ⊥ be the dual A -module structure on B ∨ ∼ = A ⊥ . Then for any ω ∈ ker( j ∨ ) ⊂ O ( L ) , we have J ( ω ) ∈ O ( A ) ⊗ A ⊥ and ∂ ⊥ A ( J ( ω )) = J ( Q L ( ω )) . (2.8) Proof.
Let ( B, D B ∼ m • ) be the A -module structure on B in Lemma 2.2. We first show that D ⊥ ( ξ ) = J ( Q L ( ξ )) , ∀ ξ ∈ A ⊥ . (2.9)In fact, using Equation (1.34), we have, for all p ( l ) ∈ B, a ⊙ · · · ⊙ a k ∈ S k ( A ) , h D ⊥ ( ξ ) , p ( l ) i ( a ⊙ · · · ⊙ a k ) = ( − | ξ | +1 h ξ, D B ( p ( l ))( a ⊙ · · · ⊙ a k ) i = ( − | ξ | + | l |∗ k + k h ξ, m k +1 ( a , · · · , a k , p ( l )) = ( − | ξ | + | l |∗ k + k h ξ, p ◦ λ k +1 ( j ( a ) , · · · , j ( a k ) , l ) i = ( − k +1 h ( j ∨ ◦ I L ◦ λ ∨ k +1 )( ξ ) , p ( l ) i ( a ⊙ · · · ⊙ a k )= h ( J ◦ Q L )( ξ ) , p ( l ) i ( a ⊙ · · · ⊙ a k ) , where ∗ k = P ki =1 | a i | , since Q L = P k ≥ ( − k λ ∨ k . This proves Equation (2.9).To prove Equation (2.8), it suffices to consider elements of the form ω ⊙ ξ ∈ O ( L ) ⊙ A ⊥ ∼ = ker( j ∨ ) , where ω ∈ O ( L ) , ξ ∈ A ⊥ . Then using Equation (2.9), we have J ( Q L ( ω ⊙ ξ )) = J ( Q L ( ω ) ⊙ ξ + ( − | ω | ω ⊙ Q L ( ξ ))= j ∨ ( Q L ( ω )) ⊙ J ( ξ ) + J ( Q L ( ω )) ⊙ j ∨ ( ξ ) + ( − | ω | (cid:0) J ( ω ) ⊙ j ∨ ( Q L ( ξ )) + j ∨ ( ω ) ⊙ J ( Q L ( ξ )) (cid:1) = Q A (cid:0) j ∨ ( ω ) (cid:1) ⊙ ξ + ( − | ω | j ∨ ( ω ) ⊙ D ⊥ ( ξ ) = ∂ ⊥ A ( J ( ω ⊙ ξ )) . This completes the proof. (cid:3)
Construction of Atiyah classes.
In this part, besides the SH Lie pair ( L, A ) , we fix an A -module ( E, m • ∼ D A,E ) . As usual, assume that the differential is of the form ∂ EA = Q A + D A,E , where D A,E ∈ ( O ( A ) ⊗ End( E )) is the A -module structure on E , and it will be treated as an O ( A ) -linearmap O ( A ) ⊗ E → O ( A ) ⊗ E .Meanwhile, A ⊥ ⊗ End( E ) carries an A -module structure, its differential ∂ A ⊥ ⊗ End( E ) A : O ( A ) ⊗ ( A ⊥ ⊗ End( E )) → O ( A ) ⊗ ( A ⊥ ⊗ End( E )) is expressed by ∂ A ⊥ ⊗ End( E ) A = Q A + D ⊥ + [ D A,E , − ] . The corresponding cohomology space is denoted by H • CE ( A, A ⊥ ⊗ End( E )) .Since j ∨ : O ( L ) → O ( A ) is surjective, one is able to find some D L,E ∈ ( O ( L ) ⊗ End( E )) such that ( j ∨ ⊗ D L,E ) = D A,E . Thus we get an L -connection ∇ = Q L + D L,E on E subject to the commutativediagram O ( L ) ⊗ E ∇ / / j ∨ ⊗ (cid:15) (cid:15) O ( L ) ⊗ E j ∨ ⊗ (cid:15) (cid:15) O ( A ) ⊗ E ∂ EA / / O ( A ) ⊗ E. We call ∇ an L -connection extending ( A, E ) . However, it is not necessarily flat. The curvature of ∇ iseasily available: R ∇ = Q L ( D L,E ) + ( D L,E ) ∈ O ( L ) ⊗ End( E ) . TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 13
We observe the following commutative diagram: E R ∇ / / O ( L ) ⊗ E J ⊗ / / j ∨ ⊗ (cid:15) (cid:15) O ( A ) ⊗ L ∨ ⊗ E ⊗ j ∨ ⊗ (cid:15) (cid:15) E ( ∂ EA ) =0 / / O ( A ) ⊗ E I A ⊗ / / O ( A ) ⊗ A ∨ ⊗ E, which implies that (1 ⊗ j ∨ ⊗ J ⊗ R ∇ ) = 0 . Hence, we get an element α E ∇ := ( J ⊗ R ∇ ) ∈ O ( A ) ⊗ A ⊥ ⊗ End( E ) (2.10)of degree . Theorem-Definition 2.11. (1)
The element α E ∇ defined by Equation (2.10) is a cocycle of the Chevalley-Eilenberg complex of A with coefficient in A ⊥ ⊗ End( E ) , which will be called the Atiyah cocycle of theSH Lie pair ( L, A ) with respect to the A -module E and the L -connection ∇ extending ( A, E ) . (2) The cohomology class [ α E ] = [ α E ∇ ] ∈ H ( A, A ⊥ ⊗ End( E )) does not depend on the choice of the L -connection ∇ extending ( A, E ) . We call it the Atiyah class of the SH Lie pair ( L, A ) with respect tothe A -module E . (3) For the canonical A -module L/A , there associates a canonical Atiyah class [ α L/A ] ∈ H ( A, A ⊥ ⊗ End(
L/A )) = H ( A, Hom(
L/A ⊗ L/A, L/A )) . Before giving a proof of the above theorem, we prove the following
Lemma 2.12. If X ∈ O ( L ) ⊗ End( E ) satisfying ( j ∨ ⊗ X ) = 0 , then [ D A,E , ( J ⊗ X )] = ( J ⊗ D L,E , X ] . Proof.
Without lose of generality, we may assume that X is homogeneous. Note that ( j ∨ ⊗ D L,E ) = D A,E . We have ( J ⊗ D L,E , X ] = ( J ⊗ D L,E ◦ X − ( − | X | X ◦ D L,E )= ( j ∨ ⊗ D L,E ) ◦ ( J ⊗ X ) + ( J ⊗ D L,E ) ◦ ( j ∨ ⊗ X ) − ( − | X | (( j ∨ ⊗ X ) ◦ ( J ⊗ D L,E ) + ( J ⊗ X ) ◦ ( j ∨ ⊗ D L,E )) ( by Equation (2.6) )= D A,E ◦ ( J ⊗ X ) − ( − | X | ( J ⊗ X ) ◦ D A,E = [ D A,E , ( J ⊗ X )] . (cid:3) Proof of Theorem-Definition 2.11. (1) . Note that ( j ∨ ⊗ R ∇ ) = ( ∂ EA ) = 0 . It follows that ( J ⊗ R ∇ ) ∈O ( A ) ⊗ A ⊥ ⊗ End( E ) . Thus ∂ A ⊥ ⊗ End( E ) A ( α E ∇ ) = ( Q A + D ⊥ + [ D A,E , − ])(( J ⊗ R ∇ ))= ( Q A + D ⊥ )(( J ⊗ R ∇ )) + [ D A,E , ( J ⊗ R ∇ )]= ( J ⊗ Q L ( R ∇ ) + [ D L,E , R ∇ ]) = 0 , where we have used Equation (2.8), Lemma 2.12, and the Bianchi identity (1.21) in the last two steps. (2) . Let ˜ ∇ = Q L + ˜ D L,E be another L -connection extending ( A, E ) . Then φ = ∇ − ˜ ∇ = D L,E − ˜ D L,E ∈ ( O ( L ) ⊗ End( E )) satisfies ( j ∨ ⊗ φ ) = 0 , ( J ⊗ φ ) ∈ O ( A ) ⊗ A ⊥ ⊗ End( E ) . It follows from Equation (2.6) that ( J ⊗ φ ) = 0 . Therefore, we have α E ∇ − α E ˜ ∇ = ( J ⊗ R ∇ − R ˜ ∇ ) = ( J ⊗ (cid:16) Q L ( D L,E ) + ( D L,E ) − Q L ( ˜ D L,E ) − ( ˜ D L,E ) (cid:17) = ( J ⊗ Q L ( φ ) + [ ˜ D L,E , φ ] + φ ) = ( J ⊗ Q L ( φ ) + [ ˜ D L,E , φ ])= ( Q A + D ⊥ )(( J ⊗ φ )) + [ D A,E , ( J ⊗ φ )] ( by Equation (2.8) and Lemma 2.12 )= ( Q A + D ⊥ + [ D A,E , − ])(( J ⊗ φ )) = ∂ A ⊥ ⊗ End( E ) A (( J ⊗ φ )) , which implies that [ α E ∇ ] = [ α E ˜ ∇ ] .Finally, statement (3) follows from the standard identification A ⊥ ∼ = ( L/A ) ∨ . (cid:3) We now characterize the Atiyah cocycle α E ∇ ∈ O ( A ) ⊗ A ⊥ ⊗ End( E ) in terms of the brackets λ • comingfrom Q L . Recall that we started from D L,E ∈ ( O ( L ) ⊗ End( E )) which extends D A,E ∈ ( O ( A ) ⊗ End( E )) . This can also be interpreted by a family of degree linear maps { ˜ m k : S k − ( L ) ⊗ E → E } k ≥ extending { m k : S k − ( A ) ⊗ E → E } k ≥ (see Equation (1.34) for the relation between D L,E and ˜ m • ).We further assume that α E ∇ = X k ≥ α k , where α k ∈ S k ( A ∨ ) ⊗ ( L/A ) ∨ ⊗ End( E ) . Below is the explicit formula of α k . Proposition 2.13.
For all a , · · · , a k ∈ A, e ∈ E, b ∈ B = L/A , we have ( − k +1 α k ( a , · · · , a k , b, e )= k X p =0 X σ ∈ sh( p,k − p ) ǫ ( σ ) ˜ m k − p +3 ( λ p ( a σ (1) , · · · , a σ ( p ) ) , · · · , a σ ( k ) , l, e )+ k X p =0 X σ ∈ sh( p,k − p ) ( − | b | ( ∗ k −† σp ) ǫ ( σ ) ˜ m k − p +2 ( λ p +1 ( a σ (1) , · · · , a σ ( p ) , l ) , a σ ( p +1) , · · · , a σ ( k ) , e )+ k X p =0 X σ ∈ sh( p,k − p ) ( − † σp ǫ ( σ ) m p +1 ( a σ (1) , · · · , a σ ( p ) , ˜ m k − p +2 ( a σ ( p +1) , · · · , a σ ( k ) , l, e ))+ k X p =0 X σ ∈ sh( p,k − p ) ( − † σp + | b | ( ∗ k −† σp +1) ǫ ( σ ) ˜ m p +2 ( a σ (1) , · · · , a σ ( p ) , l, m k − p +1 ( a σ ( p +1) , · · · , a σ ( k ) , e )) , where l ∈ L satisfies p ( l ) = b , ∗ k = P ki =1 | a i | and † σp = P pi =1 | a σ ( i ) | . The proof follows from some straightforward computations and thus is omitted.
Remark . To construct the Atiyah cocycle α E ∇ , we need D L,E , or ˜ m k : S k − ( L ) ⊗ E → E, k ≥ .Nevertheless, Proposition 2.13 implies that the only information we need is the behavior of ˜ m k restrictedto S k − ( A ) ⊗ L ⊗ E . In other words, to compute α E ∇ , it is enough to do first order extensions ˜ m (1) k : S k − ( A ) ⊗ L → End( E ) of m Ek , for all k ≥ . For this reason, we believe that there should exist other,perhaps “higher” Atiyah classes.A more convenient way to get α E ∇ is to find a complementary subspace to A in L . In doing so, one maysimply assume that L = A ⊕ B , where B is only a sub-vector space, not necessarily a subalgebra of L . TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 15
Then O ( L ) ∼ = O ( A ) ⊗ O ( B ) . Let ( E, D
A,E ∼ m E • ) be an A -module, where D A,E ∈ O ( A ) ⊗ End( E ) ⊂O ( L ) ⊗ End( E ) . Thus ∇ = Q L + D A,E is an L -connection on E extending ( A, E ) . Equivalently, ∇ isdetermined by { ˜ m Ek } k ≥ : S k − ( L ) ⊗ E → E : ˜ m Ek = X p ≥ ˜ m Ek | S k − − p ( A ) ⊗ S p ( B ) ⊗ E = ( m Ek , p = 0 , , p > . Then the Atiyah cocycle becomes much simpler: ( − k +1 α k ( a , · · · , a k , b, e )= k X p =0 X σ ∈ sh( p,k − p ) ( − | b | ( ∗ k −† σp ) ǫ ( σ ) m Ek − p +2 (Pr A ( λ p +1 ( a σ (1) , · · · , a σ ( p ) , b )) , a σ ( p +1) , · · · , a σ ( k ) , e ) , where Pr A : L → A is the projection.From now on, when we talk about the Atiyah cocycle of an SH Lie pair ( L, A ) with respect to an A -module E , we always assume a splitting of sequence (2.3) and that the Atiyah cocycle is obtained via the trivial L -connection on E extending ( A, E ) as in Remark 2.14. Example 2.15.
Let ( g , h ) be an ordinary Lie algebra pair and E an h -module, where g , h and E are all usualungraded vector spaces. The Atiyah class in [12] can be recovered as follows: In fact, setting L = g [1] , A = h [1] , we get an SH Lie pair ( L, A ) with the obvious A -module structure on E . Applying Proposition 2.13,we get the Atiyah cocycle α E ∇ ( a, b, e ) = ∇ [ a,l ] ( e ) − ∇ a ∇ l ( e ) + ∇ l ∇ a ( e ) = − R ∇ ( a, l )( e ) , where a ∈ A, b ∈ L/A, e ∈ E , l ∈ L such that p ( l ) = b and ∇ : L ⊗ E → E is an L -connection extending ( A, E ) . Comparing with the Atiyah cocycle defined in [12], the only difference is a minus sign.A nontrivial example of Atiyah classes of this type can be found in [9] (see also [12, Example 22]). Example 2.16.
Let ( L = L − , A = A − ) be a one-term SH Lie pair and E = ⊕ n ∈ Z E n be an A -module, ora Lie algebra representation up to homotopy [1] of A [ − on E . Assume that L = A ⊕ B , where B is alsoconcentrated in degree ( − . If the A -module structure of E is given by m k : S k − A ⊗ E → E, k ≥ ,then the Atiyah cocycle α E = P k ≥ α k ∈ O ( A ) ⊗ B ∨ ⊗ End( E ) is given by ( − k +1 α k ( a , · · · , a k , b, e ) = k X i =1 ( − k + i m k +1 (Pr A λ ( a i , b ) , · · · , ˆ a i , · · · , e ) , for a i ∈ A, e ∈ E, b ∈ B = L/A .In particular, if the A -module structure on E has only two nontrivial actions m : E → E and m : A ⊗ E → E , then the Atiyah cocycle α E = α ∈ A ∨ ⊗ B ∨ ⊗ End( E ) reads α ( a, b, e ) = m (Pr A λ ( a, b ) , e ) . Example 2.17.
Let ( L = L − ⊕ L − , A = A − ⊕ A − ) be a Lie 2-algebra [4] pair with brackets λ , λ , λ and E = E − ⊕ E − be an A -module. Let us fix a splitting L = A ⊕ B . The Atiyah cocycle α E = α + α + α ( α i ∈ S i ( A ∨ ) ⊗ B ∨ ⊗ End( E ) ) is given by: α ( b, e ) = − m (Pr A λ ( b ) , e ) α ( a, b, e ) = ( − | b || a | m (Pr A λ ( b ) , a, e ) + m (Pr A λ ( a, b ) , e ) α ( a , a , b, e ) = − m (Pr A λ ( a , a , b ) , e ) − ( − | b || a | m (Pr A λ ( a , b ) , a , e ) − ( − ( | b | + | a | ) | a | m (Pr A λ ( a , b ) , a , e ) , for a i ∈ A, e ∈ E, b ∈ B = L/A . Example 2.18.
Let ( L, A, λ , λ ) be a DG Lie algebra pair. Suppose that L = A ⊕ B . Then the associated A -module structure on B consists of two actions: m B and m B . Assume that E is an A -module with onlytwo nontrivial actions from A : m and m . Then the Atiyah cocycle has two terms α E = α + α ∈ ( B ∨ ⊗ End( E )) ⊕ ( A ∨ ⊗ B ∨ ⊗ End( E )) , where − α ( b, e ) = m (Pr A λ ( b ) , e ) , α ( a, b, e ) = m (Pr A λ ( a, b ) , e ) . For the A -module F = B ∨ ⊗ End( E ) , we are able to split the differential operator ∂ A = ∂ + ∂ : O ( A ) ⊗ F → O ( A ) ⊗ F, where ∂ : S • ( A ∨ ) ⊗ F → S • ( A ∨ ) ⊗ F, ∂ : S • ( A ∨ ) ⊗ F → S • +1 ( A ∨ ) ⊗ F. Now the Chevalley-Eilenberg cochain complex ( O ( A ) ⊗ F, ∂ A ) associated to the A -module F becomes adouble complex: D p,q = ( S p ( A ∨ ) ⊗ F ) p + q , p ≥ , q ∈ Z with differentials ∂ : D p,q → D p,q +1 , ∂ : D p,q → D p +1 ,q . As for the Atiyah cocycle α E = α + α , it sits in D , ⊕ D , . So, the Atiyah class yields two canonicalelements: [ α ] in H ( D , • , ∂ ) and [ α ] in H ( D • , , ∂ ) .We present a particular example with nontrivial Atiyah classes. Example 2.19.
Let A = span { a , a } be a -dimensional vector space concentrating in degree ( − suchthat A ∨ = span { a ∨ , a ∨ } , and B = span { b } an ordinary -dimensional vector space concentrating indegree with dual space B ∨ = span { b ∨ } . Then L = A ⊕ B together with the homological vector field Q L = δ : A ∨ → S ( A ∨ ) ⊗ B ∨ defined by δ ( a ∨ ) = k a ∨ ⊙ a ∨ ⊗ b ∨ , δ ( a ∨ ) = k a ∨ ⊙ a ∨ ⊗ b ∨ , k , k ∈ K determine an SH Lie pair ( L, A ) such that A ⊂ L is abelian. Note that the only nontrivial structure map is λ : S ( A ) ⊗ B → A by λ ( a , a , b ) = − k a − k a . Let E be another -dimensional vector space. Then D E : E → A ∨ ⊗ E defined by, for all e ∈ E , D E ( e ) = ( k a ∨ + k a ∨ ) ⊗ e, where k , k ∈ K such that k k + k k = 0 , determines an A -module structure on E . Equivalently, we have m ( a , e ) = − k e, m ( a , e ) = − k e. The only nontrivial part of the Atiyah cocycle is α ( a , a , b, e ) = − m (Pr A λ ( a , a , b ) , e ) = − m ( − k a − k a , e ) = − ( k k + k k ) e. Note that the A -module structure ∂ B ∨ ⊗ End( E ) A on B ∨ ⊗ End( E ) is trivial in this case. Thus the Atiyah class [ α ] ∈ H ( A, B ∨ ⊗ End( E )) is nontrivial. TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 17
An equivalent description of Atiyah classes.
Let ( L, A ) be an SH Lie pair. Then there is a coadjoint A -module structure ∂ A ∨ A = Q A + D A ∨ on A ∨ , which is the dual of the adjoint A -module structure on A (Example 1.35). Moreover, we have the following commutative diagram: A ∨ Q A / / D A ∨ (cid:15) (cid:15) O ( A ) I A x x rrrrrrrrrr O ( A ) ⊗ A ∨ , where I A is defined in Equation (1.3). In fact, we recall that Q A = P k ≥ ( − k λ ∨ k , and for all ξ ∈ A ∨ , a ⊙ · · · ⊙ a k ⊗ a k +1 ∈ S k ( A ) ⊗ A , h ( I A ◦ Q A )( ξ ) , a ⊙ · · · ⊙ a k ⊗ a k +1 i = ( − k +1 h λ ∨ k +1 ( ξ ) , µ Ak +1 ( a ⊙ · · · ⊙ a k ⊗ a k +1 ) i = ( − | ξ | + k h ξ, m Ak +1 ( a , · · · , a k , a k +1 ) i , where m Ak +1 = λ k +1 ◦ µ Ak +1 : S k ( A ) ⊗ A → A is the adjoint A -module structure on A . Using Equa-tion (1.34), we have h ( I A ◦ Q A )( ξ ) , a k +1 i = ( − | ξ | +1 h ξ, D A ( a k +1 ) i = h D A ∨ ( ξ ) , a k +1 i ∈ S k ( A ∨ ) . Hence, we have D A ∨ = I A ◦ Q A , as desired.Similarly, L carries a natural A -module structure m Lk = λ k ◦ µ Lk ◦ ( j ⊙ ( k − ⊗
1) : S k − A ⊗ L → L, k ≥ , where µ Lk is the operator defined by Equation (1.1). And the dual A -module structure ∂ L ∨ A = Q A + D L ∨ on L ∨ fits into the commutative diagram: L ∨ Q L / / D L ∨ (cid:15) (cid:15) O ( L ) J y y rrrrrrrrrr O ( A ) ⊗ L ∨ , where J = ( j ∨ ⊗ ◦ I L . Moreover, it follows from a simple induction argument that ∂ L ∨ A ◦ J = J ◦ Q L : O ( L ) → O ( A ) ⊗ L ∨ , (2.20) ∂ A ∨ A ◦ I A = I A ◦ Q A : O ( A ) → O ( A ) ⊗ A ∨ . (2.21)Using Equations (2.8), (2.20) and (2.21), it can be verified that the linear dual of Sequence (2.3) of gradedvector spaces / / A ⊥ / / L ∨ j ∨ / / A ∨ / / is also a short exact sequence of A -modules.Let ( E, m • ∼ D A,E ) be an A -module. We have a companion exact sequence of A -modules: / / A ⊥ ⊗ End( E ) / / L ∨ ⊗ End( E ) j ∨ ⊗ / / A ∨ ⊗ End( E ) / / , as well as a short exact sequence of cochain complexes: / / O ( A ) ⊗ A ⊥ ⊗ End( E ) / / O ( A ) ⊗ L ∨ ⊗ End( E ) ⊗ j ∨ ⊗ / / O ( A ) ⊗ A ∨ ⊗ End( E ) / / . (2.22)A long exact sequence on the cohomology level follows: · · · → H ( A, A ⊥ ⊗ End( E )) −→ H ( A, L ∨ ⊗ End( E )) −→ H ( A, A ∨ ⊗ End( E )) δ −→ H ( A, A ⊥ ⊗ End( E )) −→ H ( A, L ∨ ⊗ End( E )) −→ · · · . (2.23) Lemma 2.24.
The element ( I A ⊗ D A,E ) ∈ O ( A ) ⊗ A ∨ ⊗ End( E ) is a degree cocycle.Proof. Since I A is a derivation valued in the O ( A ) -bimodule O ( A ) ⊗ A ∨ , ( I A ⊗ is also a derivation onthe associative algebra ( O ( A ) ⊗ End( E ) , ◦ ) . Thus ( I A ⊗ D A,E ) ) = ( I A ⊗ D A,E ) ◦ D A,E + D A,E ◦ ( I A ⊗ D A,E ) = [ D A,E , ( I A ⊗ D A,E ] . Using Equation (2.21), we have ( ∂ A ∨ A ⊗ I A ⊗ D A,E )) = ( I A ⊗ Q A ( D A,E )) . Hence, ∂ A ∨ ⊗ End( E ) A (( I A ⊗ D A,E )) = ( ∂ A ∨ A + [ D A,E , − ])(( I A ⊗ D A,E ))= ( I A ⊗ Q A ( D A,E )) + ( I A ⊗ D A,E ) ) = ( I A ⊗ Q A ( D A,E ) + ( D A,E ) ) = 0 , where the last equality follows from the Maurer-Cartan equation (1.27). (cid:3) It turns out that the element ( I A ⊗ D A,E ) gives the Atiyah class: Theorem 2.25.
The cohomology class δ [( I A ⊗ D A,E )] ∈ H ( A, A ⊥ ⊗ End( E )) coincides with the Atiyah class [ α E ] .Proof. We chase the connecting map δ in Equation (2.22): Starting with ( I A ⊗ D A,E ) , one first choosesa degree element β ∈ O ( A ) ⊗ L ∨ ⊗ End( E ) such that (1 ⊗ j ∨ ⊗ β ) = ( I A ⊗ D A,E ) . Then one is able to find a unique degree element α ∈ O ( A ) ⊗ A ⊥ ⊗ End( E ) such that ∂ L ∨ ⊗ End( E ) A ( β ) = ( Q A + D L ∨ + [ D A,E , − ])( β ) = α. The cohomology class [ α ] is the upshot of δ [( I A ⊗ D A,E )] . We now show that there exists an L -connection ∇ = Q L + D L,E extending ( A, E ) , i.e., ( j ∨ ⊗ D L,E ) = D A,E , and the resulting Atiyahcocycle α E ∇ equals α .First of all, there exists an element D L,E ∈ O ( L ) ⊗ End( E ) such that ( J ⊗ D L,E ) = β, ( j ∨ ⊗ D L,E ) = D A,E . In fact, as J : O ( L ) → O ( A ) ⊗ L ∨ is surjective, we can find some K L,E ∈ O ( L ) ⊗ End( E ) such that ( J ⊗ K L,E ) = β . Then by Equation (2.4), we have ( I A ⊗ j ∨ ⊗ K L,E ) = (1 ⊗ j ∨ ⊗ J ⊗ K L,E ) = (1 ⊗ j ∨ ⊗ β ) = ( I A ⊗ D A,E ) . Note that ker( I A ) ∼ = ker( r + ) = K . Thus ( j ∨ ⊗ K L,E ) − D A,E = ϕ for some ϕ ∈ End( E ) . It followsthat D L,E = K L,E − ϕ satisfies the above requirements.Then, using Equation (2.20), α E ∇ = ( J ⊗ Q L ( D L,E ) + ( D L,E ) ) = ( ∂ L ∨ A ⊗ ◦ ( J ⊗ D L,E ) + [( j ∨ ⊗ D A,E , ( J ⊗ D L,E )]= ( Q A + D L ∨ )( J ⊗ D L,E ) + [ D A,E , ( J ⊗ D L,E )]= ( Q A + D L ∨ + [ D A,E , − ])( β ) = α, as required. (cid:3) TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 19
Vanishing of Atiyah classes.
Let A be an L ∞ [1] -algebra and B an A -module. Then the associatedabelian extension L = A ⊕ B of A along B (see Proposition 1.28) gives rise to an SH Lie pair ( L, A ) ,while the Atiyah class α E with respect to any A -module E is always trivial. So apparently the Atiyah classmeasures the nontriviality of the extension of A to L .It is natural to ask what we can say in general when the Atiyah class vanishes. The following facts are somefirst stage results. Further investigations of this question will be shown somewhere else. Theorem 2.26.
Let ( L, A ) be an SH Lie pair and ( E, ( ∂ EA = Q A + D A,E ) ∼ f E • ) an A -module. Then thefollowing four statements are equivalent:(1) The Atiyah class [ α E ] ∈ H ( A, A ⊥ ⊗ End( E )) vanishes;(2) There exists a degree cocycle φ ∈ O ( A ) ⊗ L ∨ ⊗ End( E ) such that (1 ⊗ j ∨ ⊗ φ ) = ( I A ⊗ D A,E ); (2.27) (3) There exists an A -module morphism { φ k : S k ( A ) ⊗ L → End( E )[1] } k ≥ from L to End( E )[1] extending the canonical A -module morphism φ A,E defined in Equation (1.40) from A to End( E )[1] ,i.e., φ k ◦ (1 ⊗ j ) = φ A,Ek = f Ek +1 ◦ µ Ak +1 : S k ( A ) ⊗ A → End( E )[1]; (2.28) (4) There exists an L -connection ∇ on E extending ( A, E ) such that the Atiyah cocycle α E ∇ of E relativeto ∇ vanishes.Proof. (1) ⇒ (2) . Assume that [ α E ] = 0 . It follows from Theorem 2.25 that δ [( I A ⊗ D A,E )] = 0 . Bychasing the long exact sequence (2.23), there exists some ∂ L ∨ ⊗ End( E ) A -cocycle ˜ φ ∈ O ( A ) ⊗ L ∨ ⊗ End( E ) of degree such that [(1 ⊗ j ∨ ⊗ φ )] = [( I A ⊗ D A,E )] ∈ H ( A, A ∨ ⊗ End( E )) . It follows that there is a degree element β ∈ O ( A ) ⊗ A ∨ ⊗ End( E ) such that (1 ⊗ j ∨ ⊗ φ ) − ( I A ⊗ D A,E ) = ∂ A ∨ ⊗ End( E ) A ( β ) . By exactness of Sequence (2.22), one can choose an element γ ∈ ( O ( A ) ⊗ L ∨ ⊗ End( E )) such that (1 ⊗ j ∨ ⊗ γ ) = β .Let φ = ˜ φ − ∂ L ∨ ⊗ End( E ) A ( γ ) ∈ ( O ( A ) ⊗ L ∨ ⊗ End( E )) . Then (1 ⊗ j ∨ ⊗ φ ) = (1 ⊗ j ∨ ⊗ φ ) − ((1 ⊗ j ∨ ⊗ ◦ ∂ L ∨ ⊗ End( E ) A )( γ )= ( I A ⊗ D A,E ) + ∂ A ∨ ⊗ End( E ) A ( β ) − ( ∂ A ∨ ⊗ End( E ) A ◦ (1 ⊗ j ∨ ⊗ γ )= ( I A ⊗ D A,E ) . (2) ⇔ (3) . Note that a degree element φ ∈ O ( A ) ⊗ L ∨ ⊗ End( E ) consists of a family of degree maps φ k : S k ( A ) ⊗ L → End( E )[1] . It can be easily seen that Equation (2.27) is equivalent to Equation (2.28). (2) ⇒ (4) . Given a degree cocycle φ ∈ O ( A ) ⊗ L ∨ ⊗ End( E )[1] satisfying Equation (2.27), by theargument in the proof of Theorem 2.25, we can find D L,E ∈ O ( L ) ⊗ End( E ) such that ( J ⊗ D L,E ) = φ, ( j ∨ ⊗ D L,E ) = D A,E . Then ∇ = Q L + D L,E is an L -connection on E extending ( A, E ) . The associated Atiyah cocycle α E ∇ = ( J ⊗ R ∇ ) = ( J ⊗ Q L ( D L,E ) + D L,E ◦ D L,E )= ∂ L ∨ A (( J ⊗ D L,E )) + [( j ∨ ⊗ D L,E ) , ( J ⊗ D L,E )] ( by Equations (2.20) and (2.6) )= ∂ L ∨ A ( φ ) + [ D A,E , φ ] = ∂ L ∨ ⊗ End( E )[1] A ( φ ) = 0 . Finally, (4) ⇒ (1) is obvious. This completes the proof. (cid:3)
3. A
TIYAH CLASSES AS FUNCTORS
Atiyah operators.
Let ( L, A ) be an SH Lie pair and denote by B = L/A the standard A -module.The identification of B ∨ and A ⊥ is assumed. For simplicity, from now on, we will denote the Chevalley-Eilenberg differential ∂ EA of any A -module E by ∂ A .Recall that the Atiyah cocycle α E ∇ ∈ O ( A ) ⊗ A ⊥ ⊗ End( E ) = O ( A ) ⊗ B ∨ ⊗ Hom(
E, E ) = O ( A ) ⊗ Hom( B ⊗ E, E ) , where ∇ is an L -connection on E extending ( A, E ) . The associated O ( A ) -linear maps α E ∇ : O ( A ) ⊗ E −→ O ( A ) ⊗ B ∨ ⊗ E ; α E ∇ ( x ) : O ( A ) ⊗ E → O ( A ) ⊗ E, where x ∈ O ( A ) ⊗ B, will be called Atiyah operators. As α E ∇ is a cocycle, we have Lemma 3.1. If x ∈ O ( A ) ⊗ B is a cocycle, i.e. ∂ A ( x ) = 0 , then ∂ End( E ) A ( α E ∇ ( x )) = 0 , i.e., α E ∇ ( x ) is an A -module morphism from E to itself. Let us fix a splitting of the short exact sequence (2.3) so that L ∼ = A ⊕ B and O ( L ) ∼ = O ( A ) ⊙ O ( B ) ∼ = O ( A ) ⊗ O ( B ) . And the associated homological vector field Q L ∈ Der( O ( L )) decomposes into a sum of derivations Q L = Q A + δ + R + D ⊥ + X i ≥ T i , (3.2)where Q A = Q L | : A ∨ −→O ( A ) ; δ = Q L | : A ∨ −→O ( A ) ⊗ B ∨ ; R = Q L | : A ∨ −→O ( A ) ⊗ b S ≥ ( B ∨ ) ; D ⊥ = Q L | : B ∨ −→O ( A ) ⊗ B ∨ ; T i = Q L | : B ∨ −→O ( A ) ⊗ S i ( B ∨ ) , i ≥ , and they are extended as a derivation of O ( L ) in a natural manner.In this situation, there is an L -connection ∇ = Q L + D A,E extending ( A, E ) (see Remark 2.14). Theassociated Atiyah cocycle is denoted by α E , where the subscript is omitted. Similarly, the associated Atiyahoperator will be denoted by α E .We extend the operator δ to a K -linear and degree map for any graded vector space Eδ = ( δ ⊗
1) : O ( A ) ⊗ E → O ( A ) ⊗ B ∨ ⊗ E, such that the Leibniz rule δ ( ξ ⊙ η ⊗ e ) = ( − ( | ξ | +1) | η | η ⊙ δ ( ξ ) ⊗ e + ( − | ξ | ξ ⊙ δ ( η ) ⊗ e holds for all ξ, η ∈ O ( A ) , e ∈ E . Lemma 3.3.
As a map O ( A ) ⊗ E → O ( A ) ⊗ B ∨ ⊗ E , the Atiyah operator α E = [ δ, ∂ A ] = δ ◦ ∂ A + ∂ A ◦ δ. TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 21
Proof.
Recall the definition of Atiyah cocycles: α E = ( J ⊗ ∇ ) = ( J ⊗ (cid:0) Q L ( D A,E ) + ( D A,E ) (cid:1) = ∂ L ∨ A (( J ⊗ D A,E )) + [ D A,E , ( J ⊗ D A,E )] ( by Equations (2.20) and (2.6) )= δ ( D A,E ) + ∂ A ∨ A (( I A ⊗ D A,E )) + [ D A,E , ( I A ⊗ D A,E )] ( by Equation (1.27) )= δ ( D A,E ) + ( I A ⊗ Q A ( D A,E ) + 12 [ D A,E , D
A,E ]) ( by Equations (2.21) and (1.4) )= δ ( D A,E ) ∈ O ( A ) ⊗ B ∨ ⊗ End( E ) . By observing Q L = 0 on the O ( A ) to O ( A ) ⊗ B ∨ -part, we have ∂ ⊥ A ◦ δ + δ ◦ Q A = 0 : O ( A ) → O ( A ) ⊗ B ∨ . Thus, for all ω ⊗ e ∈ O ( A ) ⊗ E , ∂ A ( δ ( ω ⊗ e )) + δ ( ∂ A ( ω ⊗ e ))= ∂ ⊥ A ( δ ( ω )) ⊗ e + ( − | ω | +1 δ ( ω ) ⊙ ( D A,E ( e )) + δ ( Q A ( ω )) ⊗ e + ( − | ω | δ ( ω ⊙ D A,E ( e ))= ω ⊙ δ ( D A,E )( e ) = δ ( D A,E )( ω ⊗ e ) = α E ( ω ⊗ e ) . (cid:3) Applying Lemma 3.3, one can easily get the following properties of Atiyah operators:
Lemma 3.4.
For all x ∈ O ( A ) ⊗ B , the Atiyah operator α • ( x ) has the following properties:(1) For any A -modules E and F , α E ⊗ F ( x )( r ⊗ O ( A ) s ) = ( α E ( x ) r ) ⊗ O ( A ) s + ( − | x || r | r ⊗ O ( A ) ( α F ( x ) s ) , for all r ∈ O ( A ) ⊗ E , s ∈ O ( A ) ⊗ F ;(2) For any A -module E with dual module E ∨ , and for all ϕ ∈ O ( A ) ⊗ E ∨ , r ∈ O ( A ) ⊗ E , h α E ∨ ( x ) ϕ, r i = − ( − | x || ϕ | h ϕ, α E ( x ) r i ; (3) For A -modules E and F , ( α Hom(
E,F ) ( x ) κ )( r ) = α F ( x )( κ ( r )) − ( − | x || κ | κ ( α E ( x ) r ) , for all κ ∈ O ( A ) ⊗ Hom(
E, F ) , r ∈ O ( A ) ⊗ E . Atiyah classes as Lie structures.
Let ( L, A ) be an SH Lie pair, and suppose that L = A ⊕ B asgraded vector spaces, and E an A -module. Note that the Atiyah cocycle α E is a degree 2 element in O ( A ) ⊗ B ∨ ⊗ End( E ) . If we set B = B [ − , then the associated Atiyah operators are of degree : α B ( − ) − : ( O ( A ) ⊗ B ) ⊗ O ( A ) ( O ( A ) ⊗ B ) → O ( A ) ⊗ B ; α E ( − ) − : ( O ( A ) ⊗ B ) ⊗ O ( A ) ( O ( A ) ⊗ E ) → O ( A ) ⊗ E. Here are the main results in this section:
Theorem 3.5.
Let ( L, A ) be an SH Lie pair with the quotient space L/A = B . Then the graded vector space H • CE ( A, B ) with the binary operation induced by the Atiyah operator α B is a Lie algebra. Furthermore, if E is an A -module, then H • CE ( A, E ) is a Lie algebra module over H • CE ( A, B ) , with the action induced bythe Atiyah operator α E .In particular, H ( A, B ) is an ordinary Lie algebra and H ( A, E ) is an ordinary Lie algebra module over H ( A, B ) . Remark . By Theorem-Definition 2.11, the Lie algebra and Lie algebra module structures on the coho-mology level are all canonical, i.e., they do not depend on the choice of the splitting L = A ⊕ B .To proceed the proof, we need some preparations. Let τ : B ∨ ⊗ B ∨ → B ∨ ⊗ B ∨ , ξ ⊗ η ( − | ξ || η | η ⊗ ξ. Lemma 3.7.
The symmetrization of the Atiyah cocycle α B ∨ vanishes up to homotopy, i.e., (1 ⊗ τ ) α B ∨ + α B ∨ = ∂ A P, (3.8) for some P ∈ O ( A ) ⊗ B ∨ ⊗ B ∨ ⊗ B .Proof. First, observe the following commutative diagram B ∨ ⊗ B ∨ (1+ τ ) / / s ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ B ∨ ⊗ B ∨ s (cid:15) (cid:15) S ( B ∨ ) , s − O O where the operations s and s − are defined by s : ξ ⊗ η ξ ⊙ η, s − : ξ ⊙ η
12 (1 + τ )( ξ ⊗ η ) , ∀ ξ, η ∈ B ∨ . It is clear that s − is right inverse of the symmetrization operator s , i.e., s ◦ s − = id : S ( B ∨ ) → S ( B ∨ ) .We introduce δ : O ( A ) ⊗ B ∨ → O ( A ) ⊗ S ( B ∨ ) , δ = (1 ⊗ s ) ◦ δ. Then we get the following commutative diagram B ∨ ( α B ∨ +(1 ⊗ τ ) α B ∨ ) / / δ ◦ D ⊥ * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ O ( A ) ⊗ ( B ∨ ⊗ B ∨ ) ⊗ s (cid:15) (cid:15) O ( A ) ⊗ S ( B ∨ ) . ⊗ s − O O Namely, (1 ⊗ s ) ◦
12 ( α B ∨ + (1 ⊗ τ ) α B ∨ ) = δ ◦ D ⊥ . In fact, it amounts to check α B ∨ = δ ◦ D ⊥ , as a map B ∨ → O ( A ) ⊗ B ∨ ⊗ B ∨ , which follows from Lemma 3.3 and ∂ B ∨ A | B ∨ = D ⊥ . Hence (1 ⊗ τ ) α B ∨ + α B ∨ = 2(1 ⊗ s − ⊗ δ ◦ D ⊥ ) . To prove Equation (3.8), it suffices to show that δ ◦ D ⊥ ∈ Hom( B ∨ , O ( A ) ⊗ S ( B ∨ )) ∼ = O ( A ) ⊗ S ( B ∨ ) ⊗ B is a coboundary.Restricting the condition Q L = 0 on the B ∨ to O ( A ) ⊗ S ( B ∨ ) -part, we get δ ◦ D ⊥ + Q A ◦ T + D ⊥ ◦ T + T ◦ D ⊥ = 0 , where T ∈ O ( A ) ⊗ S ( B ∨ ) ⊗ B is defined in Equation (3.2). Hence, we have δ ◦ D ⊥ = − ( Q A + D ⊥ ) ◦ T − T ◦ D ⊥ = − [ Q A + D ⊥ , T ] = − ∂ A T , as desired. (cid:3) TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 23
Lemma 3.9.
For all x, y ∈ O ( A ) ⊗ B, r ∈ O ( A ) ⊗ E , we have α E ( x )( α E ( y ) r ) − ( − | x || y | α E ( y )( α E ( x ) r ) = α E ( α B ( x ) y ) r + ( ∂ A T ) x ( x ⊗ O ( A ) y ⊗ O ( A ) r ) , where T = δ ( α E ) ∈ O ( A ) ⊗ B ∨ ⊗ B ∨ ⊗ End( E ) .Proof. Since ∂ A ( α E ) = 0 , it follows from Lemma 3.3 that ∂ A T = ∂ A ( δ ( α E )) = ( ∂ A ◦ δ + δ ◦ ∂ A )( α E ) = α B ∨ ⊗ End( E ) ( α E ) = α Hom( B ⊗ E,E ) ( α E ) . Applying Lemma 3.4, we have α Hom( B ⊗ E,E ) ( α E ) x ( x ⊗ O ( A ) y ⊗ O ( A ) r ) = (cid:0) α Hom( B ⊗ E,E ) ( x )( α E ) (cid:1) x ( y ⊗ O ( A ) r )= α E ( x )( α E ( y ⊗ O ( A ) r )) − ( − | x | α E ( α B ⊗ E ( x )( y ⊗ O ( A ) r ))= α E ( x )( α E ( y ) r ) − α E (( α B ( x ) y ) ⊗ O ( A ) r + ( − | x || y | y ⊗ O ( A ) ( α E ( x ) r ))= α E ( x )( α E ( y ) r ) − α E ( α B ( x ) y ) r − ( − | x || y | α E ( y )( α E ( x ) r ) . (cid:3) We are now ready to turn to
Proof of Theorem 3.5.
Lemma 3.7 implies that the bracket [ x, y ] = α B ( x ) y is skew-symmetric on thecohomology level. When E is taken as B in Lemma 3.9, we see that the [ − , − ] -bracket satisfies Jacobiidentity on the cohomology level. Again by Lemma 3.9, the operation x ⊲ r = α E ( x ) r defines a Lie algebraaction of H • CE ( A, B ) on H • CE ( A, E ) . (cid:3) We remark that similar results appeared in [12] for that of Lie pairs, and in [7] of relative Lie algebroids. Inthe meantime, the work related to some facts in the derived categories claimed in [11] is still going on.
Example 3.10.
Take L = A ⊕ B , where A is spanned by three vectors a , a and c , B by one vector b . Thedegrees are assigned: | a | = | a | = | b | = − , | c | = 0 . Let the dual vectors be a ∨ i , c ∨ and b ∨ , with degrees | a ∨ | = | a ∨ | = | b ∨ | = 1 , | c ∨ | = 0 . The Q -structure on L is the sum of three parts Q L = Q A + D ⊥ + δ. Here Q A is determined by Q A ( a ∨ ) = Q A ( c ∨ ) = 0 , Q A ( a ∨ ) = − a ∨ ⊙ a ∨ . The A -module structure on B ∨ is given by D ⊥ ( b ∨ ) = a ∨ ⊗ b ∨ . The δ -operator is given by δ ( ξ ) = ( − | ξ | ∆( ξ ) ⊗ b ∨ , ∀ ξ ∈ O ( A ) , where ∆ is a degree derivation on O ( A ) determined by ∆( a ∨ ) = a ∨ , ∆( a ∨ ) = a ∨ ⊙ c ∨ , ∆( c ∨ ) = 0 . It follow from some direct computations that ( L, A ) is an SH Lie pair. The Atiyah operator α B ∨ now reads α B ∨ ( b ∨ ) = δ ◦ D ⊥ ( b ∨ ) = δ ( a ∨ ⊗ b ∨ ) = − ∆( a ∨ ) ⊗ b ∨ ⊗ b ∨ = − a ∨ ⊗ b ∨ ⊗ b ∨ . Or, the Atiyah cocycle is spelled as α B = − a ∨ ⊗ b ∨ ⊗ b ∨ ⊗ b. The Atiyah class [ α B ] = 0 . In fact, any attempt to make [ α B ] = 0 yields the equation Q A ( ξ ) + ξ ⊙ a ∨ = − a ∨ , for ξ ∈ O ( A ) with | ξ | = 0 . It obviously has no solution.The space O ( A ) ⊗ B is generated by one element b [ − , and the Lie bracket on H • ( O ( A ) ⊗ B ) can beexplicitly expressed: (cid:2) b [ − , b [ − (cid:3) = α B ( b [ − b [ −
2] = a ∨ ⊗ b [ − . Atiyah functors.
Let A be an L ∞ [1] -algebra. Then taking the Chevalley-Eilenberg cohomology H • CE ( A, − ) defines a functor H • CE ( A ; − ) : Mod A → GVS K , E H • CE ( A, E ) = H • ( O ( A ) ⊗ E, ∂ EA ) . For a morphism φ ∈ Hom A ( E, F ) , the functor sends φ to [ φ ] : H • CE ( A, E ) → H • CE ( A, F ) .Recall that given an SH Lie pair ( L, A ) with quotient space B , we get a Lie algebra object B = H • CE ( A, B ) whose Lie bracket is induced by the Atiyah operator α B (Theorem 3.5).Let Mod B denote the category of B -modules. According to Theorem 3.5 again, we are able to introduce Definition 3.11.
The Atiyah functor is A : ( E, ∂ EA ) → (cid:0) H • CE ( A, E ) , α E (cid:1) from the category Mod A of A -modules to the category Mod B of B -modules. And for all φ ∈ Hom A ( E, F ) , we have A ( φ ) = [ φ ] : H • CE ( A, E ) → H • CE ( A, F ) . That A is well-defined relies on the following fact: given any φ ∈ Hom A ( E, F ) , the associated [ φ ] preservesthe B -actions, i.e., for all x ∈ O ( A ) ⊗ B , r ∈ O ( A ) ⊗ E , α F ( x )( φ ( r )) = φ ( α E ( x ) r ) + ( ∂ A W )( x ⊗ O ( A ) r ) , for some W ∈ O ( A ) ⊗ Hom( B ⊗ E, F ) . In fact, we have W = δ ( φ ) . The proof of this fact is similar tothat of Lemma 3.9, and thus omitted. Remark . Inspired by Lemma 3.4, we may expect the Atiyah functor to enjoy the following naturalproperties: A ( E ⊗ F ) ∼ = A ( E ) ⊗ H A ( F ) , A ( E ∨ ) ∼ = Hom H ( A ( E ) , H ) , and A (Hom( E, F )) ∼ = Hom H ( A ( E ) , A ( F )) . However, some condition is needed to fulfill these isomorphisms. Further investigations of this question willbe dealt with somewhere else.4. I
NVARIANCE OF A TIYAH CLASSES UNDER INFINITESIMAL DEFORMATIONS
In this section, let us fix an SH Lie pair ( L, A ; Q L ∼ λ • ) with the quotient space B = L/A , and an A -module E . We study infinitesimal deformations of the L ∞ [1] -structure Q L on L , and how the associatedAtiyah classes [ α E ] would be affected. TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 25
Compatible infinitesimal deformations.
In what follows, ~ denotes a square zero formal parameter.An infinitesimal deformation, or a first order deformation, of the L ∞ [1] -algebra structure on L , namely thatof Q L , is a differential of the form Q ( ~ ) = Q L + ~ Q + : O ( L )[ ~ ] → O ( L )[ ~ ] . Here Q + is a degree derivation of O ( L ) , and both Q L and Q + are K [ ~ ] -linear. It follows that [ Q L , Q + ] = Q L ◦ Q + + Q + ◦ Q L = 0 . In this circumstance, L [ ~ ] has an L ∞ [1] -structure Q ( ~ ) which is deformed from Q L .As our motivation is to regard L as a larger object extended from A , we only consider deformations of thefollowing type: Definition 4.1.
An infinitesimal deformation Q ( ~ ) of Q L is said to be A -compatible, if it is subject to thefollowing two conditions:(1) The L ∞ [1] -structure on A is not deformed, i.e., the following diagram commutes: O ( L )[ ~ ] j ∨ / / Q ( ~ ) (cid:15) (cid:15) O ( A )[ ~ ] Q A (cid:15) (cid:15) O ( L )[ ~ ] j ∨ / / O ( A )[ ~ ] . (2) The A -module structure on B is not deformed. This means the commutativity of A ⊥ [ ~ ] = B ∨ [ ~ ] D ⊥ / / (cid:127) _ (cid:15) (cid:15) O ( A ) ⊗ B ∨ [ ~ ] (cid:127) _ (cid:15) (cid:15) L ∨ [ ~ ] Q ( ~ ) / / O ( L )[ ~ ] J / / O ( A ) ⊗ L ∨ [ ~ ] . By choosing a splitting of Sequence (2.3), so that L ∼ = A ⊕ B and that O ( L ) is identified with O ( A ) ⊗O ( B ) ,the two compatible conditions are unraveled: if Q ( ~ ) = Q L + ~ Q + , then the aboveCondition (1) ⇔ Q + ( A ∨ ) ⊂ O ( A ) ⊗ O + ( B ) , Condition (2) ⇔ Q + ( B ∨ ) ⊂ O ( A ) ⊗ b S ≥ ( B ∨ ) . Similar to the decomposition of Q L in Equation (3.2), we denote the part of Q + that sends A ∨ into O ( A ) ⊗ B ∨ by δ + , the part that sends A ∨ into O ( A ) ⊗ b S ≥ ( B ∨ ) by R + , and the part that sends B ∨ into O ( A ) ⊗ S i ( B ∨ ) by T i + , i ≥ . Then the A -compatible infinitesimal deformation Q ( ~ ) has the form Q ( ~ ) = Q L + ~ δ + + ~ R + + ~ X i ≥ T i + . (4.2) Definition 4.3. (1) A gauge equivalence of O ( L )[ ~ ] is an automorphism σ = 1+ ~ λ of the graded com-mutative algebra O ( L )[ ~ ] , where λ : O ( L ) → O ( L ) is K -linear, such that the following diagramcommutes: O ( B )[ ~ ] / / (cid:15) (cid:15) O ( L )[ ~ ] j ∨ / / σ =1+ ~ λ (cid:15) (cid:15) O ( A )[ ~ ] (cid:15) (cid:15) O ( B )[ ~ ] / / O ( L )[ ~ ] j ∨ / / O ( A )[ ~ ] . (4.4) (2) Two A -compatible infinitesimal deformations Q ( ~ ) and ¯ Q ( ~ ) of Q L are said to be gauge equivalentif there exists a gauge equivalence σ = 1 + ~ λ such that the following diagram commutes: O ( L )[ ~ ] σ =1+ ~ λ / / Q ( ~ ) (cid:15) (cid:15) O ( L )[ ~ ] ¯ Q ( ~ ) (cid:15) (cid:15) O ( L )[ ~ ] σ =1+ ~ λ / / O ( L )[ ~ ] , i.e., σ is an isomorphism of L ∞ [1] -algebras ( L [ ~ ] , Q [ ~ ]) ∼ = ( L [ ~ ] , ¯ Q [ ~ ]) .Assume that Q ( ~ ) and ¯ Q ( ~ ) are connected by the gauge equivalence σ = 1 + ~ λ . Since σ is an algebraautomorphism, it follows that λ is a degree derivation of O ( L ) . Note that σ − = 1 − ~ λ . It follows froma simple computation that Q ( ~ ) − ¯ Q ( ~ ) = ~ [ Q L , λ ] . Recall that O ( L ) = O ( A ) ⊗ O ( B ) . The commutative property of Diagram (4.4) implies that we can write λ = X k ≥ Ψ k , where Ψ k : A ∨ → O ( A ) ⊗ S k ( B ∨ ) . (4.5)All these Ψ k are treated as degree derivations of O ( L ) which act trivially on B ∨ .4.2. Gauge invariance of Atiyah classes.
Let Q ( ~ ) = Q L + ~ Q + be an A -compatible infinitesimal defor-mation of Q L . Consider the associated Atiyah cocycle α E [ ~ ] of the SH Lie pair ( L [ ~ ] , A [ ~ ]) with respect tothe A [ ~ ] -module E [ ~ ] . By Lemma 3.3, α E [ ~ ] = [ ∂ A , δ + ~ δ + ] = α E + ~ [ ∂ A , δ + ] , where δ and δ + are the components of Q L and Q + specified respectively in Equations (3.2) and (4.2).The main result in this section is the gauge invariance of the Atiyah class [ α E [ ~ ] ] . Theorem 4.6.
Let Q ( ~ ) = Q L + ~ Q + and ¯ Q ( ~ ) = Q L + ~ ¯ Q + be two gauge equivalent A -compatibleinfinitesimal deformations of Q L . Then the associated Atiyah classes coincide: [ α E [ ~ ] ] = [ α E [ ~ ] ] ∈ H ( A [ ~ ] , ( B [ ~ ]) ∨ ⊗ End( E [ ~ ])) ∼ = H ( A, B ∨ ⊗ End( E ))[ ~ ] . Proof.
Let the gauge equivalence σ be as in Definition 4.3. The λ operator is defined in Equation (4.5).Further assume that Q ( ~ ) = Q A + δ + D ⊥ + R + X j ≥ T j + ~ δ + + ~ R + + ~ X i ≥ T i + , ¯ Q ( ~ ) = Q A + δ + D ⊥ + R + X j ≥ T j + ~ ¯ δ + + ~ ¯ R + + ~ X i ≥ ¯ T i + , are explained as earlier. Applying the equation σ ◦ Q ( ~ ) = ¯ Q ( ~ ) ◦ σ : O ( L )[ ~ ] → O ( L )[ ~ ] to an element ξ ∈ A ∨ , we have σ ( Q ( ~ )( ξ )) = ~ X k Ψ k ! ( Q A ( ξ ) + δ ( ξ ) + R ( ξ ) + ~ δ + ( ξ ) + ~ R + ( ξ ))= Q A ( ξ ) + δ ( ξ ) + R ( ξ ) + ~ δ + ( ξ ) + ~ R + ( ξ ) + ~ X k Ψ k ( Q A ( ξ ) + δ ( ξ ) + R ( ξ )) , TIYAH CLASSES OF STRONGLY HOMOTOPY LIE PAIRS 27 and ¯ Q ( ~ )( σ ( ξ )) = Q A + δ + D ⊥ + R + X j ≥ T j + ~ ¯ δ + + ~ ¯ R + + ~ X i ≥ ¯ T i + ξ + ~ X k Ψ k ( ξ ) ! = Q A ( ξ ) + δ ( ξ ) + R ( ξ ) + ~ ¯ δ + ( ξ ) + ~ ¯ R + ( ξ ) + ~ X k Q A + δ + D ⊥ + R + X j ≥ T j (Ψ k ( ξ )) . Comparing the ~ O ( A ) ⊗ B ∨ -component of both sides, one gets Ψ ( Q A ( ξ )) + δ + ( ξ ) = ¯ δ + ( ξ ) + Q A (Ψ ( ξ )) + D ⊥ (Ψ ( ξ )) , which implies that δ + − ¯ δ + = Q A ◦ Ψ − Ψ ◦ Q A + D ⊥ ◦ Ψ : A ∨ → O ( A ) ⊗ B ∨ . Hence, we have, for all e ∈ E , α E [ ~ ] ( e ) − α E [ ~ ] ( e ) = ~ [ ∂ EA , δ + − ¯ δ + ]( e ) = ~ ( δ + − ¯ δ + )( ∂ EA ( e ))= ~ ( Q A ◦ Ψ − Ψ ◦ Q A + D ⊥ ◦ Ψ ) (cid:0) D E ( e ) (cid:1) = ~ ( Q A ◦ Ψ ◦ D E + Ψ ◦ D E ◦ D E + D ⊥ ◦ Ψ ◦ D E )( e ) . ( by Equation (1.27) ) Here D E : E → O ( A ) ⊗ E defines the A -module structure on E .Now let W = [Ψ , D E ] : O ( A ) ⊗ E → O ( A ) ⊗ B ∨ ⊗ E be the graded commutator of Ψ and D E . One easily finds it is an O ( A ) -linear map.Then we have α E [ ~ ] ( e ) − α E [ ~ ] ( e ) = ~ (( Q A + D ⊥ + D E )((Ψ ◦ D E )( e )) + ( W ◦ D E )( e ))= ~ ( ∂ A ◦ W + W ◦ ∂ A )( e ) = ~ ∂ A W ( e ) , which implies that α E [ ~ ] − α E [ ~ ] = ~ ∂ A W. This completes the proof. (cid:3)
5. A
PPENDIX : M
ORPHISMS OF
SH L
IE ALGEBRAS
We prove the equivalence of the two definitions of morphisms of SH Lie algebras as in Definition 1.14.Let φ : O ( L ′ ) → O ( L ) be a morphism of K -algebras such that φ ◦ Q L ′ = Q L ◦ φ : O ( L ′ ) → O ( L ) . Assume that φ = P k ≥ φ k , where φ k : ( L ′ ) ∨ → S k ( L ∨ ) . Define a family of degree zero linear maps f k = ( − k +1 φ ∨ k : S k ( L ) → L ′ , k ≥ . We show that { f k } satisfies the two requirements as in Definition 1.14.By assumption, we have φ ◦ Q L ′ ( ξ ) = Q L ◦ φ ( ξ ) ∈ O ( L ) (5.1)for all homogeneous ξ ∈ ( L ′ ) ∨ . Note thatLHS of Equation (5.1) = φ (cid:0) h λ ′ , ξ i + Q ′ ( ξ ) + · · · (cid:1) = h λ ′ , ξ i + X n ≥ h ( φ ) ⊙ n , Q ′ n ( ξ ) i + · · · ; and RHS of Equation (5.1) = Q L ( h φ , ξ i + φ ( ξ ) + · · · ) = Q ( φ ( ξ )) + · · · = h λ , φ ( ξ ) i + · · · . Comparing the K = S ( L ∨ ) -component of both sides, one gets h λ ′ , ξ i + X n ≥ h φ ⊙ n , Q ′ n ( ξ ) i = h λ , φ ( ξ ) i , which is equivalent to h λ ′ + X n ≥ n ! ( − n λ ′ n ( φ , · · · , φ ) , ξ i = h λ , φ ( ξ ) i , or Equation (1.16).We further investigate the S n ( L ∨ )( n ≥ -component of Equation (5.1). For all u i ∈ L, i = 1 , · · · , n , wehave LHS = h φ ◦ Q L ′ ( ξ ) , u ⊙ · · · ⊙ u n i = h Q L ′ ( ξ ) , X i , ··· ,i r ≥ i + ··· + i r = n X τ ∈ sh( i , ··· ,i r ) X j ≥ ǫ ( τ )( − n + r + j r ! f ⊙ j ⊙ f i ⊙ · · · ⊙ f i r ( u τ (1) , · · · , u τ ( n ) ) i = h ξ, X i , ··· ,i r ≥ i + ··· + i r = n X τ ∈ sh( i , ··· ,i r ) X j ≥ ǫ ( τ )( − | ξ | + n r + j )! λ ′ r + j ( f , · · · , f , f i ( · · · ) , · · · , f i r ( · · · )) i and RHS = h Q L ◦ φ ( ξ ) , u ⊙ · · · ⊙ u n i = h φ ( ξ ) , X k,l ≥ k + l = n X σ ∈ sh( l,k ) ǫ ( σ )( − | ξ | + l λ l ( u σ (1) , · · · , u σ ( l ) ) ⊙ · · · ⊙ u σ ( n ) i = h ξ, X k,l ≥ k + l = n X σ ∈ sh( l,k ) ǫ ( σ )( − | ξ | + n f k +1 ( λ l ( u σ (1) , · · · , u σ ( l ) ) , · · · , u σ ( n ) ) i . Thus Equation (1.17) also holds once we assume Equation (1.15).The inverse implication “Equations (1.16)+(1.17) = ⇒ Equation (5.1)” is also clear from the previous argu-ment. This completes the proof.
Acknowledgments.
We would like to thank Camille Laurent-Gengoux, Xiaobo Liu and Ping Xu for usefuldiscussions and comments. The authors are indebted to an anonymous referee who provided many sugges-tions which helped us a lot to improve an early version of this paper. Zhuo Chen is also grateful to SheffieldUniversity for its hospitality. Honglei Lang and Maosong Xiang also would like to thank the Departmentof Mathematics at Penn State for its hospitality and China Scholarship Council for the funding during theirvisit at Penn State. R
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