Kapranov's construction of sh Leibniz algebras
aa r X i v : . [ m a t h . QA ] J un KAPRANOV’S CONSTRUCTION OF SH LEIBNIZ ALGEBRAS
ZHUO CHEN, ZHANGJU LIU, AND MAOSONG XIANG
Abstract.
Motivated by Kapranov’s discovery of an sh Lie algebra structure on the tangent complex ofa K¨ahler manifold and Chen-Sti´enon-Xu’s construction of sh Leibniz algebras associated with a Lie pair,we find a general method to construct sh Leibniz algebras. Let A be a commutative dg algebra. Given aderivation of A valued in a dg module Ω, we show that there exist sh Leibniz algebra structures on the dualmodule of Ω. Moreover, we prove that this process establishes a functor from the category of dg modulevalued derivations to the category of sh Leibniz algebras over A . Keywords : sh Leibniz algebra, Atiyah class, commutative dg algebra, dg module.
MSC class : 16E45, 18G55.
Contents
1. Introduction 12. Atiyah classes of commutative dg algebras and their twists 32.1. Atiyah classes of commutative dg algebras 32.2. Dg module valued derivations and twisted Atiyah classes 42.3. Atiyah classes of dg Lie algebroids and Lie pairs 62.4. Functoriality 83. The Kapranov functor 103.1. Leibniz ∞ [1] algebras 103.2. The Kapranov functor 113.3. Applications 174. Open questions and remarks 18References 191. Introduction
Higher homotopies and higher structures are playing important roles in mathematics and some branchesof theoretical physics, such as gauge theory and topological field theory (see Huebschmann [14]). Higherhomotopies, as explained by Huebschmann in [13], often arise from the process of transferring certain strictgeometric or algebraic structure on a huge chain complex to a smaller but chain homotopic complex. Forinstance, an sh Lie algebra (also known as L ∞ -algebra [21]) yields from a dg Lie algebra by applyinghomological perturbation theory [12]. Here and in the sequel, sh is short for strongly homotopy and dg isshort for differential graded . Sh Leibniz algebras, also known as sh Loday algebras, Loday infinity algebras Research partially supported by NSFC grant 11471179. or Leibniz ∞ algebras [1], are also examples of higher structures. In fact, the notion of Leibniz ∞ algebras isa generalization of L ∞ algebras where the skew-symmetricity constraint on multibrackets is discarded.In this note, we use the notion of Leibniz ∞ [1] algebras (see Definition 3.1), which is equivalent to the notionof sh Leibniz algebras, and study a particular method to construct Leibniz ∞ [1] algebras. This method firstappeared in Kapranov’s approach to Rozansky-Witten theory [15]: Given a K¨ahler manifold X , Kapranovdiscovered an L ∞ algebra structure on Ω , •− X ( T X ) via the Atiyah class α X . More precisely, let ∇ be theChern connection on the holomorphic tangent bundle T X . Then the curvature R ∇ ∈ Ω , X (Hom( S ( T X ) , T X ))is a Dolbeault representative of the Atiyah class α X . The L ∞ brackets { λ k } k ≥ on Ω , •− X ( T X ) are definedby • λ = ¯ ∂ . • λ = R ∇ . • λ k +1 = ∇ , ( λ k ) ∈ Ω , X (Hom( S k +1 ( T X ) , T X )), for k = 2 , , , ... .Kapranov’s construction of L ∞ algebras is generalized in Chen, Sti´enon and Xu’s work [8] where the settingis a Lie algebroid pair (Lie pair, for short) ( L, A ). It is shown that the graded vector space Γ( ∧ • A ∨ ⊗ L/A )admits a Leibniz ∞ [1] algebra structure ([8, Theorem 3.13]) via the Atiyah class of the Lie pair ( L, A ). TheAtiyah class of Lie pairs encompasses the original Atiyah class [2] of holomorphic vector bundles and theMolino class [24] of foliations as special cases. This construction of Leibniz ∞ [1] algebra structures is similarto that of Kapranov — First, we choose a splitting j : L/A → L of vector bundles so that L ∼ = A ⊕ L/A .Second, choose an L -connection ∇ on L/A extending the A -module structure. Then the Leibniz ∞ [1] brackets { λ k } k ≥ on Γ( ∧ • A ∨ ⊗ L/A ) are determined as follows: • λ = d CE is the Chevalley-Eilenberg differential of the Bott representation of A on L/A . • Define a bundle map R : L/A ⊗ L/A → A ∨ ⊗ L/A via the Atiyah cocycle α ∇ L/A (see Section 2.3.2): R ( b , b ) = α ∇ L/A ( − , b ) b , ∀ b , b ∈ Γ( L/A ) . The second structure map λ is specified by λ ( ξ ⊗ b , ξ ⊗ b ) = ( − | ξ | + | ξ | ξ ∧ ξ ∧ R ( b , b ) , for all ξ , ξ ∈ Γ( ∧ • A ∨ ) and b , b ∈ Γ( L/A ). • Define a sequence of bundle maps R k : ( L/A ) ⊗ k → A ∨ ⊗ L/A, k ≥ R n +1 = ∇ R n ,i.e., R n +1 ( b ⊗ · · · ⊗ b n ) = R n ( ∇ j ( b ) ( b ⊗ · · · ⊗ b n )) − ∇ j ( b ) R n ( b ⊗ · · · ⊗ b n ) . The k -th structure map is specified by λ k ( ξ ⊗ b , · · · , ξ k ⊗ b k ) = ( − | ξ | + ··· + | ξ k | ξ ∧ · · · ∧ ξ k ∧ R k ( b , · · · , b k ) , for all ξ i ∈ Γ( ∧ • A ∨ ) , b i ∈ Γ( L/A ) , ≤ i ≤ k .We call (Γ( ∧ • A ∨ ⊗ L/A ) , { λ k } k ≥ ) a Kapranov Leibniz ∞ [1] algebra. Its construction needs, a priori , someextra choices (a splitting j and an L -connection ∇ on L/A ). Then one asks a natural question ([8, Remark3.19]) — how does the Leibniz ∞ [1] algebra structure on Γ( ∧ • A ∨ ⊗ L/A ) depend on the choice of splittingdata and connections? The main goal of this note is to answer this question—Kapranov Leibniz ∞ [1] algebrastructures on Γ( ∧ • A ∨ ⊗ L/A ) associated with different choices of j and ∇ , are mutually isomorphic in thecategory of Leibniz ∞ [1] algebras over Γ( ∧ • A ∨ ) (see Theorem 1.3 or Theorem 3.32).We adopt an algebraic approach to achieve this goal. The algebraic notion we need is a dg module valuedderivation of a commutative differential graded algebra (cdga for short) A (see Definition 2.4). As animmediate example from complex geometry, consider a complex manifold X . The Dolbeault dg algebra A = (Ω , • X , ¯ ∂ ) is a cdga. Let Ω = (Ω , • X (( T , X ) ∨ ) , ¯ ∂ ) be the dg A -module generated by the section spaceof holomorphic cotangent bundle ( T , X ) ∨ . Then ∂ : A → Ω is an Ω-valued derivation of A .We now explain how Kapranov’s original method and Chen-Sti´enon-Xu’s construction can be further gen-eralized in the setting of a dg module valued derivation A δ −→ Ω. Consider the dual dg A -module B = Ω ∨ of Ω. First, one chooses a δ -connection ∇ on B , i.e., a map ∇ : B → Ω ⊗ A B that extends the δ -map (seeDefinition 2.9). Then one can define a sequence of degree 1 maps {R ∇ k : ⊗ k B → B} k ≥ as follows: APRANOV’S CONSTRUCTION OF SH LEIBNIZ ALGEBRAS 3 • R ∇ = ∂ A is the differential on B . • R ∇ = At ∇B : B ⊗ A B → B is the twisted Atiyah cocycle (see Definition 2.14). • R ∇ k for k ≥ R ∇ k = ∇R ∇ k − (see Equation (3.9)).Our first result is the following Theorem 1.1.
When endowed with structure maps {R ∇ k } k ≥ , the dg A -module B becomes a Leibniz ∞ [1] A -algebra. Here by saying that B is a Leibniz ∞ [1] A -algebra, we mean that its higher structure maps {R ∇ k } k ≥ areall A -multilinear. We emphasise that the Kapranov Leibniz ∞ [1] algebra ( B , {R ∇ k } k ≥ ) should be treatedas an object in the category of Leibniz ∞ [1] A -algebras. In fact, if we treat ( B , {R ∇ k } k ≥ ) merely as aLeibniz ∞ [1] algebra over K , it is always isomorphic to a trivial one (see Remark 3.21). We call ( B , {R ∇ k } k ≥ )the Kapranov Leibniz ∞ [1] A -algebra associated with the dg module valued derivation A δ −→ Ω and the δ -connection ∇ .Our second result is that the above construction is functorial: Theorem 1.2.
The above construction defines a functor
Kap , called Kapranov functor, from the category ofdg module valued derivations of a cdga A to the category of Leibniz ∞ A -algebras. Moreover, the Kapranovfunctor Kap is homotopy invariant, i.e., if δ and δ ′ are two homotopic derivations of A valued in the samedg module, then Kap( δ ) is isomorphic to Kap( δ ′ ) . Applying Theorem 1.2 to dg module valued derivations arising from Lie pairs, we obtain the answer of ourmotivating question:
Theorem 1.3.
Let ( L, A ) be a Lie pair. The Kapranov Leibniz ∞ [1] algebra structure on the graded vectorspace Γ( ∧ • A ∨ ⊗ L/A ) is unique up to isomorphisms in the category of Leibniz ∞ [1] Γ( ∧ • A ∨ ) -algebras. This note is organized as follows: Section 2 consists of our conventions, notations, and the notion of twistedAtiyah classes. We will see that twisted Atiyah classes encompass Atiyah classes of Lie pairs and dg Liealgebroids as special cases. Section 3 contains a brief summary of sh Leibniz algebras, the construction ofthe Kapranov functor, and its applications. Finally, we present some relevant remarks and open questionsin Section 4.
Acknowledgements.
We would like to thank Bangming Deng, Wei Hong, Kai Jiang, Honglei Lang, CamilleLaurent-Gengoux, Mathieu Sti´enon, Jim Stasheff, Yannick Voglaire, and Ping Xu for useful discussions andcomments. Xiang is grateful to Penn State University, Peking University and Tsinghua University, for theirhospitality during his visits.2.
Atiyah classes of commutative dg algebras and their twists
In [2], Atiyah introduced a cohomology class, which has come to be known as Atiyah class, to characterizethe obstruction to the existence of holomorphic connections on a holomorphic vector bundle. The notion ofAtiyah classes have been developed in the past decades for diverse purposes (see [5–8, 10, 19, 20, 23]). In thissection, we recall Atiyah classes of commutative dg algebras defined by Costello [10] and introduce a versionof twisted Atiyah classes.2.1.
Atiyah classes of commutative dg algebras.
Throughout this paper, K denotes a field of charac-teristic zero and graded means Z -graded. A commutative differential graded algebra (cdga for short) over K is a pair ( A , d A ), where A is a commutative graded K -algebra, and d A : A → A , usually called thedifferential, is a homogeneous degree one derivation of square zero. We also write A for a cdga withoutmaking its differential explicitly.An A -module is a representation of the underlying commutative graded algebra of A by forgetting thedifferential d A . By a dg A -module, we mean an A -module E , together with a degree one and square zero ZHUO CHEN, ZHANGJU LIU, AND MAOSONG XIANG endomorphism ∂ EA of the graded K -vector space E , called the differential, such that ∂ EA ( ae ) = ( d A a ) e + ( − | a | a∂ EA ( e ) , for all a ∈ A , e ∈ E . To work with various different dg A -modules, the differential ∂ EA of any dg A -module E will be denoted by the same notation ∂ A . A dg A -module ( E , ∂ A ) will also be simply denoted by E .The dg A -module of K¨ahler differentials is the graded A -moduleΩ A | K = span { d dR a : a ∈ A } / { d dR ( ab ) − ( d dR a ) b − ( − | a | ad dR b : a, b ∈ A } , together with the differential ∂ A such that the algebraic de Rham operator d dR : A → Ω A | K is a cochainmap, i.e., ∂ A ( d dR a ) = d dR ( d A a ) for all a ∈ A . In the sequel, we assume that Ω A | K is projective as an A -module.A degree r morphism of dg A -modules, denoted by α ∈ Hom r dg A ( E , F ), is a degree r A -module morphism α : E → F , which is also compatible with differentials: ∂ A ( α ) := ∂ A ◦ α − ( − r α ◦ ∂ A = 0 : E → F . Definition 2.1 (Costello [10]) . Let A be a cdga and E an A -module.(1) A connection on E is a (degree ) map of graded K -vector spaces H : E → Ω A | K ⊗ A E , satisfying the Leibniz rule H ( ae ) = ( d dR a ) ⊗ e + a H ( e ) , ∀ a ∈ A , e ∈ E . (2) Assume that ( E , ∂ A ) is a dg A -module. Given a connection H on E , At H E := [ H , ∂ A ] = H ◦ ∂ A − ∂ A ◦ H ∈ Ω A | K ⊗ A End A ( E ) is a closed element of degree which measures the failure of H to be a cochain map. Its cohomologyclass At E ∈ H ( A , Ω A | K ⊗ A End A ( E )) is independent of the choice of connections, and is calledthe Atiyah class of the dg A -module E . The existence of connections on E is guaranteed if E is a projective A -module. Hence we make the following Convention 2.2.
In this note, all A -modules are assumed to be projective. Example 2.3 (Mehta-St´enon-Xu [23]) . Let ( M , Q M ) be a smooth dg manifold, where M = ( M, O M ) is asmooth Z -graded manifold, and Q M is a homological vector field on M . Then A = ( C ∞ ( M ) , Q M ) is a cdga.For each dg vector bundle ( E , Q E ) over ( M , Q M ) , its space of sections E = (Γ( E ) , Q E ) is a dg A -module.The Atiyah class At E of E coincides, up to a minus sign, with the Atiyah class At E of the dg vector bundle E with respect to the dg Lie algebroid T M defined by Mehta-St´enon and Xu. This is a particular instanceof Atiyah classes of dg vector bundles with respect to a general dg Lie algebroid (see Section 2.3). Dg module valued derivations and twisted Atiyah classes.
A key notion in this note is dgmodule valued derivation (dg derivation for short):
Definition 2.4.
Let ( A , d A ) be a cdga and (Ω , ∂ A ) a dg A -module. • A dg derivation of A valued in (Ω , ∂ A ) is a degree derivation δ : A → Ω of the commutativegraded algebra A valued in the A -module Ω , δ ( ab ) = δ ( a ) b + aδ ( b ) , ∀ a, b ∈ A , which commutes with the differentials as well: δ ◦ d A = ∂ A ◦ δ : A → Ω . (2.5) Such a dg derivation is simply denoted by A δ −→ Ω . • Let δ and δ ′ be two (Ω , ∂ A ) -valued dg derivations of A . They are said to be homotopic, written as δ ∼ δ ′ , if there exists a degree ( − derivation h of A valued in the A -module Ω such that δ ′ − δ = [ ∂ A , h ] = ∂ A ◦ h + h ◦ d A : A → Ω . APRANOV’S CONSTRUCTION OF SH LEIBNIZ ALGEBRAS 5
An immediate example of dg derivations is Ω , • X ∂ −→ Ω , • X (( T , X ) ∨ ) arising from a complex manifold X , whichhas already been explained in Section 1. Another fundamental example is the dg derivation A d dR −−→ Ω A | K ,which is universal in the following sense: For any generic dg derivation A δ −→ Ω, there exists a unique dg A -module morphism ¯ δ : Ω A | K → Ω such that the following diagram commutes: A d dR (cid:15) (cid:15) δ / / ΩΩ A | K . ∃ ! ¯ δ = = ③③③③③③③③ Thus, ¯ δ ⊗ A id End A ( E ) : Ω A | K ⊗ A End A ( E ) → Ω ⊗ A End A ( E ) (2.6)is a dg A -module morphism as well. Definition 2.7 (Twisted Atiyah class) . Let E be a dg A -module and A δ −→ Ω a dg derivation. The dg A -module morphism ¯ δ ⊗ A id End A ( E ) in Equation (2.6) sends the Atiyah class At E ∈ H ( A , Ω A | K ⊗ A End A ( E )) of E to a cohomology class At δ E ∈ H ( A , Ω ⊗ A End A ( E )) , which is called the δ -twisted Atiyah class of E . It follows immediately that twisted Atiyah classes are homotopic invariant:
Proposition 2.8. If δ ∼ δ ′ , then for any dg A -module E , At δ E = At δ ′ E ∈ H ( A , Ω ⊗ A End A ( E )) . Below we give a different characterization of the twisted Atiyah class At δ E . We need another key notion inthis note — δ -connections, which can be thought of as operations extending δ . Definition 2.9.
Let A δ −→ Ω be a dg derivation and E an A -module. A δ -connection on E is a degree , K -linear map of graded K -vector spaces ∇ : E → Ω ⊗ A E satisfying the following Leibniz rule: ∇ ( ae ) = δ ( a ) ⊗ e + a ∇ ( e ) , ∀ a ∈ A , e ∈ E . (2.10) Remark 2.11.
A connection H as in Definition 2.1 induces a δ -connection ∇ as in Definition 2.9 via thefollowing triangle E H / / ∇ $ $ ■■■■■■■■■■■ Ω A | K ⊗ A E ¯ δ ⊗ A id E (cid:15) (cid:15) Ω ⊗ A E . (2.12) It follows that δ -connections always exist on projective A -modules. However, δ -connections do not necessarilyarise in this manner. Proposition 2.13.
Let E = ( E , ∂ A ) be a dg A -module.1) For any δ -connection ∇ on E , the degree element At ∇ E := [ ∇ , ∂ A ] = ∇ ◦ ∂ A − ∂ A ◦ ∇ ∈ Ω ⊗ A End A ( E ) is a cocycle.2) The cohomology class [At ∇ E ] ∈ H ( A , Ω ⊗ A End A ( E )) coincides with the δ -twisted Atiyah class At δ E of E . ZHUO CHEN, ZHANGJU LIU, AND MAOSONG XIANG
Proof.
The first statement is clear. It only suffices to prove the second one: Observe that the differenceof two δ -connections is a degree zero element in Ω ⊗ A End A ( E ). Hence, the cohomology class [At ∇ E ] isindependent of the choice of δ -connections. We choose a particular δ -connection ∇ induced by a connection H on E as in the commutative triangle (2.12).Since the map ¯ δ ⊗ A id End A ( E ) defined in Equation (2.6) is a dg A -module morphism, it follows thatAt ∇ E = [ ∇ , ∂ A ] = [(¯ δ ⊗ A id End A ( E ) ) ◦ H , ∂ A ] = (¯ δ ⊗ A id End A ( E ) )[ H , ∂ A ] = (¯ δ ⊗ A id End A ( E ) )(At H E ) . Passing to the cohomology, we have[At ∇ E ] = (¯ δ ⊗ A id End A ( E ) )(At E ) = At δ E . (cid:3) Definition 2.14.
We call At ∇ E the δ -twisted Atiyah cocycle of E with respect to the δ -connection ∇ on E . Denote the A -dual Ω ∨ of Ω by B , which is also a dg A -module. Given a δ -connection ∇ : E → Ω ⊗ A E ofan A -module E , the covariant derivation along b ∈ B is ∇ b : E → E , ∇ b ( e ) := ι b ∇ ( e ) , ∀ e ∈ E . The δ -twisted Atiyah cocycle At ∇ E could be viewed as a degree 1 element in Hom A ( B ⊗ A E , E ) by settingAt ∇ E ( b, e ) = ( − | b | ι b At ∇ E ( e ) = ( − | b | ι b ( ∇ ( ∂ A ( e )) − ∂ A ( ∇ ( e )))= − ∂ A ( ∇ b e ) + ∇ ∂ A ( b ) e + ( − | b | ∇ b ∂ A ( e )= ∇ ∂ A ( b ) e − [ ∂ A , ∇ b ]( e ) , (2.15)for all b ∈ B and e ∈ E . Moreover, as At ∇ E is a 1-cocycle, it is a morphism of dg A -modules, i.e., At ∇ E ∈ Hom A ( B ⊗ A E , E ).As an immediate consequence of Proposition 2.13 and Equation (2.15), we have the following Proposition 2.16.
Let A δ −→ Ω be a dg derivation and E a dg A -module. Then the δ -twisted Atiyahclass At δ E vanishes if and only if there exists a δ -connection ∇ on E such that the associated twisted Atiyahcocycle At ∇ E vanishes, i.e., the map ∇ : E → Ω ⊗ A E is compatible with the differentials. In this case, forall ∂ A -closed elements b ∈ B and e ∈ E , ∇ b e is also ∂ A -closed. Atiyah classes of dg Lie algebroids and Lie pairs.
In this section, we briefly recall Atiyah classesof dg vector bundles with respect to a dg Lie algebroid defined in [23] and Atiyah classes of Lie pairs definedin [8] (see [9] for the equivalence between the two types of Atiyah classes arising from integrable distributions),and show that both of them can be viewed as twisted Atiyah classes.2.3.1.
Dg Lie algebroids.
A dg Lie algebroid can be thought of as a Lie algebroid object in the category ofsmooth dg manifolds. The precise description is as follows.
Definition 2.17.
A dg Lie algebroid over a dg manifold ( M , Q M ) is a quadruple ( L , Q L , ρ L , [ − , − ] L ) , where1) ( L , Q L ) is a dg vector bundle over ( M , Q M ) ;2) ( L , ρ L , [ − , − ] L ) is a graded Lie algebroid over M ;3) The anchor map ρ L : ( L , Q L ) → ( T M , L Q M ) is a morphism of dg vector bundles;4) Q L : Γ( L ) → Γ( L ) is a derivation with respect to the bracket [ − , − ] L , i.e., Q L ([ X, Y ] L ) = [ Q L ( X ) , Y ] L + ( − | X | [ X, Q L ( Y )] L , ∀ X, Y ∈ Γ( L ) . Given a dg Lie algebroid ( L , Q L , ρ L , [ − , − ] L ) and a dg vector bundle ( E , Q E ) over ( M , Q M ), Mehta, Sti´enonand Xu constructed the Atiyah class At E of E with respect to L as follows: Choose a Lie algebroid L -connection ∇ E on the vector bundle E , i.e., a degree 0 K -bilinear map ∇ E : Γ( L ) × Γ( E ) → Γ( E ) APRANOV’S CONSTRUCTION OF SH LEIBNIZ ALGEBRAS 7 subject to the relations ∇ E fX e = f ∇ E X e, ∇ E X ( f e ) = ( ρ L ( X ) f ) e + ( − | f || X | f ∇ E X e, for all f ∈ C ∞ ( M ) , X ∈ Γ( L ) and e ∈ Γ( E ). There associates a degree 1 cocycle At ∇ E E ∈ Γ( L ∨ ⊗ End( E ))defined by At ∇ E E ( X, e ) = Q E ( ∇ E X e ) − ∇ E Q L ( X ) e − ( − | X | ∇ E X ( Q E e ) , ∀ X ∈ Γ( L ) , e ∈ Γ( E ) . (2.18)Its cohomology class At E ∈ H (Γ( L ∨ ⊗ End( E ))), which is independent of the choice of L -connections, iscalled the Atiyah class of the dg vector bundle E with respect to the dg Lie algebroid L [23].Meanwhile, there associates a (Γ( L ∨ ) , Q L ∨ )-valued derivation of the cdga ( C ∞ ( M ) , Q M ) defined by δ L : C ∞ ( M ) d dR −−→ Ω ( M ) ρ ∨L −−→ Γ( L ∨ ) , (2.19)where Q L ∨ is induced from the differential Q L on L . The fact that δ L commutes with the two differentials Q M and Q L ∨ follows from (3) of Definition 2.17. The section space of a dg vector bundle E gives rise to a dg( C ∞ ( M ) , Q M )-module E := (Γ( E ) , Q E ). It is obvious that a δ L -connection ∇ δ L on E = Γ( E ) is equivalentlyto a Lie algebroid L -connection ∇ L on the graded vector bundle E . Comparing Equations (2.15) and (2.18),we have the following Proposition 2.20.
The Atiyah class At E of the dg vector bundle E with respect to the dg Lie algebroid L coincides, up to a minus sign, with the twisted Atiyah class At δ L E of the dg ( C ∞ ( M ) , Q M ) -module E =(Γ( E ) , Q E ) , where the dg derivation δ L is given by Equation (2.19) . Lie pairs.
By a Lie pair (
L, A ), we mean two Lie algebroids L and A over the same smooth manifold M such that A ⊂ L is a Lie subalgebroid. The quotient bundle B = L/A carries a natural flat (Lie algebroid) A -connection, called the Bott A -module structure.Let us recall the Atiyah class of the Lie pair ( L, A ) defined in [8]. First of all, there is a short exact sequenceof vector bundles over M , 0 → A i −−→ L pr B −−→ B → . (2.21)Choose a splitting of Sequence (2.21), i.e., a vector bundle injection j : B → L , which determines a bundleprojection pr A : L → A such thatpr A ◦ i = id A , pr B ◦ j = id B , i ◦ pr A + j ◦ pr B = id L . Using this splitting, one could identify L with A ⊕ B . Meanwhile, for each A -module ( E, ∂ EA ), where E is avector bundle over M and ∂ EA is a flat A -connection on E , choose an L -connection ∇ L on E extending thegiven flat A -connection. Then there associates a 1-cocycle α ∇ L E ∈ Γ( A ∨ ⊗ B ∨ ⊗ End( E )), called the Atiyahcocycle, of the Lie algebroid A valued in the A -module B ∨ ⊗ End( E ): α ∇ L E ( a, b ) e := ∇ a ∇ Lj ( b ) e − ∇ Lj ( b ) ∇ a e − ∇ L [ a,j ( b )] e, for all a ∈ Γ( A ) , b ∈ Γ( B ) and e ∈ Γ( E ). The cohomology class α E = [ α ∇ L E ] ∈ H ( A, B ∨ ⊗ End( E ))does not depend on the choice of j and ∇ L , and is called the Atiyah class of the A -module E with respectto the Lie pair ( L, A ).From the Lie pair (
L, A ), we get a cdga Ω • A = (Γ( ∧ • A ∨ ) , d A ), and a dg Ω • A -module Ω • A ( B ∨ ) := (Γ( ∧ • A ∨ ⊗ B ∨ ) , ∂ A ), where ∂ A is the A -module structure dual to the Bott A -module structure on B . Here the degreeconvention is that Γ( B ∨ ) concentrates in degree zero.Fixing a splitting j of Sequence (2.21), we construct an Ω • A ( B ∨ )-valued derivation δ j of Ω • A , i.e., a map δ j : Γ( ∧ • A ∨ ) → Γ( ∧ • A ∨ ⊗ B ∨ ) . (2.22)As a degree zero derivation of the graded K -algebra Γ( ∧ • A ∨ ), δ j is fully determined by its action on itsgenerators, i.e. elements in C ∞ ( M ) and Γ( A ∨ ) — Define δ j : C ∞ ( M ) d L −−→ Γ( L ∨ ) j ∨ −→ Γ( B ∨ ) , ZHUO CHEN, ZHANGJU LIU, AND MAOSONG XIANG δ j : Γ( A ∨ ) pr ∨ A −−→ Γ( L ∨ ) d L −−→ Γ( ∧ L ∨ ) −→ Γ( L ∨ ⊗ L ∨ ) i ∨ ⊗ j ∨ −−−−→ Γ( A ∨ ⊗ B ∨ ) , where d L : Γ( ∧ • L ∨ ) → Γ( ∧ • +1 L ∨ ) is the Chevalley-Eilenberg differential of the Lie algebroid L . A straight-forward verification shows that δ j is compatible with the differentials and thus is an Ω • A ( B ∨ )-valued dgderivation of Ω • A .Note that δ j depends on a choice of a splitting j of Sequence (2.21). However, we have Proposition 2.23.
The Ω • A ( B ∨ ) -valued dg derivations δ j of Ω • A associated with different splittings of Se-quence (2.21) are homotopic to each other.Proof. Given two splittings j and j ′ of Sequence (2.21), their difference is a bundle map j ′ − j : B → A .Define a degree ( −
1) derivation h : Γ( ∧ • A ∨ ) → Γ( ∧ •− A ∨ ⊗ B ∨ ) by setting h | C ∞ ( M ) = 0 , h | Γ( A ∨ ) = ( j ′ − j ) ∨ . It follows from direct verifications that δ j ′ − δ j = [ ∂ A , h ] : Γ( ∧ • A ∨ ) → Γ( ∧ • A ∨ ⊗ B ∨ ) . This proves that δ j ∼ δ j ′ . (cid:3) Let (
E, ∂ EA ) be an A -module. There induces a dg Ω • A -module E := (Γ( ∧ • A ∨ ⊗ E ) , ∂ EA ). Proposition 2.24.
The Atiyah class α E of the A -module E with respect to the Lie pair ( L, A ) coincideswith the twisted Atiyah class At δ j E of the dg Ω • A -module E , where the dg derivation δ j is given as in Equation (2.22) .Proof. First of all, the spaces where the two Atiyah classes live are exactly the same, i.e.,( α E ∈ ) H ( A, B ∨ ⊗ End( E )) = H (Ω • A , Ω • A ( B ∨ ) ⊗ Ω • A End Ω • A ( E )) ( ∋ At δ j E ) . According to Proposition 2.13, to find the twisted Atiyah class At δ j E , one may use a δ j -connection ∇ δ j on E ,which is determined by its restriction on Γ( E ): ∇ δ j | E : Γ( E ) → Γ( B ∨ ) ⊗ Γ( E ) . This is equivalent to an L -connection ∇ L on E extending the given flat A -connection by setting ∇ La + b = ∇ a + ∇ δ j b | E , ∀ a + b ∈ L ∼ = A ⊕ B. (2.25)The two associated Atiyah cocycles coincide by straightforward computations, i.e., At ∇ δj E = α ∇ L E . (cid:3) As a consequence of Propositions 2.20 and 2.24, both Atiyah classes of dg Lie algebroids and those of Liepairs arise from Atiyah classes of cdgas. In particular, we have
Corollary 2.26.
Let A be a Lie algebroid and E an A -module. Denote by E the corresponding dg vectorbundle over ( A [1] , d A ) . If the Atiyah class of the dg vector bundle E with respect to the dg Lie algebroid T ( A [1]) vanishes, then the Atiyah class of E with respect to any Lie pair ( L, A ) vanishes. Functoriality.
We now study functorial properties of twisted Atiyah classes. Let H(dg A ) denote thehomology category of dg A -modules: Objects in H(dg A ) are dg A -modules, and morphisms in H(dg A )are dg A -module morphisms modulo homotopy [16].Let A δ −→ Ω be a dg derivation. For each object E in H(dg A ), by Definition 2.7, the twisted Atiyah classAt δ E ∈ H ( A , Ω ⊗ A End A ( E )) ∼ = Hom A ) ( E , Ω ⊗ A E )is a degree 1 morphism in the category H(dg A ). This identification defines a functorial transformation. Infact, when the dg derivation δ is fixed, the δ -twisted Atiyah class is a functorial transformation on H(dg A )from the identity functor id to the tensor functor Ω ⊗ A − : APRANOV’S CONSTRUCTION OF SH LEIBNIZ ALGEBRAS 9
Proposition 2.27.
Let E and F be dg A -modules, λ ∈ Hom
H(dg A ) ( E , F ) . The following diagram com-mutes in the category H(dg A ) : E At δ E / / ( − | λ | λ (cid:15) (cid:15) Ω ⊗ A E id Ω ⊗ A λ (cid:15) (cid:15) F At δ F / / Ω ⊗ A F . (2.28) Proof.
Let us first show the non-twisted case. Namely, the following diagram commutes in H(dg A ): E At E / / ( − | λ | λ (cid:15) (cid:15) Ω A | K ⊗ A E id ⊗ A λ (cid:15) (cid:15) F At F / / Ω A | K ⊗ A F . (2.29)In fact, this can be directly verified. We choose connections H E and H F , respectively, on E and F . Forsimplicity, they are both denoted by H . Then(id ⊗ A λ ) ◦ At H E − ( − | λ | At H F ◦ λ = (id Ω ⊗ A λ ) ◦ ( H ◦ ∂ A − ∂ A ◦ H ) − ( − | λ | ( H ◦ ∂ A − ∂ A ◦ H ) ◦ λ = ((id ⊗ A λ ) ◦ H − H ◦ λ ) ◦ ∂ A − ( − | λ | ∂ A ◦ ((id ⊗ A λ ) ◦ H − H ◦ λ ) . The map (id ⊗ A λ ) ◦ H − H ◦ λ : E → Ω A | K ⊗ A F is actually A -linear, by direct verifications. Thusthe two maps (id ⊗ A λ ) ◦ At H E and ( − | λ | At H F ◦ λ are only differed by an exact term. Composing with thedg A -module morphism ¯ δ : Ω A | K → Ω induced from the dg derivation δ , we accomplish a commutativediagram in H(dg A ): E At E / / ( − | λ | λ (cid:15) (cid:15) Ω A | K ⊗ A E id ⊗ A λ (cid:15) (cid:15) ¯ δ ⊗ A id / / Ω ⊗ A E id ⊗ A λ (cid:15) (cid:15) F At F / / Ω A | K ⊗ A F ¯ δ ⊗ A id / / Ω ⊗ A F . (cid:3) Now we study how Atiyah classes vary when twisted by different dg derivations. So we need the categoryof dg derivations, denoted by dgDer A , whose objects are dg derivations A δ −→ Ω as in Definition 2.4, andwhose morphisms are defined as follows:
Definition 2.30.
A morphism φ from A δ −→ Ω to A δ ′ −→ Ω ′ is a morphism φ : Ω → Ω ′ of dg A -modulessuch that δ ′ = φ ◦ δ : A → Ω ′ . Now let us fix a dg A -module E . We have the constant functor ( − 7→ E ) and the tensor functor − ⊗ A E ,both from the category dgDer A of dg derivations to the homology category H(dg A ) of dg A -modules. TheAtiyah class is a functorial transformation from ( − 7→ E ) to − ⊗ A E : Proposition 2.31.
Given a morphism φ : ( A δ −→ Ω) → ( A e δ −→ e Ω) of dg derivations and a dg A -module E ,let At δ E and At ˜ δ E be the Atiyah classes of E twisted, respectively, by A δ −→ Ω and A e δ −→ e Ω . Then the followingdiagram commutes in H(dg A ) : E At δ E / / id E (cid:15) (cid:15) Ω ⊗ A E φ ⊗ A id E (cid:15) (cid:15) E At ˜ δ E / / e Ω ⊗ A E . The proof is easy and thus omitted. Combining the previous two propositions, we have
Theorem 2.32.
With the same assumptions as in Propositions 2.27 and 2.31, the following diagram in
H(dg A ) commutes: E At δ E / / ( − | λ | λ (cid:15) (cid:15) Ω ⊗ A E φ ⊗ A λ (cid:15) (cid:15) F At ˜ δ F / / e Ω ⊗ A F . The Kapranov functor
In this section, we explore higher algebraic structures, called Kapranov Leibniz ∞ [1] algebras, induced froma dg derivation of a cdga A . Our main goal is to show that there exists a contravariant functor, called theKapranov functor, from the category of dg derivations to the category of Leibniz ∞ [1]-algebras over A .3.1. Leibniz ∞ [1] algebras. We recall some basic notions of homotopy Leibniz algebras (c.f.[1, 8]). In whatfollows, all tensor products ⊗ without adoration are assumed to be over K . Definition 3.1.
A Leibniz ∞ [1] algebra (over K ) is a graded K -vector space V = ⊕ n ∈ Z V n , together with asequence { λ k : ⊗ k V → V } k ≥ of degree , K -multilinear maps satisfying X i + j = n +1 n X k = j X σ ∈ sh( k − j,j − ǫ ( σ )( − | v σ (1) | + ··· + | v σ ( k − j ) | λ i ( v σ (1) , · · · , v σ ( k − j ) , λ j ( v σ ( k − j +1) , · · · , v σ ( k − , v k ) , v k +1 , · · · , v n ) = 0 , (3.2) for all n ≥ and all homogeneous elements v i ∈ V , where sh( p, q ) denotes the set of ( p, q ) -shuffles ( p, q ≥ ),and ǫ ( σ ) is the Koszul sign of σ . Definition 3.3.
A morphism of Leibniz ∞ [1] algebras from ( V, { λ k } k ≥ ) to ( V ′ , { λ ′ k } k ≥ ) is a sequence { f k : V ⊗ k → V ′ } k ≥ of degree , K -multilinear maps, satisfying the following compatibility condition: X k + p ≤ n − X σ ∈ sh( k,p ) ǫ ( σ )( − † σk f n − p ( b σ (1) , · · · , b σ ( k ) , λ p +1 ( b σ ( k +1) , · · · , b σ ( k + p ) , b k + p +1 ) , · · · , b n )= X q ≥ X I ∪···∪ I q = N ( n ) I , ··· ,I q = ∅ i | I | < ···
Let ( V, { λ k } k ≥ ) be a Leibniz ∞ [1] algebra. A ( V, { λ k } k ≥ ) -module is a graded K -vectorspace W together with a sequence { µ k : V ⊗ ( k − ⊗ W → W } k ≥ of degree , K -multilinear maps satisfyingthe identities X i + j = n +1 n X k = j X σ ∈ sh( k − j,j − ǫ ( σ )( − † σk − j µ i ( v σ (1) , · · · , v σ ( k − j ) , λ j ( v σ ( k − j +1) , · · · , v σ ( k − , v k ) , v k +1 , · · · , v n − , w )+ X ≤ j ≤ n X σ ∈ sh( k,j ) ǫ ( σ )( − † σn − j µ i ( v σ (1) , · · · , v σ ( n − j ) , µ j ( v σ ( n − j +1) , · · · , v σ ( n − , w )) = 0 , APRANOV’S CONSTRUCTION OF SH LEIBNIZ ALGEBRAS 11 for all n ≥ and all homogeneous vectors v , · · · , v n − ∈ V, w ∈ W , where † σj = | v σ (1) | + · · · + | v σ ( j ) | for all j ≥ . Definition 3.6.
Let ( V, { λ k } k ≥ ) be a Leibniz ∞ [1] algebra. A morphism of ( V, { λ k } k ≥ ) -modules from ( W, { µ k } k ≥ ) to ( W ′ , { µ ′ k } k ≥ ) is a sequence { ψ k : V ⊗ ( k − ⊗ W → W ′ } k ≥ of degree , K -multilinear mapssatisfying the identity X i + j = n +1 n X k = j X σ ∈ sh( k − j,j − ǫ ( σ )( − | v σ (1) | + ··· + | v σ ( k − j ) | ψ i ( v σ (1) , · · · , v σ ( k − j ) , λ j ( v σ ( k − j +1) , · · · , v σ ( k − , v k ) , v k +1 , · · · , v n − , w )+ X ≤ j ≤ n X σ ∈ sh( k,j ) ǫ ( σ )( − | v σ (1) | + ··· + | v σ ( n − j ) | ψ i ( v σ (1) , · · · , v σ ( n − j ) , µ j ( v σ ( n − j +1) , · · · , v σ ( n − , w ))= X p ≥ X I ∪···∪ I p +1 = N ( n − I , ··· ,I p +1 = ∅ i | I | < ···
A Leibniz ∞ [1] A -algebra is a Leibniz ∞ [1] algebra ( V, { λ k } k ≥ ) (in the category of Leibniz ∞ [1] algebras over K ) such that the cochain complex ( V, λ ) is a dg A -module and all higher brackets λ k : ⊗ k V → V ( k ≥ ) are A -multilinear.A morphism of Leibniz ∞ [1] A -algebras from ( V, { λ k } k ≥ ) to ( V ′ , { λ ′ k } k ≥ ) is a morphism { f k : V ⊗ k → V ′ } k ≥ (in the category of Leibniz ∞ [1] algebras over K ) such that all structure maps { f k } k ≥ are A -multilinear. In particular, its tangent morphism f : ( V, λ ) → ( V ′ , λ ′ ) is a dg A -module morphism.Such a morphism f • : ( V, λ • ) → ( V ′ , λ ′• ) is called a quasi-isomorphism (resp. an isomorphism) if its tangentmorphism f is a quasi-isomorphism (resp. an isomorphism) of dg A -modules. Denote the category of Leibniz ∞ [1] A -algebras by Leib ∞ ( A ). It is a subcategory of the category ofLeibniz ∞ [1] algebras over K . There are analogous notions of modules of a Leibniz ∞ [1] A -algebra andtheir morphisms: Definition 3.8.
Let ( V, { λ k } k ≥ ) be a Leibniz ∞ [1] A -algebra. A ( V, { λ k } k ≥ ) A -module is a ( V, { λ k } k ≥ ) -module ( W, { µ k } k ≥ ) (in the category of Leibniz ∞ [1] modules over K ) such that ( W, µ ) is a DG A -moduleand all higher structure maps { µ k } k ≥ are A -multilinear.A morphism of ( V, { λ k } k ≥ ) A -modules from ( W, { µ k } k ≥ ) to ( W ′ , { µ ′ k } k ≥ ) is a morphism of ( V, { λ k } k ≥ ) -modules { ψ k : V ⊗ ( k − ⊗ W → W ′ } k ≥ (in the category of Leibniz ∞ [1] modules over K ) such that all maps { ψ k } k ≥ are A -multilinear. It follows that the collection of ( V, { λ k } k ≥ ) A -modules and their morphisms form a category.3.2. The Kapranov functor.
In this section, we generalize Kapranov’s construction of an L ∞ algebrastructure [15] and Chen-Sti´enon-Xu’s construction of a Leibniz ∞ [1] algebra structure [8] in the setting of adg derivation A δ −→ Ω of a cdga A .3.2.1. Kapranov Leibniz ∞ [1] algebras. Let E be a graded A -module with a δ -connection ∇ . For each ho-mogeneous b ∈ B , there is a degree | b | derivation on the reduced tensor algebra T ( E ) (over A ) definedby ∇ b ( e ⊗ · · · ⊗ e n ) = n X i =1 ( − | b |∗ i − e ⊗ · · · ∇ b e i ⊗ · · · e n , for all homogeneous e i ∈ E , where ∗ i = P ij =1 | e j | .Let E and F be two graded A -modules with δ -connections ∇ E and ∇ F , respectively. For b ∈ B and λ ∈ Hom A ( E , F ), there associates the derivation ∇ b ( λ ) = [ ∇ b , λ ] = ∇ F b ◦ λ − ( − | b || λ | λ ◦ ∇ E b : E → F . It follows from a direct verification that ∇ b ( λ ) ∈ Hom A ( E , F ).Choose a δ -connection on B . There associates a sequence of degree 1 maps R ∇ k : B ⊗ k → B , k ≥ • R ∇ = ∂ A : B → B ; • R ∇ is specified by the associated twisted Atiyah cocycles At ∇B ; • {R ∇ k +1 : B ⊗ ( k +1) → B} k ≥ are defined recursively by R ∇ k +1 = ∇ ( R ∇ k ). Explicitly, we have R ∇ k +1 ( b , b , · · · , b k ) = ( − | b | [ ∇ b , R ∇ k ]( b , · · · , b k ) , ∀ b i ∈ B . (3.9) Proposition 3.10.
The A -module B , together with the sequence of operators {R ∇ k } k ≥ , is a Leibniz ∞ [1] A -algebra.Proof. The 2-bracket R ∇ is the twisted Atiyah cocycles At ∇B , which is certainly A -bilinear. By the recursiveconstruction of higher brackets R ∇ k +1 ( k ≥
2) in Equation (3.9), they are all A -multilinear as well. So itsuffices to verify that {R ∇ k } k ≥ satisfies Equation (3.2). We argue by induction.The n = 1 case follows from the fact that R ∇ = ∂ A is a differential, and the n = 2 case follows from thefact that the δ -twisted Atiyah cocycle At ∇B is a ∂ A -cocycle.Now assume that the identity (3.2) holds for some n ≥
2, i.e., − ∂ A ( R ∇ n )( b , · · · , b n ) = − ∂ A ( R ∇ n ( b , · · · , b n )) + n X i =1 ( − ∗ i − R ∇ n ( b , · · · , ∂ A b i , · · · , b n )= X i,j ≥ ,i + j = n +1 n X k = j X σ ∈ sh( k − j,j − ǫ ( σ )( − | b σ (1) | + ··· + | b σ ( k − j ) | R ∇ i ( b σ (1) , · · · , b σ ( k − j ) , R ∇ j ( b σ ( k − j +1) , · · · , b σ ( k − , b k ) , b k +1 , · · · , b n ) . (3.11)Consider the ( n + 1) case: We first compute − ∂ A ( R ∇ n +1 )( b , b , · · · , b n ) = ∂ A ([ R ∇ n , ∇ b ])( b , · · · , b n ) + [ R ∇ n , ∇ ∂ A b ]( b , · · · , b n )= [ ∂ A ( R ∇ n ) , ∇ b ]( b , · · · , b n ) + [ R ∇ n , R ∇ ( b , − )]( b , · · · , b n ) . (3.12)Here we have used the recursive definition (3.9) in the first equality and Equation (2.15) in the second one.We introduce b ( i ) k = ( b k , if k = i ∇ b b i , if k = i. Then the first summand in Equation (3.12) is,[ ∂ A ( R ∇ n ) , ∇ b ]( b , b , · · · , b n )= n X i =1 ( − | b |∗ i − ∂ A ( R ∇ n )( b , · · · , ∇ B b b i , · · · , b n ) − ∇ b ( ∂ A ( R ∇ n )( b , · · · , b n ))by assumption (3.11)= − n X i =1 X p,q ≥ ,p + q = n +1 n X k = q X σ ∈ sh( k − q,q − ǫ ( σ )( − | b |∗ i − + | b ( i ) σ (1) | + ··· + | b ( i ) σ ( k − q ) | R ∇ p ( b ( i ) σ (1) , · · · , b ( i ) σ ( k − q ) , R ∇ j ( b ( i ) σ ( k − q +1) , · · · , b ( i ) σ ( k − , b ( i ) k ) , b ( i ) k +1 , · · · , b ( i ) n ) APRANOV’S CONSTRUCTION OF SH LEIBNIZ ALGEBRAS 13 + X i,j ≥ ,i + j = n +1 n X k = j X σ ∈ sh( k − j,j − ǫ ( σ )( − | b σ (1) | + ··· + | b σ ( k − j ) | ∇ b ( R ∇ i ( b σ (1) , · · · , b σ ( k − j ) , R ∇ j ( b σ ( k − j +1) , · · · , b σ ( k − , b k ) , b k +1 , · · · , b n ))= X i,j ≥ ,i + j = n +1 n X k = j X σ ∈ sh( k − j,j − ǫ ( σ )( − | b | + | b σ (1) | + ··· + | b σ ( k − j ) | R ∇ i +1 ( b , b σ (1) , · · · , b σ ( k − j ) , R ∇ j ( b σ ( k − j +1) , · · · , b σ ( k − , b k ) , b k +1 , · · · , b n )+ X i,j ≥ ,i + j = n +1 n X k = j X σ ∈ sh( k − j,j − ǫ ( σ )( − ( | b σ (1) | + ··· + | b σ ( k − j ) | )( | b | +1) R ∇ i ( b σ (1) , · · · , b σ ( k − j ) , R ∇ j +1 ( b , b σ ( k − j +1) , · · · , b σ ( k − , b k ) , b k +1 , · · · , b n ) . (3.13)Here in the last step we used the recursive definition of {R ∇ i } .Meanwhile, the second summand in Equation (3.12) is[ R ∇ n , R ∇ ( b , − )]( b , b , · · · , b n )= n X i =1 ( − ( | b | +1) ∗ i − R ∇ n ( b , · · · , R ∇ ( b , b i ) , · · · , b n ) + ( − | b | R ∇ ( b , R ∇ n ( b , · · · , b n )) . (3.14)Substituting Equations (3.13) and (3.14) into Equation (3.12), we see that Equation (3.11) holds for the case( n + 1). This proves that {R ∇ k } k ≥ satisfies Equation (3.2) for all n ≥ (cid:3) The Leibniz ∞ [1] A -algebra ( B , {R ∇ k } k ≥ ) will be denoted by Kap c ( δ ). Here the superscript c is to remindthe reader that this Leibniz ∞ [1] A -algebra is defined via a particular δ -connection on B . Remark 3.15.
This method is originated from Kapranov’s construction of L ∞ algebra structure on theshifted tangent complex Ω , •− X ( T , X ) of a compact K¨ahler manifold X [15] . For this reason, we call ( B , {R ∇ k } k ≥ ) the Kapranov Leibniz ∞ [1] A -algebra. Functoriality.
Next, we show that the assignment of a Leibniz ∞ [1] A -algebra to each pair of dgderivation A δ −→ Ω and a δ -connection ∇ on B is functorial: Proposition 3.16.
Let φ be a morphism from A δ ′ −→ Ω ′ to A δ −→ Ω in the category dgDer A of dg derivations(see Definition 2.30). Let B = Ω ∨ and B ′ = (Ω ′ ) ∨ be their dual dg A -modules. For a δ -connection ∇ on B and a δ ′ -connection ∇ ′ on B ′ , there exists a morphism f • = { f k } k ≥ of Kapranov Leibniz ∞ [1] A -algebrasfrom ( B , {R ∇ k } k ≥ ) to ( B ′ , {R ∇ ′ k } k ≥ ) , whose first map is f = φ ∨ . In other words, we have the followingcommutative diagram ( A δ ′ −→ Ω ′ ) φ (cid:15) (cid:15) Kap c / / ( B ′ , {R ∇ ′ k } k ≥ )( A δ −→ Ω) Kap c / / ( B , {R ∇ k } k ≥ ) . Kap c ( φ )= f • O O Proof.
Define a sequence of A -multilinear maps f k : B ⊗ k → B ′ recursively by setting f ( b ) = φ ∨ ( b ) , f k +1 ( b , · · · , b k ) = ∇ ′ f ( b ) f k ( b , · · · , b k ) − f k ( ∇ b ( b , · · · , b k )) , (3.17)for all k ≥ b i ∈ B . It is easy to verify that all the maps { f k } k ≥ are A -multilinear. Now we show that { f k } k ≥ is a morphism of Leibniz ∞ [1] A -algebras from ( B , R ∇ k ) to ( B ′ , R ∇ ′ k ).We argue by induction: First of all, the n = 1 case is obvious, since f = φ ∨ : B → B ′ is a morphism of dgmodules. Now assume that Equation (3.4) holds for some n ≥
1, i.e., X k + p ≤ n − X σ ∈ sh( k,p ) ǫ ( σ )( − † σk f n − p ( b σ (1) , · · · , b σ ( k ) , R ∇ p +1 ( b σ ( k +1) , · · · , b σ ( k + p ) , b k + p +1 ) , · · · , b n ) = X q ≥ X I ∪···∪ I q = N ( n ) I , ··· ,I q = ∅ i | I | < ···
1. This proves that f = { f k } k ≥ is a morphism ofLeibniz ∞ [1] A -algebras. (cid:3) APRANOV’S CONSTRUCTION OF SH LEIBNIZ ALGEBRAS 15
As a consequence, the Kapranov’s construction defines a contravariant functorKap c : dgDer A → Leib ∞ ( A )from the category dgDer A of dg derivations of A to the category Leib ∞ ( A ) of Leibniz ∞ [1]-algebras. Remark 3.21.
The reason that we restrict to work in the category
Leib ∞ ( A ) of Leibniz ∞ [1] A -algebras isas follows: If we treat ( B , {R ∇ k } k ≥ ) merely as a Leibniz ∞ [1] algebra over K , it is always isomorphic to thetrivial one ( B, { ∂ A , , , · · · } ) (all higher brackets are zero). In fact, one can build a sequences of degree maps φ k : B ⊗ k → B , k ≥ , where φ = id B , and { φ k +1 } k ≥ are defined recursively by φ k +1 ( b , · · · , b k ) = ∇ b ◦ φ k ( b , · · · , b k ) , ∀ b i ∈ B . The set { φ k : B ⊗ k → B} k ≥ defines an isomorphism of Leibniz ∞ [1] algebras from ( B , { ∂ A , , , · · · } ) to ( B , {R ∇ k } k ≥ ) in the category of Leibniz ∞ [1] algebras over K . The proof is similar to that of Proposition 3.16.However, the maps { φ k } k ≥ are not A -multilinear. Next, we stress the independence from the choice of connections in the definition of Kapranov functors. Fora dg derivation A δ −→ Ω, suppose that we have another δ -connection e ∇ on B = Ω ∨ . Denote the correspondingKapranov Leibniz ∞ [1] A -algebra by Kap ˜ c ( δ ) = ( B , R e ∇ k ). By Proposition 3.16, there exists an isomorphism g ∇ , e ∇• : Kap c ( δ ) → Kap ˜ c ( δ ) of Leibniz ∞ [1] A -algebras, where g ∇ , e ∇ = id B , and { g ∇ , e ∇ k +1 } k ≥ are definedrecursively as follows: g ∇ , e ∇ k +1 ( b , · · · , b k ) = ( e ∇ b ◦ g ∇ , e ∇ k − g ∇ , e ∇ k ◦ ∇ b )( b , · · · , b k ) , ∀ b i ∈ B . Moreover, via a straightforward verification, we have
Lemma 3.22.
There exists a natural equivalence between Kapranov functors
Kap c and Kap ˜ c with respectto different connections. In other words, for any morphism φ : ( A δ ′ −→ Ω ′ ) → ( A δ −→ Ω) of dg derivations of A , we have the following commutative diagram Kap c ( δ ) Kap c ( φ ) (cid:15) (cid:15) g ∇ , f ∇• / / Kap ˜ c ( δ ) Kap ˜ c ( φ ) (cid:15) (cid:15) Kap c ( δ ′ ) g ∇′ , g ∇′ / / Kap ˜ c ( δ ′ ) . By this natural equivalence, we are allowed to drop the superscript c to obtain the following Theorem 3.23.
The Kapranov’s construction defines a contravariant functor
Kap : dgDer A → Leib ∞ ( A ) from the category dgDer A of dg derivations of A to the category Leib ∞ ( A ) of Leibniz ∞ [1] -algebras. Remark 3.24.
By the universal property, the K¨ahler differential A d dR −−→ Ω A | K is the initial object in thecategory dgDer A of dg derivations. Thus the corresponding Kapranov Leibniz ∞ [1] A -algebra on the tangentcomplex T A | K = (Ω A | K ) ∨ of A is the final object of the subcategory in Leib ∞ ( A ) consisting of KapranovLeibniz ∞ [1] A -algebras arising from dg derivations of A . Let A δ −→ Ω be a dg derivation of A and E a dg A -module. By a similar argument, E carries a Leibniz ∞ [1] A -module structure over Kap( δ ). Moreover, we have Theorem 3.25.
Given a dg derivation A δ −→ Ω of A , there exists a functor from the category dg A of dg A -modules to the category of Leibniz ∞ [1] A -modules over Kap( δ ) . Leibniz algebra structures.
Let ( V, { λ k } k ≥ ) be a Leibniz ∞ [1] A -algebra as in Definition 3.7. Then( V, λ = ∂ A ) is a dg A -module. Its cohomology H • ( V ) is called the tangent cohomology of the Leibniz ∞ [1] A -algebra ( V, { λ k } k ≥ ). According to [8, Proposition 3.10], the (degree ( −
1) shifted) tangent cohomology H • ( V [ − K ), when equipped with the bracketˇ λ : H • ( V [ − × H • ( V [ − → H • ( V [ − λ ([ x ] , [ y ]) := ( − | x | [ λ ( x, y )] , where x, y ∈ V are λ -closed.In a similar fashion, if ( W, { µ k } k ≥ ) is a ( V, { λ k } k ≥ ) A -module as in Definition 3.8, then ( W, µ = ∂ A )is also a dg A -module. The cohomology H • ( W ) is a Leibniz module over the aforesaid Leibniz algebra H • ( V [ − K ), when equipped with the actionˇ µ : H • ( V [ − × H • ( W ) → H • ( W )ˇ µ ([ x ] , [ w ]) := ( − | x | [ µ ( x, w )] , where x ∈ V, w ∈ W are, respectively, λ - and µ -closed elements.As a consequence of Theorem 2.32, Theorems 3.10 and 3.25, we have the following Corollary 3.26.
Let φ be a morphism of dg derivations from A δ ′ −→ Ω ′ to A δ −→ Ω and let B and B ′ be thedual dg A -modules of Ω and Ω ′ , respectively.(1) The (degree ( − shifted) cohomology space H • ( A , B [ − is a Leibniz algebra, whose bracket h − , − i B is induced by the δ -twisted Atiyah class of B : h [ b ] , [ b ] i B = ( − | b | At δ B ([ b ] , [ b ]) , where b , b ∈ B are ∂ A -closed elements. Moreover, φ ∨ : B → B ′ induces a morphism of Leibnizalgebras, i.e., h φ ∨ ( b ) , φ ∨ [ b ] i B ′ = φ ∨ ( h [ b ] , [ b ] i B ) . (2) For any dg A -module E , there exists a representation of H • ( A , B [ − on the cohomology space H • ( A , E ) , with the action map − ⊲ − induced by the δ -twisted Atiyah class of E : [ b ] ⊲ [ e ] = ( − | b | At δ E ([ b ] , [ e ]) , where b ∈ B , e ∈ E are both ∂ A -closed elements. Moreover, this assignment is functorial, i.e., foreach dg A -module morphism λ : E → F (of degree ), [ b ] ⊲ λ ( e ) = λ ([ b ] ⊲ [ e ]) . Remark 3.27.
According to [8, Theorem 3.4] , the Atiyah class of a Lie pair ( L, A ) induces a Lie algebrastructure on the cohomology H • CE ( A, L/A [ − . A similar result holds for L ∞ algebra pairs [7] . However, itis not the case in general (see an example below). It is natural to ask when the Leibniz algebra structure inCorollary 3.26 could be refined to a Lie algebra structure. We will investigate this question somewhere else. Example 3.28.
Let LM be the category of linear maps [22] . A Lie algebra object in LM is a triple E ψ −→ g ,where g is a Lie algebra, E is a left g -module, and ψ is a g -equivariant linear map. Consider the cdga A = C • ( g ) = ( ∧ • g ∨ , d CE ) and dg C • ( g ) -module Ω = C • ( g , E ∨ [ − ∧ • g ∨ ⊗ E ∨ [ − , d CE ) , i.e., theChevalley-Eilenberg cochain complex of the dual g -module E ∨ [ − . The g -equivariant map E ψ −→ g gives riseto a dg derivation of C • ( g ) : C • ( g ) δ = ψ ∨ −−−−→ C • ( g , E ∨ [ − . The dual module of
Ω = C • ( g , E ∨ [ − is B = C • ( g , E [1]) . One can take the trivial δ -connection on B : ∇ : B = C • ( g , E [1]) → Ω ⊗ A B = C • ( g , E ∨ [ − ⊗ E [1]) , defined by ∇ ( ω ⊗ e ) = δ ( ω ) ⊗ e, ∀ ω ∈ ∧ • g ∨ , e ∈ E. By Equation (2.15) , the associated Atiyah cocycle is a degree element At ∇B ∈ E ∨ [ − ⊗ E ∨ [ − ⊗ E [1] specified by R ∇ = At ∇B ( e , e ) = − ψ ( e ) e , ∀ e , e ∈ E. APRANOV’S CONSTRUCTION OF SH LEIBNIZ ALGEBRAS 17
It can be easily seen that higher structures R ∇ j = 0 for all j ≥ . Hence, the Kapranov Leibniz ∞ [1] C • ( g ) -algebra B = C • ( g , E [1]) is simply a dg Leibniz [1] algebra in this case, or equivalently, B [ −
1] = C • ( g , E ) is adg Leibniz algebra. In particular, the subspace E is a Leibniz algebra, recovering the result in [22] .By Corollary 3.26, there is a Leibniz algebra structure on the graded vector space H • ( A , B [ − H • CE ( g , E ) ,whose bracket is given by h [ e ] , [ e ] i = ( − | e | +1 [At ∇B ( e , e )] = ± [ ψ ( e ) e ] , for all d CE -closed elements e , e ∈ C • ( g , E ) . Here the last term ψ ( − )( − ) : C • ( g , E ) × C • ( g , E ) → C • ( g , E ) is a ( ∧ • g ∨ ) -bilinear map naturally extended from ψ ( − )( − ) : E × E → E . In general, the Leibniz structureon ( H • CE ( g , E ) , [ − , − ]) is not skewsymmetric. Homotopic invariance.
In this section, we prove that the isomorphism class of Kapranov Leibniz ∞ [1] A -algebras arising from dg derivations only depends on their homotopy classes. Proposition 3.29.
Let δ ∼ δ ′ be homotopic Ω -valued dg derivations of A .Then there exists an isomor-phism { g k } k ≥ sending the Kapranov Leibniz ∞ [1] A -algebra Kap( δ ′ ) = ( B , {R ∇ ′ k } k ≥ ) (with respect to a δ ′ -connection ∇ ′ ) to Kap( δ ) = ( B , {R ∇ k } k ≥ ) (with respect to a δ -connection ∇ ).Proof. By assumption, there exists a degree ( −
1) Ω-valued derivation h : A → Ω of A such that δ ′ = δ + [ ∂ A , h ] = δ + ∂ A ◦ h + h ◦ d A . We choose an h -connection on B , i.e. a degree ( −
1) linear map b ∇ : B → Ω ⊗ A B satisfying b ∇ ( ab ) = h ( a ) ⊗ b + ( − | a | a b ∇ ( b ) , ∀ a ∈ A , b ∈ B . For each δ -connection ∇ on B , it can be easily verified that ∇ ′′ := [ ∂ A , b ∇ ] is a [ ∂ A , h ]-connection on B , andthus ∇ ′ = ∇ + ∇ ′′ = ∇ + [ ∂ A , b ∇ ] : B → Ω ⊗ A B is a δ ′ -connection on B . It follows that R ∇ ′ = [ ∇ ′ , ∂ A ] = [ ∇ + [ ∂ A , b ∇ ] , ∂ A ] = [ ∇ , ∂ A ] = R ∇ . Define a family of A -multilinear maps g k : B ⊗ k → B inductively by setting g = id B , g = 0, and g k +1 ( b , · · · , b k ) = ( − | b | k X p =2 X I ∪···∪ I p = N ( k ) I , ··· ,I p = ∅ i | I | < ···
2. It follows from a straightforward inductive argument that { g k } k ≥ is a morphism of Leibniz ∞ [1] A -algebras from ( B , R ∇ ′ n ) to ( B , R ∇ n ). (cid:3) Remark 3.31.
Although the Kapranov functor
Kap maps homotopic derivations to isomorphic Leibniz ∞ [1] A -algebra, it does not reduce to a functor from the category consisting of homology classes of dg derivationsof A to the category Leib ∞ ( A ) of Leibniz ∞ [1] A -algebras. Applications.
We first consider a Lie pair (
L, A ), and let B = L/A be the Bott A -module. In theintroduction, we explained that for each splitting j : B → L of the short exact sequence (2.21) and forany L -connection ∇ on B extending the Bott A -module structure, there associates a Leibniz ∞ [1] algebrastructure { λ k } k ≥ on the graded K -vector space Γ( ∧ • A ∨ ⊗ B ). As all { λ k } k ≥ are Ω • A -multilinear, it is aLeibniz ∞ [1] Ω • A -algebra.Recall that we have a Ω • A ( B ∨ )-valued dg derivation δ j of the cdga Ω • A as in Equation (2.22). By Proposi-tion 2.24, the Atiyah cocycle α ∇ B of the Lie pair coincides with the Atiyah cocycle At ∇ δj B of the dg Ω • A -module B := Ω • A ( B ) with respect to a δ j -connection ∇ δ j as in Equation (2.25). Comparing definitions of { λ k } k ≥ in the introduction and {R ∇ δj k } as in Equation (3.9), we see that the two Leibniz ∞ [1] Ω • A -algebras ( B , { λ k } k ≥ )and ( B , R ∇ δj k ) are exactly the same.Applying Proposition 2.23, Theorem 3.23, Theorem 3.25, and Proposition 3.29, we have the following Theorem 3.32.
Let ( L, A ) be a Lie pair over a smooth manifold M . Then the Leibniz ∞ [1] algebra structureconstructed in [8, Theorem 3.13] on the graded K -vector space Γ( ∧ • A ∨ ⊗ L/A ) is unique up to isomorphismsin the category of Leibniz ∞ [1] Ω • A -algebras.Moreover, if ( E, ∂ EA ) is an A -module, then the representation of the above Leibniz ∞ [1] algebra on the graded K -vector space Γ( ∧ • A ∨ ⊗ E ) is also unique up to isomorphisms in the category of Leibniz ∞ [1] Ω • A -modules. Finally, we consider another interesting application: Let X be a complex manifold and A = (Ω , • X , ¯ ∂ ) itsDolbeault dg algebra. Let Ω = (Ω , • X ( T , X ) , ¯ ∂ ) be the dg A -module generated by the smooth sectionspace Γ( T , X ) of the holomorphic tangent bundle T , X . Note that each holomorphic bivector field π ∈ Γ( ∧ T , X ) determines an Ω-valued dg derivation of A , denoted by δ π , which is the composition A ∂ −→ Ω , • X = Ω , • X (( T , X ) ∨ ) π ♯ −→ Ω . Here π ♯ is the contraction along π from ( T , X ) ∨ to T , X .In fact, π ♯ is a morphism of dg derivations of A (from A ∂ −→ Ω , • X to A δ π −→ Ω). It sends the Atiyahclass α E ∈ H ( X, ( T , X ) ∨ ⊗ End( E )) of any holomorphic vector bundle E to the δ π -twisted Atiyah classAt δ π E ∈ H ( X, T , X ⊗ End( E )) of the associated dg A -module E = Ω , • X ( E ).By Proposition 2.16, the δ π -twisted Atiyah class At δ π E measures the existence of holomorphic δ π -connectionson E . In particular, if π a holomorphic Poisson bivector field, then ( T , X ) ∨ is a holomorphic Lie alge-broid [18], and At δ π E measures the existence of holomorphic ( T , X ) ∨ -connections on E .Applying Theorem 3.23, we have the following Theorem 3.33.
Let X be a complex manifold, π a holomorphic bivector field. Then, • Both Ω , • X ( T , X ) and Ω , • X (( T , X ) ∨ ) carry canonical Kapranov Leibniz ∞ [1] Ω , • X -algebra structures; • There is a morphism of Leibniz ∞ [1] Ω , • X -algebras { f k } k ≥ : Ω , • X (( T , X ) ∨ ) → Ω , • X ( T , X ) suchthat f = π ♯ . Open questions and remarks
In this note, we assume that each dg A -module E is projective in order that connections exist on E . In thenon-projective case, one can follow Calaque-Van den Bergh’s approach [6] to define the Atiyah class of E (which coincides with the Atiyah class of E in Definition 2.1 when E admits connections)— The first step isto construct a short exact sequence, called the jet sequence, of dg A -modules:0 / / Ω A | K ⊗ A E / / J E / / E / / . The Atiyah class of E is then defined to be the extension class of the above jet sequence. We would like tofollow this approach to study twisted Atiyah classes of some cases when connections do not exist (singularfoliations considered in [17] for example).Note that Kapranov’s original construction on Ω , •− X ( T X ) of a K¨ahler manifold X is an L ∞ algebra, whereasChen, Sti´enon and Xu’s construction of Γ( ∧ • A ∨ ⊗ B ) is a Leibniz ∞ [1] algebra. In fact, this is due to theexistence of Chern connection on T X which enjoys special properties (see [8, Section 3.4.4]). Meanwhile,when A = C ∞ ( M ) is the cdga of functions of a smooth dg manifold M . According to [23], the tangentcomplex T A | K = Γ( T M ) admits an L ∞ [1] algebra structure (by a construction different from the Kapranov’sconstruction we discussed). Moreover, Laurent-Gengoux, Sti´enon and Xu [20] have proved that for each Liepair ( L, A ), there exists a canonical L ∞ [1] algebra structure on the graded K -vector space Γ( ∧ • A ∨ ⊗ L/A )(which is different from the Chen-Sti´enon-Xu’s construction in [8]). It is natural to ask how to tweak the
APRANOV’S CONSTRUCTION OF SH LEIBNIZ ALGEBRAS 19
Kapranov Leibniz ∞ [1] algebra of general dg derivations so as to produce an L ∞ [1] algebra rather than amere Leibniz ∞ [1] algebra.According to the perturbation lemmas proved by Huebschmann [12, 13], many L ∞ algebras arise from dg Liealgebras or L ∞ algebras by homological perturbation theory. It is interesting to investigate whether similarperturbation lemma holds for Leibniz ∞ [1] algebras. Moreover, if this is the case, then it is natural to askfor which kind of dg derivations of a cdga A , the associated Kapranov Leibniz ∞ [1] A -algebra results fromsome perturbation.These questions will be investigated somewhere else.We would also like to mention other works that are related to the present paper: Batakidis and Voglaire [3]showed how Atiyah classes of Lie pairs [8] and of dg Lie algebroids [23] give rises to Atiyah classes ofdDG algebras [6]. Bordemann [4] studied the Atiyah class as the obstruction to the existence of invariantconnections on homogeneous spaces. Hennion [11] generalized Kapranov’s construction to algebraic derivedstack: There exists a Lie algebra structure on the shifted tangent complex T X [ −
1] of a derived Artin stack X locally of finite presentation. Moreover, given a perfect module E over X , there exists a representationof the aforesaid Lie algebra on E induced by the Atiyah class of E . References [1] Mourad Ammar and Norbert Poncin,
Coalgebraic approach to the Loday infinity category, stem differential for n -arygraded and homotopy algebras , Ann. Inst. Fourier (Grenoble) (2010), no. 1, 355–387 (English, with English and Frenchsummaries).[2] Michael Francis Atiyah, Complex analytic connections in fibre bundles , Trans. Amer. Math. Soc. (1957), 181–207.[3] Panagiotis Batakidis and Yannick Voglaire, Atiyah classes and dg-Lie algebroids for matched pairs , J. Geom. Phys. (2018), 156–172.[4] Martin Bordemann,
Atiyah classes and equivariant connections on homogeneous spaces , Travaux math´ematiques. VolumeXX, Trav. Math., vol. 20, Fac. Sci. Technol. Commun. Univ. Luxemb., Luxembourg, 2012, pp. 29–82.[5] Francesco Bottacin,
Atiyah classes of Lie algebroids , Current trends in analysis and its applications, Trends Math.,Birkh¨auser/Springer, Cham, 2015, pp. 375–393.[6] Damien Calaque and Michel Van den Bergh,
Hochschild cohomology and Atiyah classes , Adv. Math. (2010), no. 5,1839–1889.[7] Zhuo Chen, Honglei Lang, and Maosong Xiang,
Atiyah classes of Strongly homotopy Lie pairs , Algebra Colloq. (2019),no. 2, 195–230.[8] Zhuo Chen, Mathieu Sti´enon, and Ping Xu, From Atiyah classes to homotopy Leibniz algebras , Comm. Math. Phys. (2016), no. 1, 309–349.[9] Zhuo Chen, Maosong Xiang, and Ping Xu,
Atiyah and Todd classes arising from integrable distributions , J. Geom. Phys. (2019), 52–67.[10] Kevin Costello,
A geometric construction of the Witten genus II , available at arXiv:1112.0816.[11] Benjamin Hennion,
Tangent Lie algebra of derived Artin stacks , J. Reine Angew. Math. (2018), 1–45.[12] Johannes Huebschmann,
The Lie algebra perturbation lemma , Higher structures in geometry and physics, Progr. Math.,vol. 287, Birkh¨auser/Springer, New York, 2011, pp. 159–179.[13] ,
The sh-Lie algebra perturbation lemma , Forum Math. (2011), no. 4, 669–691.[14] , Origins and breadth of the theory of higher homotopies , Higher structures in geometry and physics, Progr. Math.,vol. 287, Birkh¨auser/Springer, New York, 2011, pp. 25–38.[15] Mikhail M. Kapranov,
Rozansky-Witten invariants via Atiyah classes , Compositio Math. (1999), no. 1, 71–113.[16] Bernhard Keller,
On differential graded categories , International Congress of Mathematicians. Vol. II, Eur. Math. Soc.,Z¨urich, 2006, pp. 151–190.[17] Camille Laurent-Gengoux, Sylvain Lavau, and Thomas Strobl,
The universal Lie ∞ -algebroid of a singular foliation ,available at arXiv:1806.00475.[18] Camille Laurent-Gengoux, Mathieu Sti´enon, and Ping Xu, Holomorphic Poisson manifolds and holomorphic Lie algebroids ,Int. Math. Res. Not. IMRN (2008), Art. ID rnn 088, 46.[19] ,
Exponential map and L ∞ algebra associated to a Lie pair , C. R. Math. Acad. Sci. Paris (2012), no. 17-18,817–821 (English, with English and French summaries).[20] , Poincar´e–Birkhoff–Witt isomorphisms and Kapranov dg-manifolds , available at arXiv:1408.2903.[21] Tom Lada and Jim Stasheff,
Introduction to SH Lie algebras for physicists , Internat. J. Theoret. Phys. (1993), no. 7,1087–1103.[22] Jean-Louis Loday and Teimuraz Pirashvili, The tensor category of linear maps and Leibniz algebras , Georgian Math. J. (1998), no. 3, 263–276.[23] Rajan Amit Mehta, Mathieu Sti´enon, and Ping Xu, The Atiyah class of a dg-vector bundle , C. R. Math. Acad. Sci. Paris (2015), no. 4, 357–362 (English, with English and French summaries).[24] Pierre Molino,
Classe d’Atiyah d’un feuilletage et connexions transverses projetables. , C. R. Acad. Sci. Paris S´er. A-B (1971), A779–A781 (French).
Department of Mathematics, Tsinghua University
E-mail address : [email protected] Department of Mathematics, Peking University
E-mail address : [email protected] Center for Mathematical Sciences, Huazhong University of Science and Technology
E-mail address ::