Higher derived brackets, strong homotopy associative algebras and Loday pairs
aa r X i v : . [ m a t h . QA ] J a n Higher derived brackets, strong homotopy associativealgebras and Loday pairs
K. UCHINO
Abstract
We give a quick method of constructing strong homotopy associative algebra,namely, the higher derived product construction. This method is associative ana-logue of classical higher derived bracket construction in the category of Loday al-gebras. We introduce a new type of algebra,
Loday pair , which is noncommutativeanalogue of classical Leibniz pair. We study strong homotopy Loday pairs and thehigher derived brackets on the Loday pairs.
Let ( g , d, [ , ]) be a differential graded (dg, for short) Lie algebra. We define a newproduct by [ x, y ] d := ( − | x | [ dx, y ]. This product is called a derived bracket ofKoszul-Kosmann-Schwarzbach ([15]). It is known that the algebra of the derivedbracket is a Loday algebra (so-called Leibniz algebra), i.e., [ x, y ] d satisfies the Leibnizidentity: [ x, [ y, z ] d ] d = [[ x, y ] d , z ] d + ( − | x || y | [ y, [ x, z ] d ] d . The derived bracket construction is a method of constructing new algebra structure.It plays important roles in modern analytical mechanics and in differential geometry.For instance, a Poisson bracket on a smooth manifold is a derived bracket of a gradedPoisson bracket which is called a Schouten-Nijenhuis bracket. { f, g } = ( − f [[ π, f ] SN , g ] SN , Mathematics Subject Classifications (2000): 17A32, 53D17 Keywords: strong homotopy associative algebras, higher derived brackets, Leibniz pairs here f, g are smooth functions, π is a Poisson structure tensor, [ , ] SN is theSchouten-Nijenhuis bracket and { , } is the induced Poisson bracket. Since d π :=[ π, − ] SN is a differential, the Poisson bracket is a derived bracket. We recall anotherexample of derived brackets. Let f = f ( p, q ) and g = g ( p, q ) be super functions on asuper symplectic-manifold, where ( p, q ) is a canonical cordinate of the manifold. Weconsider a Laplacian with odd degree ∆ BV := P ( ± ) ∂ ∂p∂q and define a differential d BV := [∆ BV , − ]. It is known that the derived bracket associated with d BV is aPoisson bracket (so-called BV-bracket):( f, g ) = X f ←− ∂∂p −→ ∂ g∂q − f ←− ∂∂q −→ ∂ g∂p = ( ± )[ d BV ˆ f , ˆ g ] , where ˆ f ( − ) := f × ( − ) is a scalar multiplier, ( ± ) is an appropriate sign and [ , ] is aLie bracket (commutator). Thus various bracket products (Lie algebroid brackets,Courant brackets, BV-brackets and so on) are given as derived brackets (see [16]).The idea of the derived bracket arises in several mathematical areas. We recalla derived bracket in the category of associative algebras. Let ( A, ∗ , d ) be a dgassociative algebra. We define a modified product by a ∗ d b := ( − | a | ( da ) ∗ b , a, b ∈ A . Then it is again associative. This new product is called a derivedproduct , which is used in the study of Loday type algebras (cf. Loday [17]).The derived bracket/product constructions have been extended to any algebra overbinary quadratic operad in [22].We consider n -fold derived brackets composed of Lie brackets:[ x , ..., x n ] d := ( ± )[[ ... [[ dx , x ] , x ] ... ] , x n ] . Such higher brackets were studied by several authors in various contexts (cf. Akman(1996) [1], Vallejo (2001) [24], Roytenberg (2002) [20] and Voronov (2005) [25]).Koszul’s original type higher derived brackets, which are denoted by Φ, are definedon super commutative algebras byΦ n ∆ ( a , ..., a n ) := [[ ... [ d ˆ a , ˆ a ] , ... ] , ˆ a n ](1)where d := [∆ , − ] and where ∆ is a certain differential operator like ∆ BV above.The higher brackets Φ are used to study higher order differential operators (cf.[1],[24]). In [23], the author studied a higher derived bracket construction in thecategory of Loday algebras. We briefly describe the result in [23]. Let (
V, δ ) be adg Loday algebra and let d := δ + tδ + t δ + · · · e a formal deformation of δ , where dd = 0. Define a higher derived bracket onthe Loday algebra by l n ( x , ..., x n ) := ( ± )[[ ... [[ δ n − x , x ] , x ] ... ] , x n ] , where the binary bracket [ , ] is a Loday bracket. It was shown that the collectionof the higher derived brackets ( l , l , l , ... ) provides a strong homotopy (shortly,sh) Loday algebra structure (also called Loday ∞ -algebra or sh Leibniz algebra orLeibniz ∞ -algebra). If each l n ≥ is skewsymmetric, then the sh Loday algebra is ansh Lie ( L ∞ -)algebra. This proposition is a homotopy version of the binary derivedbracket construction in [15].The first aim of this note is to study a higher version of the derived productconstruction. Let ( A, δ ) be a dg associative algebra and d = P i ≥ t i δ i be a formaldeformation of δ . Define a higher derived products by m n ( a , ..., a n ) := ( ± ) (cid:0) δ n − a (cid:1) ∗ a ∗ · · · ∗ a n . We show in Theorem 3.1 below that the system with the higher derived products(
A, m , m , ... ) becomes an sh associative algebra (or A ∞ -algebra).The second aim of this note is to unify the higher derived bracket/productconstructions. To complete this task we recall Leibniz pairs . The notion of Leibnizpair was introduced by Flato-Gerstenhaber-Voronov [5], motivated by the studyof deformation quantization. The Leibniz pairs are defined to be the pairs of Lieand associative algebras ( g , A ) equipped with derivation representations rep : g → Der( A ). The representation satisfies the following two derivation relations:[ x, [ a, b ]] = [[ x, a ] , b ] + [ a, [ x, b ]] , (1)[ x, [ y, a ]] = [[ x, y ] , a ] + [ y, [ x, a ]] , (2)where x, y ∈ g , a, b ∈ A and where [ x, a ] is the derivation action of L on A and[ a, b ] is the associative multiplication on A , i.e., [ a, [ b, c ]] = [[ a, b ] , c ]. We recall twotypical examples of Leibniz pairs.a) The self pair of a Poisson algebra P , ( P, P ), is obviously a Leibniz pair.b) Let g → M be a Lie algebroid over a smooth manifold M . Then the pair(Γ g , C ∞ ( M )) is a Leibniz pair, where Γ g is the space of sections of g .We consider the pairs of Loday algebras and associative algebras satisfying (1)and (2). We call such pairs the Loday pairs . There exists interesting examples ofLoday pairs, which are regarded as noncommutative analogues of Examples a),b).a-1) It is known that a Poisson manifold is a classical solution of a master equation ssociated with 2-dimensional topological field theory (cf. Cattaneo-Felder [3, 4]).In the 3-dimensional cases, the classical solutions are known as Courant algebroids (Ikeda [10, 11, 12], see also [21]). A Courant algebroid is a vector bundle E → M of which the space of sections is a Loday algebra satisfying some axioms. When E is a Courant algebroid, the pair (Γ E, C ∞ ( M )) is a Loday pair.b-1) Let L → M be a vector bundle over M with a bundle map ρ : L →
T M . L iscalled a Leibniz algebroid (Ibanez and collaborators [9]), if the space of sections Γ L has a Loday bracket satisfying [ X , f X ] = f [ X , X ]+ ρ ( X )( f ) X , where X , X ∈ Γ L and f ∈ C ∞ ( M ). When L is a Leibniz algebroid, the pair (Γ L , C ∞ ( M )) is aLoday pair.The place of Loday pairs among other objects may be illustrated by the followingtable. g \ A commutative noncommutative dimensioncommutative Poisson algebras classical Leibniz pairs 2Lie algebroidsnoncommutative Courant algebroids Loday pairs 3Leibniz algebroidsLoday pairs are noncommutative analogues of Courant/Leibniz algebroids.We introduce a coalgebra description of Loday pairs, and then study higher de-rived bracket construction in the category Loday pairs (see Section 4). The Leibnizpairs up to homotopy which are called sh Leibniz pairs are studied by Kajiura-Stasheff [13, 14] and by Hoefel [8] in the context of open-closed string field theory.We introduce a new type of homotopy algebra, sh Loday pair , which is consideredas a noncommutative analogue of Leibniz pair. We show that sh associative/Lodayalgebras are both subalgebras of sh Loday pairs. The higher derived brackets in thecatrogy of Loday pairs are defined by n i + j ( x , ..., x i , a , ..., a j ) := ( ± )[[ ... [[[ ... [ δ i + j − x , x ] , ... ] , x i ] , a ] , ... ] , a j ] , where x · ∈ L , a · ∈ A and [ , ] is a multiplication on a Loday pair. The higher derivedbrackets and the higher derived products are both subsystem of { n i + j } . The secondmain result of this note is as follows. Let ( L, A, δ ) be a Loday pair ( L, A ) withdifferential δ and let d := P i ≥ t i δ i be a deformation of δ . We prove in Proposition4.12 that the system with the unified higher derived brackets ( n , n , n , ... ) is ansh Loday pair. Acknowledgements . I would like to say thank to professors Jim Stasheff andAkira Yoshioka for their kind advices. otations and Assumptions . In the following, we assume that the characteristicof the ground field K is zero and that a tensor product is defined over the field, ⊗ := ⊗ K . We follow the Koszul sign convention. For instance, a linear map f ⊗ g : V ⊗ V → V ⊗ V satisfies, for any v ⊗ v ∈ V ⊗ V ,( f ⊗ g )( v ⊗ v ) = ( − | g || v | f ( v ) ⊗ g ( v ) , where | g | and | v | are degrees of g and v . We will use a degree shifting operator s (resp. s − ) with degree +1 (resp. − s ⊗ s =( s ⊗ ⊗ s ) = − (1 ⊗ s )( s ⊗ − o the sign ( − | o | without missreading. We consider the tensor space over a graded vector space V :¯ T V := V ⊕ V ⊗ ⊕ · · · . The space ¯
T V has an associative coalgebra structure, ∆ : ¯
T V → ¯ T V ⊗ ¯ T V , definedby ∆( V ) := 0 and ∆( v , ..., v n ) := n X i =1 ( v , ..., v i ) ⊗ ( v i +1 , ..., v n ) , (3)where v i ∈ V . Then ( ¯ T V, ∆) becomes a cofree coalgebra in the category of nilpotent coalgebras. Let Coder( ¯
T V ) be the space of coderivations, i.e., D c ∈ Coder( ¯
T V )satisfies the coderivation rule:( D c ⊗ ⊗ D c )∆ = ∆ D c . It is well-known that Coder( ¯
T V ) is identified with the space of the endomorphismson V : Hom( ¯ T V, V ) ∼ = Coder( ¯ T V ) . (4)We recall an explicit formula of the isomorphism. For a given i -ary endomorphism f : V ⊗ i → V , we define a coderivation f c by f c ( V n
T V, V ) has a Lie bracket which is induced by the isomorphism (4). The nduced Lie bracket on Hom( ¯ T V, V ), which is denoted by { f, g } , is well-known as a Gerstenhaber bracket on a Hochschild complex. If sV (the shifted space of V ) is anassociative algebra, then Hom( ¯ T V, V ) becomes a Hochschild complex: · · · b → Hom( V ⊗ n , V ) b → Hom( V ⊗ n +1 , V ) b → · · · . The coboundary map b is induced by the associative structure on sV (see Remark2.2).If f c , g c are coderivations associated with i -ary, j -ary endomorphisms, respec-tively, then the Lie bracket [ f c , g c ] is the coderivation associated with the Gersten-haber bracket of f and g , i.e., { f, g } c = [ f c , g c ] , where { f, g } is an ( i + j − T V ) with Hom( ¯
T V, V ). Hence we omit thesubscript “ c ” from f c without miss reading. Definition 2.1.
Let sV be the shifted space equipped with a collection of i ( ≥ -aryendomorphisms, m i : ( sV ) ⊗ i → sV . We assume that the degree of m i is − i foreach i . We set the shifted map: ∂ i := s − ◦ m i ◦ ( i z }| { s ⊗ · · · ⊗ s ) . This is an element in
Hom( ¯
T V, V ) or in Coder( ¯
T V ) up to the identification. Wedefine a coderivation by ∂ := ∂ + ∂ + · · · . The system ( sV, m , m , ... ) is called a strong homotopy (sh) associative algebra, orsometimes called an A ∞ -algebra, if ∂ is square zero, or equivalently,
12 [ ∂, ∂ ] = 0 . Remark 2.2.
The usual associative algebra can be seen as a special sh associativealgebra such that ∂ n =2 = 0 . In such a case, we put b ( − ) := [ ∂ , − ] . Then b becomesthe coboundary map of the Hochschild complex. Let ( A, ∗ , δ ) be a differential graded (dg) associative algebra. We consider a defor-mation of δ : d := δ + tδ + t δ + · · · . he deformation d is a square zero derivation on A [[ t ]]. The square zero conditionof d is equivalent to the following condition. X i + j = Const δ i δ j = 0 . (5)We define the higher derived products on sA by m i := ( − ( i − i − s ◦ M i ◦ ( i z }| { s − ⊗ · · · ⊗ s − )( sδ i − s − ⊗ i − z }| { ⊗ · · · ⊗ , where M i ( a , ..., a i ) := a ∗ a ∗···∗ a i for any a , ..., a i ∈ A . By a direct computation,we have m i ( sa , ..., sa i ) = ( ± ) s (cid:16) δ i − a ∗ a ∗ · · · ∗ a i (cid:17) , where ± = ( ( − a + a + ··· + a n +1 + ··· i = even , ( − a + a + ··· + a n + ··· i = odd . The main theorem of this note is as follows.
Theorem 3.1.
The system with the higher derived products ( sA, m , m , ... ) is ansh associative algebra. We need some lemmas in order to show this theorem.
Lemma 3.2.
Let ∂ i be the coderivation associated with the higher derived product m i . Then ∂ i has the following form. ∂ i = M i ◦ ( δ i − ⊗ i − z }| { ⊗ · · · ⊗ . Proof. ∂ i := s − ◦ m i ◦ ( s ⊗ · · · ⊗ s )= ( − ( i − i − M i ◦ ( s − ⊗ · · · ⊗ s − )( sδ i − s − ⊗ ⊗ · · · ⊗ s ⊗ · · · ⊗ s )= ( − ( i − i − M i ◦ ( s − ⊗ · · · ⊗ s − )( sδ i − ⊗ s ⊗ · · · ⊗ s )= ( − ( i − i − ( − i − M i ◦ ( s − ⊗ · · · ⊗ s − )( s ⊗ s ⊗ · · · ⊗ s )( δ i − ⊗ ⊗ · · · ⊗ − ( i − i − ( − i − ( − i ( i − M i ◦ ( δ i − ⊗ ⊗ · · · ⊗ . Let Der( A ) be the space of derivations on the algebra ( A, ∗ ). For any D ∈ Der( A ), we define an i -ary map by M i D := M i ◦ ( D ⊗ i − z }| { ⊗ · · · ⊗ , in particular, M D = D . One can identify M i D with a coderivation in Coder( ¯ T A ). emma 3.3. For any
D, D ′ ∈ Der( A ) and for any i, j , the Lie bracket of thecoderivations, [ M i D, M j D ′ ] , is compatible with the one of the derivations, [ D, D ′ ] ,namely, [ M i D, M j D ′ ] = M i + j − [ D, D ′ ] . Proof.
We assume for the sake of simplicity that the variables have no degree. Weput a = ( a , ..., a i + j − ) ∈ A ⊗ ( i + j − . Then we have M i D ◦ M j D ′ ( a ) = i − X s =0 Da ∗ · · · ∗ D ′ a s +1 ∗ · · · ∗ a i + j − = D (cid:0) D ′ a ∗ · · · ∗ a j (cid:1) ∗ · · · ∗ a i + j − + i − X s =1 Da ∗ · · · ∗ D ′ a s +1 ∗ · · · ∗ a i + j − = ( DD ′ a ) ∗ · · · ∗ a i + j − + ( − DD ′ j X t =2 D ′ a ∗ · · · ∗ Da t ∗ · · · ∗ a i + j − + i − X s =1 Da ∗ · · · ∗ D ′ a s +1 ∗ · · · ∗ a i + j − . On the other hand, we have M j D ′ ◦ M i D ( a ) = j X t =1 D ′ a ∗ · · · ∗ Da t ∗ · · · ∗ a i + j − = D ′ (cid:0) Da ∗ · · · ∗ a i (cid:1) ∗ · · · ∗ a i + j − + j X t =2 D ′ a ∗ · · · ∗ Da t ∗ · · · ∗ a i + j − = ( D ′ Da ) ∗ · · · ∗ a i + j − + ( − D ′ D i − X s =1 Da ∗ · · · ∗ D ′ a s +1 ∗ · · · ∗ a i + j − + j X t =2 D ′ a ∗ · · · ∗ Da t ∗ · · · ∗ a i + j − . Hence we obtain[ M i D, M j D ′ ]( a ) = ( DD ′ a ) ∗ · · · ∗ a i + j − − ( − DD ′ ( D ′ Da ) ∗ · · · ∗ a i + j − = M i + j − [ D, D ′ ]( a ) . We give a proof of Theorem 3.1:
Proof.
The higher derived product m i corresponds to the coderivation ∂ i = M i δ i − .The deformation condition [ d, d ] / X i + j = Const [ ∂ i , ∂ j ] = X i + j = Const [ M i δ i − , M j δ j − ] = M i + j − X i + j = Const [ δ i − , δ j − ] = 0 . e consider the special case of m n =2 = 0, namely, the case of trivial deformation: d = tδ . Corollary 3.4.
Assume that m n =2 = 0 , or equivalently, sA is the usual associativealgebra with the binary derived product. Then the collection of { M i Der( A ) } is asubcomplex of the Hochschild complex Hom( ¯
T A, A ) , where M i Der( A ) := h M i D | D ∈ Der( A ) i . Proof.
The coboundary map on Hom( ¯
T A, A ) is given by b ( − ) := [ ∂ , − ] = [ M δ , − ] . Hence we obtain b ( M i D ) = M i +1 [ δ , D ]. We discuss a relationship between deformation theory and sh associative algebras.The main result of this subsection is Proposition 3.5 below. A Loday algebra versionof this proposition was shown in [22].The deformation of δ , d = δ + tδ + · · · , is considered as a differential on A [[ t ]] which is an associative algebra of formal series with coefficients in A . Let h ( t ) := th + t h + · · · be a derivation on the associative algebra A [[ t ]] with degree | h ( t ) | := 0. We consider the second deformation d ′ = P t n δ ′ n . The deformations d and d ′ are equivalent, if they are related via the gauge transformation: d ′ := exp ( X h ( t ) )( d ) , where X h ( t ) := [ − , h ( t )]. We denote by ∂ ′ = P ∂ ′ n the induced sh associativestructure associated with d ′ . Proposition 3.5. If d and d ′ are gauge equivalent, then the sh associative structures ∂ and ∂ ′ are equivalent, namely, ∂ ′ = exp ( X Mh )( ∂ ) , where M h is a well-defined infinite sum of coderivations:
M h := M h + M h + · · · + M i +1 h i + · · · , nd the integral of M h , e Mh := 1 + M h + 12! (
M h ) + · · · , is a dg coalgebra isomorphism between ( ¯ T A, ∂ ) and ( ¯ T A, ∂ ′ ) , namely, (6) and (7)below hold. ∂ ′ = e − Mh · ∂ · e Mh , (6)∆ e Mh = ( e Mh ⊗ e Mh )∆ . (7) Proof.
The proof of this proposition is the same as the one in [22].In general, an A ∞ -morphism is defined to be a dg coalgebra morphism between( ¯ T A, ∂ ) and ( ¯
T A ′ , ∂ ′ ). Hence e Mh is an A ∞ -isomorphism. We introduce the concept “Loday pair” and study its homotopy algebras.
We recall sh Loday algebras. Let L be a graded vector space and let sL be theshifted space and let l i : ( sL ) ⊗ i → sL be a multilinear map with degree 2 − i , foreach i ≥ Definition 4.1. ([2], and see also [22]) The system with the multiplications, ( sL, l , l , ... ) ,is called a strong homotopy (sh) Loday algebra (Loday ∞ -algebra or sh Loday algebraor Loday ∞ -algebra), if the collection { l i } i ≥ satisfies (8) below. X i + j = Const i + j − X k = j X σ χ ( σ )( − ( k +1 − j )( j − ( − j ( sx σ (1) + ... + sx σ ( k − j ) ) l i ( sx σ (1) , ..., sx σ ( k − j ) , l j ( sx σ ( k +1 − j ) , ..., sx σ ( k − , sx k ) , sx k +1 , ..., sx i + j − ) = 0 , (8) where ( sx , ..., sx i + j − ) ∈ sL ⊗ ( i + j − , σ is a ( k − j, j − -unshuffle, χ ( σ ) is ananti-Koszul sign, χ ( σ ) := sgn ( σ ) ǫ ( σ ) . Sh Lie algebras are special examples of sh Loday algebras such that all l i ( i ≥ l i ≥ is skewsymmetric, then l i ( sx σ (1) , ..., sx σ ( k − j ) , l j ( sx σ ( k +1 − j ) , ..., sx σ ( k − , sx k ) , sx k +1 , ..., sx i + j − ) =( ± ) l i ( l j ( sx σ ( k +1 − j ) , ..., sx σ ( k − , sx k ) , sx σ (1) , ..., sx σ ( k − j ) , sx k +1 , ..., sx i + j − ) =( ± ) l i ( l j ( sx τ (1) , ..., sx τ ( j ) ) , sx τ ( j +1) , ..., sx τ ( i + j − ) , here τ is an unshuffle permutation. And P i + j − k = j P σ changes into P τ . Then (8)becomes, X i + j = Const X τ ( ± ) l i ( l j ( sx τ (1) , ..., sx τ ( j ) ) , sx τ ( j +1) , ..., sx τ ( i + j − ) = 0 . This is the defining relation of sh Lie algebras.The cofree nilpotent dual-Loday coalgebra over L is the tensor space ¯ T L witha comultiplication, ∆ L : ¯ T L → ¯ T L ⊗ ¯ T L , defined by ∆ L ( L ) := 0 and∆ L ( x , ..., x n +1 ) := n X i =1 X σ ǫ ( σ )( x σ (1) , x σ (2) , ..., x σ ( i ) ) ⊗ ( x σ ( i +1) , ..., x σ ( n ) , x n +1 ) , (9)where ǫ ( σ ) is a Koszul sign, σ is an ( i, n − i )-unshuffle. Let Coder( ¯ T L ) be the spaceof coderivations on ¯
T L . By a standard argument, we obtain an isomorphism:Coder( ¯
T L ) ∼ = Hom( ¯ T L, L ) . We recall an explicit formula of the isomorphism. Let f : L ⊗ i → L be an i -ary map.It is one of the generators in Hom( ¯ T L, L ). The coderivation associated with f isdefined by f c ( L ⊗ n
L, A ). Set a tensor space: LA := X n ≥ X i + j = n L ⊗ i ⊗ A ⊗ j . Define a comultiplication ∆ on LA by the same manner as (9). For instance,∆( x, a , a , a ) = x ⊗ ( a , a , a ) ± a ⊗ ( x, a , a ) ± a ⊗ ( x, a , a ) ± ( x, a ) ⊗ ( a , a ) ± ( x, a ) ⊗ ( a , a ) ± ( a , a ) ⊗ ( x , a ) ± ( x , a , a ) ⊗ a , (13)where x ∈ L and a , a , a ∈ A . The space of coderivations on ( LA, ∆), Coder(
LA, ∆),is identified with a subspace of Hom(
LA, L ⊕ A ). This identification is defined bythe same rule as (10). For instance, if a binary map f : P i + j =2 L ⊗ i ⊗ A ⊗ j → L ⊕ A corresponds to a coderivation, then it satisfies f c ( x, a , a , a ) = ( f ( x, a ) , a , a ) ± ( x, f ( a , a ) , a ) ± ( a , f ( x, a ) , a ) ± ( x, a , f ( a , a )) ± ( x, a , f ( a , a )) ± ( a , a , f ( x, a )) . (14) The word “regular” is used in the sense of σ = id , i.e., the order of variables is regular. e notice that f ( x, a ) and f ( a , a ) are A -valued, because the elements of A mustbe put on the right side of the elements of L . By a simple observation, we obtainCoder( LA, ∆) ∼ = Hom( ¯ T L, L ) ⊕ X i ≥ ,j ≥ Hom( L ⊗ i ⊗ A ⊗ j , A ) . (15)We are regularizing ∆ with respect to the order of variables of A (we say thisoperation an A -regularization). For instance, the A -regularization of (13) is∆( x, a , a , a ) A -regular ∼ x ⊗ ( a , a , a ) ± a ⊗ ( x, a , a ) ± ( x, a ) ⊗ ( a , a ) ± ( a , a ) ⊗ ( x , a ) ± ( x , a , a ) ⊗ a , where the ordering a < a < a is preserved. We denote by reg (∆) the regularizedcomultiplication. It generally has the form: reg (∆)( x , a ) := X i + k X σ ( ± )( x σ (1) , ..., x σ ( i ) , a , ..., a k ) ⊗ ( x σ ( i +1) , ..., x σ ( m ) , a k +1 , ..., a n ) , where x = ( x , ..., x m ), a = ( a , ..., a n ) and σ is an ( i, m − i )-unshuffle. Remark thatthe restrictions reg (∆) | L and reg (∆) | A coincide with the classical comultiplicationsabove, respectively. Let D c be a coderivation on the coalgebra ( LA, ∆). We definethe A -regularization of D c , reg ( D c ), by regularizing the order of variables of A . Forinstance, the regularization of (14) becomes reg ( f c )( x, a , a , a ) = ( f ( x, a ) , a , a ) ± ( a , f ( x, a ) , a ) ± ( x, f ( a , a ) , a ) ± ( x, a , f ( a , a )) ± ( a , a , f ( x, a )) . Lemma 4.2.
The regularization reg ( D c ) is a coderivation on ( LA, reg (∆)) , thatis, (cid:0) reg ( D c ) ⊗ ⊗ reg ( D c ) (cid:1) reg (∆) = reg (∆) reg ( D c ) . Proof.
Apply reg on the both-side of (cid:0) D c ⊗ ⊗ D c (cid:1) ∆ = ∆ D c . We have reg (cid:16) ( D c ⊗ (cid:17) = reg (cid:16) ( D c ⊗ reg (∆) (cid:17) = ( reg ( D c ) ⊗ reg (∆) and reg (cid:16) (1 ⊗ D c )∆ (cid:17) = reg (cid:16) (1 ⊗ D c ) reg (∆) (cid:17) = (1 ⊗ reg ( D c )) reg (∆). Hence we obtain reg (cid:16)(cid:0) D c ⊗ ⊗ D c (cid:1) ∆ (cid:17) = (cid:0) reg ( D c ) ⊗ ⊗ reg ( D c ) (cid:1) reg (∆) . On the other hand, we obtain reg (∆ D c ) = reg (∆ reg ( D c )) = reg (∆) reg ( D c ).Therefore we get the identity of the lemma.The space of coderivations on the regularized coalgebra ( LA, reg (∆)) also cor-responds to the same homomorphism space as (15). The above lemma implies thatthe correspondence is the A -regularization of the isomorphism in (15). .4 Unified derived brackets Let L be a Loday algebra and let A be an associative algebra. We assume that thedegrees of the multiplications on ( L, A ) are both zero (or even).
Definition 4.3.
The pair ( L, A ) equipped with a binary multiplication [ , ] : L ⊗ A → A is called a left-Loday pair, or simply, Loday pair, if it satisfies [ x, [ y, a ]] = [[ x, y ] , a ] + ( − xy [ y, [ x, a ]] , (16)[ x, [ a, b ]] = [[ x, a ] , b ] + ( − xa [ a, [ x, b ]] , (17) where x, y ∈ L , a, b ∈ A and where [ x, y ] is the Loday bracket on L and [ a, b ] is theassociative multiplication on A , i.e., [ a, [ b, c ]] = [[ a, b ] , c ] for any a, b, c ∈ A . The Loday pairs are algebraizations of Leibniz algebroids ([9]). The classicalLeibniz pair in [5] is the Lie version of our noncommutative Leibniz pair (i.e. Lodaypair). We give two geometric examples of Loday pairs.
Example 4.4. (Courant bracket) Let M be a smooth manifold. Consider a bundle T M := T M ⊕ T ∗ M (so-called generalized tangent bundle). The Courant bracket isdefined on the space of sections of T M , Γ T M , by [ ξ + θ , ξ + θ ] := [ ξ , ξ ]+ L ξ θ − i ξ dθ , where ξ , ξ ∈ Γ T M and θ , θ ∈ Γ T ∗ M . Then the pair (Γ T M, C ∞ ( M )) isa Loday pair. Example 4.5.
Let ( M, π ) be a Poisson manifold equipped with a Poisson structuretensor π . The space of multivector fields Γ V · T M becomes a graded Poisson algebraof type ( − , , whose Poisson bracket is known as Schouten-Nijenhuis (SN) bracket.The Poisson tensor is a solution of Maurer-Cartan equation [ π, π ] SN = 0 . Sincethe degree of π is +2 , d := [ π, − ] SN becomes a differential with degree +1 . Thisdifferential is the coboundary operator of the Poisson cohomology. We define aLoday bracket by [ X, Y ] π := ( − X [ dX, Y ] SN for any X, Y ∈ Γ V · T M . The bracket [ , ] π is the derived bracket of SN -bracket by the Poisson structure. Then the self pair (Γ V · T M, Γ V · T M ) is a Loday pair with multiplications [ , ] π and ∧ . In particular,the sub-pair ( C ∞ ( M ) , C ∞ ( M )) is the self pair of the Poisson algebra on M . We get a natural result.
Corollary 4.6.
The structure of Loday pair on ( sL, sA ) is equivalent to a (binary)codifferential on the regularized coalgebra ( LA, reg (∆)) . This corollary leads us to efinition 4.7. An sh (left) Loday pair is, by definition, a pair ( sL, sA ) equippedwith a codifferential on ( LA, reg (∆)) . We consider a derived bracket construction in the category of Loday pairs.
Definition 4.8.
A derivation on a Loday pair ( L, A ) is, by definition, a pair ofderivations, D = ( D L , D A ) , D L ∈ Der( L ) and D A ∈ Der( A ) satisfying D [ o , o ] = [ Do , o ] + ( − Do [ o , Do ] for any o , o ∈ ( L, A ) . We assume that the parity of D L is equal with the one of D A . Example 4.9. If ( L, A ) is a Loday pair, then an adjoint action [ x, − ] , x ∈ L , is aderivation. A Loday pair (
L, A ) is called a dg Loday pair, if it has a differential δ on ( L, A ).It is easy to check that dg Loday pairs are special sh Loday pairs such that thehigher homotopies vanish. Given a dg Loday pair (
L, A, δ ), define derived bracketsby [ sx, sy ] d := ( − x s [ δx, y ] , [ sx, sa ] d := ( − x s [ δx, a ] , [ sa, sb ] d := ( − a s [ δa, b ] . Then the triple of the derived brackets provides a new structure of Loday pair on( sL, sA ). Example 4.10.
In Example 4.5, if π ′ is a second Poisson tensor which is compatiblewith π , i.e., [ π, π ′ ] SN = 0 , then δ := [ π ′ , − ] SN is a differerential on the Loday pair. We consider the higher derived bracket construction for Loday pairs. Let D bea derivation on a Loday pair ( L, A ). We put N k D ( x , a ) := [[ ... [[[ ... [ Dx , x ] , ... ] , x i ] , a i +1 ] , ... ] , a i + j ] , (18)where x := ( x , ..., x i ), a := ( a i +1 , ..., a i + j ) and k := i + j , in particular, N k D ( a ) = M k D ( a ). Lemma 4.11.
We regard N · D as a coderivation on ( LA, reg (∆)) . For any deriva-tions
D, D ′ on ( L, A ) and for any k, l ≥ , [ N k D, N l D ′ ] = N k + l − [ D, D ′ ] . e will give a proof of this lemma in the end of this section. The main resultof this section is as follows. Proposition 4.12.
Let ( L, A, δ ) be a dg Loday pair. We consider a deformationof δ , d := P i ≥ t i δ i . For each k ≥ , define a coderivation by ∂ k := N k δ k − . Then ∂ := P k ∂ k is a structure of sh Loday pair.The multilinear map N k δ k − ( x , a ) corresponds to the higher bracket on the siftedpair ( sL, sA ) : n k ( sx , ..., sx i , sa i +1 , ..., sa i + j ) := ( ± ) s [[ ... [[[ ... [ δ i + j − x , x ] , ... ] , x i ] , a i +1 ] , ... ] , a i + j ] , where k := i + j , k ≥ and ± := ( ( − o + o + ··· + o n +1 + ··· i + j = even , ( − o + o + ··· + o n + ··· i + j = odd . and where o · ∈ { x · , a · } . The restrictions ( sL, n | sL ) and ( sA, n | sA ) become an shLoday algebra and an sh associative algebra, respectively. We give a proof of Lemma 4.11. To show this lemma we use convenient symboles:[ x , ..., x i ] := [[[ x , x ] , ... ] , x i ] ,D x := [ Dx , ..., x i ] , where x := ( x , ..., x i ). The pure Loday version of the lemma was shown in [22].We consider the mixed case. Since the adjoint action b L := [ L, − ] is a derivation on A , (18) becomes N k D ( x , a ) = ( M j d D x )( a ) , We denote by | · | the length of word.
Lemma 4.13.
Assume that | ( x , a ) | := k + l − and | a | ≥ . [ N k D, N l D ′ ]( x , a ) = X ( x , x ) M | a | [ d D x , [ D ′ x ]( a ) , (19) where ( x , x ) runs over the unshuffle-permutations including ( ∅ , x ) and ( x , ∅ ) .Proof. N l D ′ ( x , a ) is decomposed into the pure Loday term (if it exists) and themixed term: N l D ′ ( x , a ) = X | x | = l ( x , D ′ x , x , a ) + X | x | A, B ∈ L and for any y := ( y , ..., y n ) ∈ L ⊗ n ,by the Leibniz rule, we have[[ A, B, y ] , − ] = X ( y , y ) [[[ A, y ] , [ B, y ]] , − ] here ( y , y ) are the unshuffle permutations of y including ( ∅ , y ) and ( y , ∅ ). Now,replace A → D x , B → D ′ x and y → x . Then X | x | = l [ D x , D ′ x , x , a ] = X | x | = l [[ D x , y ] , [ D ′ x , y ] , a ]= X | x | = l [ D ( x , y ) , D ′ ( x , y ) , a ]= X | x |≥ l [[ D x , D ′ x ] , a ]= X | x |≥ l M | a | \ [ D x , D ′ x ]( a ) = X | x |≥ l M | a | [ d D x , [ D ′ x ]( a )where x i := ( x i , y i ) redefined i ∈ { , } . The other term is, by the same manner, − X | x | = k [ D ′ x , D x , x , a ] = X | x |≥ k M | a | [ d D x , [ D ′ x ]( a ) . This implies (20). References [1] F. Akman. On Some Generalizations of Batalin-Vilkovisky Algebras. Journal ofPure and Applied Algebra. 120 (1997), no. 2, 105–141.[2] M. Ammar and N. Poncin. Coalgebraic Approach to the Loday Infinity Category,Stem Differential for 2 n -ary Graded and Homotopy Algebras. Preprint Arxive,math/0809.4328.[3] AS. Cattaneo and G. Felder. A path integral approach to the Kontsevich quan-tization formula. Comm. Math. Phys. 212 (2000), no. 3, 591–611.[4] AS. Cattaneo and G. Felder. On the AKSZ formulation of the Poisson sigmamodel. EuroConference Moshe Flato 2000, Part II (Dijon). Lett. Math. Phys. 56(2001), no. 2, 163–179.[5] M. Flato, M. Gerstenhaber and A.A. Voronov. Cohomology and deformation ofLeibniz pairs. Lett. Math. Phys. 34 (1995), no. 1, 77–90.[6] M. Doubek, M. Markl and P. Zima. Deformation theory (lecture notes). Arch.Math. (Brno) 43 (2007), no. 5, 333–371.[7] V. Ginzburg and M. Kapranov. Koszul duality for operads. Duke Math. J. 76(1994), no. 1, 203–272.[8] E. Hoefel. On the Coalgebra Description of OCHA. arXiv:math/0607435v2. 9] R. Ibanez, M. de Leon, J.C. Marrero and E. Padron. Leibniz algebroid associatedwith a Nambu-Poisson structure. J. Phys. A 32 (1999), no. 46, 8129–8144.[10] N. Ikeda. Topological Field Theories and Geometry of Batalin-Vilkovisky Al-gebras. J. High Energy Phys. 0210 (2002) 076.[11] N. Ikeda. On the construction of topological field theory and quantization.Seminar at Kagoshima University. (2007/11/22).[12] N. Ikeda. Topological Field Theory, AKSZ-formalism and Higher Poisson Struc-ture. Seminar at Akita University. (2008/11/12).[13] H. Kajiura and J. Stasheff. Homotopy algebras inspired by classical open-closedstring field theory. Comm. Math. Phys. 263 (2006), no. 3, 553–581.[14] H. Kajiura and J. Stasheff. Homotopy algebras of open-closed strings. Groups,homotopy and configuration spaces, 229–259, Geom. Topol. Monogr., 13, Geom.Topol. Publ., Coventry, 2008.[15] Y. Kosmann-Schwarzbach. From Poisson algebras to Gerstenhaber algebras.(English, French summary) Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1243–1274.[16] Y. Kosmann-Schwarzbach. Derived brackets. Lett. Math. Phys. 69 (2004), 61–87.[17] J-L. Loday. Dialgebras. Lecture Notes in Mathematics, 1763. Springer-Verlag,Berlin, (2001), 7–66.[18] M. Markl. Homotopy algebras via resolutions of operads. Preprint Arxive,math/9808101.[19] M. Markl, S. Shnider and J. Stasheff. Operads in algebra, topology and physics.Mathematical Surveys and Monographs, 96. American Mathematical Society,Providence, RI, (2002). x+349 pp.[20] D. Roytenberg. Quasi-Lie bialgebroids and twisted Poisson manifolds. Lett.Math. Phys. 61 (2002), no. 2, 123–137.[21] D. Roytenberg. AKSZ-BV formalism and Courant algebroid-induced topolog-ical field theories. Lett. Math. Phys. 79 (2007), no. 2, 143–159.[22] K. Uchino. Derived brackets and sh Leibniz algebras. (submitted).arXiv:0904.1961.[23] K. Uchino. Derived bracket construction and Manin products. (submitted).arXiv:0902.0044. 24] J-A. Vallejo. Nambu-Poisson manifolds and associated n -ary Lie algebroids. J.Phys. A. 34 (2001), no. 13, 2867–2881.[25] T. Voronov. Higher derived brackets and homotopy algebras. J. Pure Appl.Algebra 202 (2005), no. 1-3, 133–153.Post doctoral.Tokyo University of Science.3-14-1 Shinjyuku Tokyo Japan.e-mail: K Uchino[at]oct.rikadai.jp-ary Lie algebroids. J.Phys. A. 34 (2001), no. 13, 2867–2881.[25] T. Voronov. Higher derived brackets and homotopy algebras. J. Pure Appl.Algebra 202 (2005), no. 1-3, 133–153.Post doctoral.Tokyo University of Science.3-14-1 Shinjyuku Tokyo Japan.e-mail: K Uchino[at]oct.rikadai.jp