Uniqueness and intrinsic properties of non-commutative Koszul brackets
aa r X i v : . [ m a t h . QA ] M a y UNIQUENESS AND INTRINSIC PROPERTIES OFNON-COMMUTATIVE KOSZUL BRACKETS
MARCO MANETTI
Abstract.
There exists a unique natural extension of higher Koszul brackets to everyunitary associative algebra in a way that every square zero operator of degree 1 gives acurved L ∞ structure. Introduction
The name (higher, commutative) Koszul brackets is usually referred to the sequence ofgraded symmetric maps Ψ nf : A ⊙ n → A , n ≥
0, defined in [14] for every graded commutativeunitary algebra A and every linear endomorphism f : A → A by the formula:Ψ f = f (1) , Ψ f ( a ) = f ( a ) − f (1) a, Ψ f ( a, b ) = f ( ab ) − f ( a ) b − ( − | a || b | f ( b ) a + f (1) ab ...Ψ nf ( a , . . . , a n ) = n X k =0 ( − n − k k !( n − k )! X π ∈ Σ n ǫ ( π ) f (1 · a π (1) · · · a π ( k ) ) a π ( k +1) · · · a π ( n ) , (0.1)where ǫ ( π ) is the Koszul sign of the permutation π with respect to the sequence of homo-geneous elements a , . . . , a n . As proved in [6, 15, 22] they have the remarkable propertyof satisfying the generalized Jacobi identities of Lada and Stasheff [16], and therefore theyare applied in the study of L ∞ -algebras, of (commutative) Batalin-Vilkovisky algebras andtheir deformations.The question of extending their definition to every unitary graded associative algebra,preserving generalized Jacobi identities, is a nontrivial task and has been first answered byBering [7] about ten years ago. Very recently, other solutions, quite different in their originand presentation, are proposed by Bandiera [3, 4] and by Manetti and Ricciardi [17].Apart from the natural question whether the above mentioned non-commutative exten-sion formulas coincide or not, the main goal of this paper is to determine a minimal set ofconditions which implies existence and unicity of non-commutative Koszul brackets.A similar goal has been recently achieved by Markl in the paper [18], where it is provedthat both hierarchies of B¨orjeson brackets and commutative Koszul brackets are the uniquenatural hierarchies of brackets satisfying the technical conditions called hereditarity, recur-sivity and with fixed initial terms. Strictly speaking, this viewpoint does not apply to ourgoal since it is easy to see that a non-commutative hereditary extension of Koszul bracketscannot satisfy generalized Jacobi identities; however Markl’s work has certainly inspiredthis paper. Date : May 6, 2016.2010
Mathematics Subject Classification.
Key words and phrases.
Graded Lie algebras, Koszul brackets.
The point of view which we adopt in this paper is based on the slogan that “the mostimportant properties for a hierarchy of brackets are naturality, base change and generalizedJacobi formulas”. Precise definitions will be given in Theorem 2.1; here we only mentionthat Markl’s notion of naturality is essentially equivalent to the join of our notions ofnaturality and base change.Whereas in the previous literature on the subject the starting point is Koszul’s definitionin the commutative case, the middle point is the proposal of a non-commutative extensionand the conclusive point is the proof of generalized Jacobi identities, in this paper we reversethe logical path: we start with a generic hierarchy satisfying naturality, base change andgeneralized Jacobi, and then we add assumptions on the initial terms until we reach theunicity. Quite surprisingly, this approach goes very smoothly and provides, in our opinion,a simplification of the theory also when restricted to the classical commutative case.The paper is organized as follows: in Section 1 we fix notation and we recall the definitionof the Nijenhuis-Richardson bracket, in terms of which the generalized Jacobi identities canbe expressed in their simplest form. Section 2 is completely devoted to the proof of theuniqueness theorem of Koszul brackets, whose first properties, including their restrictionto the commutative case, are studied in Section 3. In Section 4 we shall prove that thenon-commutative Koszul brackets may be also explicitly described by the formulas givenin [3, 7]. The last section is devoted to a discussion about the reduction of Koszul bracketsto non-unitary graded associative algebras.
Acknowledgments.
The author thanks the referee for several useful comments and ac-knowledges partial support by Italian MIUR under PRIN project 2012KNL88Y “Spazi dimoduli e teoria di Lie”. 1.
General setup
The symmetric group of permutations of n elements is denoted by Σ n . Every gradedvector space, every graded algebra and every tensor product is intended Z -graded and overa fixed field K of characteristic 0. For every graded vector space V we shall denote by V ⊙ n its n th symmetric power: for simplicity of notation we always identify a linear map f : V ⊙ n → W with the corresponding graded symmetric operator f : V × · · · × V | {z } n factors → W, f ( v , . . . , v n ) = f ( v ⊙ · · · ⊙ v n ) . To every graded vector space V we shall consider the following graded Lie algebras:(1) the algebra of linear endomorphisms:End ∗ ( V ) = Hom ∗ K ( V, V ) = M n ∈ Z Hom n K ( V, V ) , equipped with the graded commutator bracket;(2) the space of affine endomorphisms:Aff ∗ ( V ) = M n ∈ Z Aff n ( V ) = { f ∈ End ∗ ( V ⊕ K ) | f ( V ⊕ K ) ⊆ V } , considered as a graded Lie subalgebra of End ∗ ( V ⊕ K ).Thus, giving an element f ∈ Aff n ( V ) = { f ∈ End n ( V ⊕ K ) | f ( V ⊕ K ) ⊆ V } is the sameas giving a linear map g i : V i → V i + n , g i ( v ) = f ( v ), for every i = 0 and an affine map g : V → V n , g ( v ) = f ( v + 1). NIQUENESS AND INTRINSIC PROPERTIES OF NON-COMMUTATIVE KOSZUL BRACKETS 3
It is useful to consider both End ∗ ( V ) and Aff ∗ ( V ) as graded Lie subalgebras of D ( V ) = Y n ≥− D n ( V ) , D − ( V ) = V, D n ( V ) = Hom ∗ K ( V ⊙ n +1 , V ) , where the Lie structure on D ( V ) is given by the Nijenhuis-Richardson bracket, inducedby the right pre-Lie product ⊼ defined in the following way [20]: given f ∈ D n ( V ) and g ∈ D m ( V ) the operator f ⊼ g ∈ D n + m ( V )is equal to:(1) f ⊼ g = 0 whenever f ∈ D − ( V ) = V ;(2) f ⊼ g ( v , . . . , v n ) = f ( g, v , . . . , v n ) whenever g ∈ D − ( V ) = V ;(3) when n, m ≥ f ⊼ g ( v , . . . , v n + m ) = X σ ∈ S ( m +1 ,n ) ǫ ( σ ) f ( g ( v σ (0) , . . . , v σ ( m ) ) , v σ ( m +1) , . . . , v σ ( m + n ) ) . Here S ( m + 1 , n ) ⊂ Σ n + m +1 is the set of shuffles of type ( m + 1 , n ), i.e., the set ofpermutations σ of 0 , . . . , n + m such that σ (0) < · · · < σ ( m ) and σ ( m +1) < · · · < σ ( m + n ).The Koszul sign ǫ ( σ ) is equal to ( − α , where α is the number of pairs ( i, j ) such that i < j , σ ( i ) > σ ( j ) and | v i || v j | is odd. The Nijenhuis-Richardson bracket is defined as the gradedcommutator of ⊼ : [ f, g ] = f ⊼ g − ( − | f || g | g ⊼ f . Notice that, since [ D i ( V ) , D j ( V )] ⊆ D i + j ( V ) we have that D ( V ) , D − ( V ) × D ( V ) and D ≥ ( V ) = Q n ≥ D n ( V ) are graded Lie subalgebras of D ( V ); notice also that [ f, Id V ] = nf for every f ∈ D n ( V ), where Id V is the identity on V .By definition End ∗ ( V ) = D ( V ) and there exists a natural isomorphism of graded Lie al-gebras Aff ∗ ( V ) ∼ = D − ( V ) × D ( V ), where every pair ( x, f ) ∈ D − ( V ) × D ( V ) correspondsto the linear map( x, f ) : V ⊕ K → V, ( x, f )( v + t ) = f ( v ) + tx, v ∈ V, t ∈ K . Remark . It is well known, and in any case easy to prove, that the Nijenhuis-Richardsonproduct ⊼ is the symmetrization of the Gerstenhaber productHom ∗ K ( V ⊗ p − n +1 , V ) × Hom ∗ K ( V ⊗ n +1 , V ) ◦ −→ Hom ∗ K ( V ⊗ p +1 , V ) ,f ◦ g ( v , . . . , v p ) = p − n X i =0 ( − | g | ( | v | + ··· + | v i − | ) f ( v , . . . , v i − , g ( v i , . . . , v i + n ) , v i + n +1 , . . . , v p ) . More precisely, denoting by N : V ⊙ n +1 → V ⊗ n +1 the map N ( v ⊙ · · · ⊙ v n ) = X σ ∈ Σ n +1 ǫ ( σ ) v σ (0) ⊗ · · · ⊗ v σ ( n ) , we have ( f ◦ g ) N = f N ⊼ gN . Remark . Although not relevant for this paper, it is useful to point out that the gradedLie algebra D ( V ) is naturally isomorphic to the graded Lie algebra of coderivations of thesymmetric coalgebra S c ( V ) = L n ≥ V ⊙ n . The isomorphismCoder ∗ ( S c ( V ) , S c ( V )) → D ( V ) ∼ = Hom ∗ K ( S c ( V ) , V )is induced by taking composition with the projection map S c ( V ) → V , see e.g., [12, 16, 21]. MARCO MANETTI
Definition 1.3.
For a unitary graded associative algebra A we consider the sequence ofmaps µ n ∈ D n ( A ), n ≥ − µ − = 1 , µ = Id A , µ n ( a , . . . , a n ) = 1( n + 1)! X σ ∈ Σ n +1 ǫ ( σ ) a σ (0) a σ (1) · · · a σ ( n ) . When A is graded commutative we recover the multiplication maps µ n ( a , . . . , a n ) = a · · · a n . When the algebra A is clear from the context, we shall simply denote by Id the identity map Id A : A → A . In order to avoid possible confusion with the Nijenhuis-Richardson bracket we shall denote the graded commutator of a, b ∈ A by { a, b } = ab − ( − | a || b | ba ∈ A .The following lemma is a straightforward consequence of Remark 1.1. Lemma 1.4.
In the above setup, for every n, m ≥ − we have µ n ⊼ µ m = (cid:18) n + m + 1 m + 1 (cid:19) µ n + m , [ µ n , µ m ] = ( n − m ) ( n + m + 1)!( n + 1)!( m + 1)! µ n + m . The uniqueness theorem
Theorem 2.1.
There exists a unique way to assign to every unitary graded associativealgebra A a morphism of graded vector spaces Ψ : Aff ∗ ( A ) → D ( A ) , x Ψ x = ∞ X n =0 Ψ nx , Ψ nx ∈ D n − ( A ) , such that the following conditions are satisfied: (1) generalized Jacobi: Ψ is a morphism of graded Lie algebras; (2) naturality: for every morphism α : A → B of unitary graded algebras, for every x ∈ A and every pair of linear maps f : A → A , g : B → B such that gα = αf , wehave α Ψ nx = Ψ nα ( x ) α ⊙ n , α Ψ nf = Ψ ng α ⊙ n : A ⊙ n → B . (3) base change : the operators Ψ n , Ψ nId are multilinear over the centre of A . More pre-cisely, if c ∈ A is homogeneous and ac = ( − | a || c | ca for every homogeneous a ∈ A ,then Ψ n ( a , . . . , a n c ) = Ψ n ( a , . . . , a n ) c, Ψ nId ( a , . . . , a n c ) = Ψ nId ( a , . . . , a n ) c, for every a , . . . , a n . (4) initial terms: for every x ∈ A , f ∈ End ∗ ( A ) , we have Ψ x = x, Ψ f = f (1) . (5) gauge fixing: for A = K we have Ψ nId = 0 for every n > .Proof. We identify Aff ∗ ( A ) with the graded Lie subalgebra D − ( A ) × D ( A ) ⊂ D ( A ); forour goals it is convenient to prove the existence following the ideas of [4, 17]. Notice firstthat every operator µ n is multilinear over the centre of A and commutes with morphismsof unitary graded associative algebras. Next, for every sequence K , K , . . . of rationalnumbers, the map b Ψ : D ( A ) → D ( A ) , b Ψ u = exp " − , ∞ X n =1 K n µ n exp([ − , µ − ]) u, NIQUENESS AND INTRINSIC PROPERTIES OF NON-COMMUTATIVE KOSZUL BRACKETS 5 is an isomorphism of graded Lie algebras which is compatible with morphisms of unitarygraded algebras and gives the required initial terms. A simple recursive argument showsthat the gauge fixing condition A = K , b Ψ µ = exp " − , ∞ X n =1 K n µ n ( µ + µ − ) = µ − , can be written as exp " ∞ X n =1 K n µ n , − µ − = µ − + µ , and determines uniquely the coefficients K n . The first terms are: K = 1 , K = − , K = 12 , K = 23 , K = 1112 , K = − , K = − , . . . . According to [17], the formal power series X n ≥ K n t n +1 ( n + 1)! ∈ Q [[ t ]]is the iterative logarithm of e t −
1, cf. [2], and the sequence K n may be also computedrecursively by the linear equations K = 1 , K n = − n + 2)( n − n − X i =1 (cid:26) n + 1 i (cid:27) K i , where (cid:8) n +1 i (cid:9) are the Stirling numbers of the second kind. It is now sufficient to define Ψas the restriction b Ψ to the graded Lie subalgebra D − ( A ) × D ( A ).Let us now prove the unicity, the first step is to prove, for every algebra A , the formulas:(2.1) Ψ = X n ≥ ( − n µ n − , Ψ Id = µ − . Assume first A = K , then µ n is a generator of D n ( K ) and therefore there exists a sequence s , s , . . . in K such that Ψ = X n ≥ s n µ n − , where s = 1 by the initial terms condition. Using the relation [Ψ Id , Ψ ] = Ψ [ Id, = Ψ weobtain Ψ n = [ µ − , Ψ n +11 ] for every n ≥ − s n µ n − = [ µ − , s n +1 µ n ] = − s n +1 µ n − , s n +1 = − s n = ( − n +1 . Consider now the polynomial algebra K [ t ], with t a central element of degree 0: by the basechange propertyΨ n ( t i , . . . , t i n ) = Ψ n (1 , . . . , t i + ··· + i n = ( − n µ n ( t i , . . . , t i n ) , Ψ nId ( t i , . . . , t i n ) = Ψ nId (1 , . . . , t i + ··· + i n , and then (2.1) holds for K [ t ]. The passage from K [ t ] to any A is done by using the standardpolarization trick: given a finite sequence of homogeneous elements a , . . . , a n ∈ A weconsider the algebra B = A [ t , . . . , t n ] , where every t i is a central indeterminate of degree | t i | = −| a i | . We have a morphism ofunitary associative algebras α : K [ t ] → B, α ( t ) = a t + · · · + a n t n , MARCO MANETTI which by naturality givesΨ n ( α ( t ) , . . . , α ( t )) = α Ψ n ( t, . . . , t ) = α (( − n t n ) = ( − n n X i =1 a i t i ! n , while by symmetry Ψ n ( α ( t ) , . . . , α ( t )) = n !Ψ n ( a t , . . . , a n t n ) . Looking at the coefficients of t · · · t n , in the first case we get n !( − n µ n − ( a t , . . . , a n t n ) = n !( − n ( − P i For reference purposes it is convenient to collect as a separate result the recursive for-mulas obtained in the proof of Theorem 2.1. NIQUENESS AND INTRINSIC PROPERTIES OF NON-COMMUTATIVE KOSZUL BRACKETS 7 Theorem 2.2. The higher brackets Ψ n , Φ n are determined by the following recursive formu-las: for every graded unitary associative algebra A , every x ∈ A and every f ∈ Hom ∗ K ( A, A ) we have Ψ nx = Φ nx , Ψ nf = Φ nf + Φ nf (1) , where Φ x = x, Φ nx = 1 n n X h =1 ( − h +1 [Φ n − hx , µ h ] , Φ f = 0 , Φ f = f, Φ n +1 f = 1 n n X h =1 ( − h +1 [Φ n − h +1 f , µ h ] . We shall prove in Proposition 3.5 that if A is graded commutative then the operatorsΨ nf reduce to the usual Koszul brackets as defined in [14]. Similarly the operators Φ nf arethe higher brackets defined in [1, 6] and called Koszul braces in [18, 19]. We shall refer tothe operators Ψ n as Koszul brackets and to the operators Φ n as reduced Koszul brackets .3. Examples and first properties of Koszul brackets The brackets Φ n , Ψ n for low values of n can be easily computed by using the recursiveformulas of Theorem 2.2. For every x ∈ A we have:Ψ x = Φ x = x, Ψ x = Φ x = [ x, µ ] , Ψ x = Φ x = 12 [[ x, µ ] , µ ] − 12 [ x, µ ] , Ψ x = Φ x = 16 [[[ x, µ ] , µ ] , µ ] − 16 [[ x, µ ] , µ ] − 13 [[ x, µ ] , µ ] + 13 [ x, µ ] . For every f ∈ Hom ∗ K ( A, A ) we have:Φ f = f, Φ f = [ f, µ ] , Φ f = 12 [[ f, µ ] , µ ] − 12 [ f, µ ] , Φ f = 16 [[[ f, µ ] , µ ] , µ ] − 16 [[ f, µ ] , µ ] − 13 [[ f, µ ] , µ ] + 13 [ f, µ ] . In the commutative case, the above formulas for Φ f and Φ f were already observed in [10].In a more explicit way, for a ∈ A we have:Φ x ( a ) = Φ x ( a ) = − 12 ( xa + ( − | x || a | ax ) , Φ f ( a ) = f ( a ) , Ψ f ( a ) = f ( a ) − 12 ( f (1) a + ( − | f || a | af (1)) . For x, a, b ∈ A we have:Φ x ( a, b ) = h ( a, b ) + ( − | a || b | h ( b, a )2 , h ( a, b ) = xab + ( − | a || x | axb + ( − | x || ab | abx , which can be written in the form:Φ x ( a, b ) = 112 ( xab + ( − | a || x | axb + ( − | x || ab | abx ) + ( − | a || b | ( a ⇄ b ) . MARCO MANETTI In a similar way, for every f ∈ Hom ∗ K ( A, A ) and every a, b ∈ A we get:Φ f ( a, b ) = f ( ab ) − f ( a ) b − ( − | f | | a | af ( b )2 + ( − | a || b | ( a ⇄ b ) , Ψ f ( a, b ) = f ( ab ) − f ( a ) b − ( − | f | | a | af ( b )2 + f (1) ab + ( − | a || f | af (1) b + ( − | f || ab | abf (1)12+ ( − | a || b | ( a ⇄ b ) . Lemma 3.1. For every x, a , . . . , a n ∈ A , every f ∈ Hom ∗ K ( A, A ) and every n > wehave: (1) Φ nx (1 , a , . . . , a n ) = − Φ n − x ( a , . . . , a n ) , (2) Φ nf (1 , a , . . . , a n ) = Φ n − f (1) ( a , . . . , a n ) , (3) Ψ nf (1 , a , . . . , a n ) = 0 .In particular Φ nf (1 , . . . , 1) = ( − n − f (1) and then Φ nf = 0 if and only if Ψ f = f (1) = 0 and Ψ nf = 0 .Proof. We have seen that Ψ Id = Φ Id + Φ = µ − and then,Φ x (1 , a , . . . , a n ) = [Φ x , µ − ]( a , . . . , a n ) = Φ [ x,Id +1] ( a , . . . , a n )= − Φ x ( a , . . . , a n ) , Ψ f (1 , a , . . . , a n ) = [Φ f , µ − ]( a , . . . , a n ) = Ψ [ f,Id ] ( a , . . . , a n ) = 0 , Φ f (1 , a , . . . , a n ) = Ψ f (1 , a . . . , a n ) − Φ f (1) (1 , a , . . . , a n )= − Φ f (1) (1 , a , . . . , a n ) = Φ f (1) ( a , . . . , a n ) . (cid:3) Example 3.2 (Derivations) . Let f : A → A be a derivation, then [ f, µ n ] = Ψ n +1 f = Φ n +1 f =0 for every n > f ( ab ) = f ( a ) b +( − | a || f | af ( b ) for every a, b ∈ A , a completely straight-forward computation gives [ f, µ ] = [ f, µ ] = 0. According to Lemma 1.4 and Jacobi iden-tity we have then [ f, µ n ] = 0 for every n > 2. The vanishing of Φ nf for n ≥ nf follows from the factthat f (1) = 0.The converse of the above implication is generally false when A is not graded commuta-tive. Consider for instance the algebra A = T ( V ) /I , where V is a vector space of dimension ≥ T ( V ) = L n ≥ V ⊗ n is the tensor algebra generated by V and I is the ideal generatedby V ⊗ . Consider now a map f : A → A such that f (1) = f ( v ) = f ( u ⊗ v + v ⊗ u ) = 0 forevery u, v ∈ V . Since f (1) = 0 we have Φ f = Ψ f and it is easy to see that [ f, µ n ] = Ψ n +1 f =Φ n +1 f = 0 for every n > Example 3.3 (Left and right multiplication maps) . For a graded associative algebra A and every x ∈ A we shall denote by L x and R x the operators of left and right multiplicationby x : L x , R x : A → A, L x ( a ) = xa, R x ( a ) = ( − | a || x | ax . Denoting by { a, b } = ab − ( − | a || b | ba the graded commutator in A , we have: L x (1) = R x (1) = x, Ψ L x ( a ) = { x, a } , Ψ R x ( a ) = { a, x } , Φ L x ( a, b ) = Φ R x ( a, b ) = − 12 (( − | a || x | axb + ( − ( | a | + | x | ) | b | bxa ) . NIQUENESS AND INTRINSIC PROPERTIES OF NON-COMMUTATIVE KOSZUL BRACKETS 9 Ψ L x ( a, b ) = Ψ R x ( a, b ) = 112 ( {{ x, a } , b } + ( − | a || b | {{ x, b } , a } ) . and then Ψ L x + R x = 0 , Ψ L x + R x ( a, b ) = 16 ( {{ x, a } , b } + ( − | a || b | {{ x, b } , a } ) . Notice that L x − R x = { x, −} is a derivation and thenΦ nL x = Φ nR x , Ψ nL x = Ψ nR x , for every n ≥ Example 3.4. As a partial converse of Example 3.2 we have that if A = T ( V ) is a tensoralgebra and f : A → A is linear, then f is a derivation if and only if Φ f = 0.In fact, if Φ f = 0, according to the formula Φ f (1 , 1) = − f (1) = 0 we have f (1) = 0;replacing f with f − δ , where δ : A → A is the (unique) derivation such that δ ( v ) = f ( v )for every v ∈ V , it is not restrictive to assume f ( V ) = 0. For every a ∈ V , since f ( a ) = 0,we have 0 = Φ f ( a, a ) = f ( a ); by the same argument 0 = Φ f ( a , a ) = f ( a ) and moregenerally f ( a n ) = 0 for every n .Next we prove by induction on n that f ( V ⊗ n +1 ) = 0; assuming f ( V ⊗ i ) = 0 for every i ≤ n , we need to prove f ( ab ) = 0 for every a ∈ V and b = v ⊗· · ·⊗ v n ∈ V ⊗ n . If ab − ba = 0then every v i is a scalar multiple of a , and therefore f ( ab ) = cf ( a n +1 ) = 0. If ab = ba , thenby the inductive assumption0 = 2Φ f ( a, b ) = f ( ab ) + f ( ba ) , f ( a , b ) = f ( a b ) + f ( ba ) . Moreover, the vanishing of Φ f ( ab, a ) and Φ f ( a, ba ) gives the equalities f ( aba ) + f ( a b ) = f ( ab ) a + af ( ab ) , f ( aba ) + f ( ba ) = f ( ba ) a + af ( ba ) , whose sum gives f ( aba ) = 0 and therefore f (( ab ) ) + f (( ba ) ) = f (( aba ) b ) + f ( b ( aba )) = 0 .abf ( ab ) + f ( ab ) ab = f (( ab ) ) = − f (( ba ) ) = − baf ( ba ) − f ( ba ) ba = baf ( ab ) + f ( ab ) ba, ( ab − ba ) f ( ab ) + f ( ab )( ab − ba ) = 0 . Since ab − ba = 0 the last equality implies f ( ab ) = 0. Proposition 3.5. Let x ∈ A be a central element. Then Ψ nxy = L x Ψ ny , Ψ nL x f = L x Ψ nf , Ψ n +1 f ( x, a , . . . , a n ) = Ψ n [ f,L x ] ( a , . . . , a n ) , for every y ∈ A , f : A → A . For every sequence of central elements x, c , . . . , c n ∈ A wehave Ψ nx ( c , . . . , c n ) = ( − n xc · · · c n , Ψ nf ( c , . . . , c n ) = [ ... [[ f, L c ] , L c ] . . . , L c n ](1) , and therefore, when A is graded commutative the Koszul brackets Ψ nf are the same of theones defined in [14] .Proof. Since x is central we have [ L x φ, µ n ] = L x [ φ, µ n ] for every n and every φ ∈ D ( A );the first two formulas follow from this and Theorem 2.2. In particular, for f = Id = L weget Ψ L x = L x Ψ Id , viz. Ψ L x = xµ − and Ψ nL x = 0 for every n > 0. This givesΨ [ f,L x ] = [Ψ f , Ψ L x ] = [Ψ f , xµ − ] . Since [ f, L ] = 0 we get Ψ nf (1 , a , . . . , a n ) = Ψ n − f,L ] ( a , . . . , a n ) = 0 and, if c , . . . , c n ∈ A are central elements, by induction on n we haveΨ nf ( c , . . . , c n ) = Ψ ... [[ f,L c ] ,L c ] ...,L cn ] = [ ... [[ f, L c ] , L c ] . . . , L c n ](1) . (cid:3) Remark . A well known consequence of Proposition 3.5 is that, in the commutative case,the Koszul brackets are hereditary: this means that if Ψ nf = 0 for some n > 0, then Ψ kf = 0for every k ≥ n . This is clear if n = 1 since Ψ f = 0 if and only if f = L f (1) . If Ψ nf = 0for some n > 1, then Ψ n − f,L x ] = 0 for every x and by induction Ψ k − f,L x ] = 0 for every x andevery k ≥ n . According to Lemma 3.1 also the reduced Koszul brackets Φ nf are hereditary.This is generally false in the non-commutative case. Consider for instance an element x ∈ A of degree 0 and the operator f = L x + R x : A → A , f ( a ) = ax + xa , for which wehave already seen in Example 3.3 thatΨ f = 0 , Ψ f ( a, b ) = 16 ( {{ x, a } , b } + ( − | a || b | {{ x, b } , a } ) . Remark . If A is not commutative and d : A → A is a derivation, then in generalΦ d = 0. Therefore the attempt to use Koszul higher brackets to define differential operatorsremains unsatisfactory, although slightly better than the trivial extension of Grothendieck’sdefinition, cf. [11, Rem. 2.3.5].4. The formulas of Bering and Bandiera For a given linear endomorphism f : A → A , the brackets Ψ nf were defined by Koszul[14] in the commutative case by the formulaΨ nf ( a , . . . , a n ) = [ ... [[ f, L a ] , L a ] . . . , L a n ](1)= n X k =0 ( − n − k k !( n − k )! X π ∈ Σ n ǫ ( π ) f (1 · a π (1) · · · a π ( k ) ) a π ( k +1) · · · a π ( n ) , where the 1 inside the argument of f is the unit of A . The generalized Jacobi identities(4.1) Ψ n [ f,g ] = n +1 X i =0 [Ψ if , Ψ n − i +1 g ]were first proved independently in [6, 15]. Extensions of the brackets Ψ nf to the non-commutative case satisfying (4.1) were first proposed by Bering [7] and later, with a com-pletely different approach, by Bandiera [3] as a consequence of a more general formulaabout derived brackets. Both approaches involve the sequence B n of Bernoulli numbers: B ( x ) = X n ≥ B n x n n ! = xe x − 1= 1 − x 1! + 16 x − x 4! + 142 x − x 8! + 566 x 10! + · · · . In order to simplify the notation it is useful to introduce the rational numbers B i,j = ( − j j X k =0 (cid:18) jk (cid:19) B i + k , i, j ≥ , together their exponential generating function B ( x, y ) = X i,j ≥ B i,j x i y j i ! j ! ∈ Q [[ x, y ]] . NIQUENESS AND INTRINSIC PROPERTIES OF NON-COMMUTATIVE KOSZUL BRACKETS 11 Since B ( x, y ) = X i,j ( − j j X k =0 B i + k (cid:18) jk (cid:19) x i y j i ! j ! = X i,j j X k =0 ( − y ) j − k ( j − k )! B i + k x i ( − y ) k i ! k != e − y B ( x − y ) = x − ye x − e y = 1 − (cid:16) x y (cid:17) + 12! (cid:18) x xy y (cid:19) − (cid:18) x y xy (cid:19) + 14! (cid:18) − x 30 + 2 x y 15 + 4 x y xy − y (cid:19) + · · · , we have B ( x, y ) = B ( y, x ) and therefore B i,j = B j,i for every i, j ; as a byproduct we havejust proved the following (well known) formulas about Bernoulli numbers:(4.2) ( − j j X k =0 (cid:18) jk (cid:19) B i + k = ( − i i X k =0 (cid:18) ik (cid:19) B j + k , i, j ≥ . (4.3) n X k =0 (cid:18) nk (cid:19) B k = ( − n X k =0 (cid:18) k (cid:19) B n + k = ( − n B n , n ≥ . For x = y we get ∞ X i,j =0 B i,j x i i ! x j j ! = e − x ∞ X n =0 B n n ! ( x − x ) n = e − x and then n X i =0 B i,n − i x n i ! x j j ! = ( − n x n n ! , n X i =0 (cid:18) ni (cid:19) B i,n − i = ( − n . Lemma 4.1. The numbers B i,j are uniquely determined by the following properties: (1) B ,n = B n ; (2) B i,j + B i +1 ,j + B i,j +1 = 0 for every i, j ≥ .Proof. The only nontrivial part is the proof that B i,j + B i +1 ,j + B i,j +1 = 0. This can bedone either by applying binomial identities to the formulas B i,j = ( − j P jk =0 (cid:0) jk (cid:1) B i + k , orby applying the differential operator ∂∂x + ∂∂y to the equality e y B ( x, y ) = B ( x − y ). (cid:3) We are now ready to prove that both Bering and Bandiera’s brackets coincide with thebrackets defined in Section 2; the key technical points of the proof will be the two lemmas3.1 and 4.1. Theorem 4.2 (Bering’s formulas) . In the above notation, for every x ∈ A and every f ∈ Hom ∗ K ( A, A ) we have Ψ nx ( a , . . . , a n ) = i + j = n X i,j ≥ B i,j i ! j ! X π ∈ Σ n ǫ ( π, i, x ) a π (1) · · · a π ( i ) x a π ( i +1) · · · a π ( n ) , Ψ nf ( a , . . . , a n ) = i + j + k = n X i,j,k ≥ B i,j i ! j ! k ! X π ∈ Σ n ǫ ( π, i, f ) a π (1) · · · a π ( i ) ·· f (1 · a π ( i +1) · · · a π ( i + k ) ) a π ( i + k +1) · · · a π ( n ) , where ǫ ( π, i, x ) = ǫ ( π ) · ( − | x | ( | a π (1) | + ··· + | a π ( i ) | ) , ǫ ( π, i, f ) = ǫ ( π ) · ( − | f | ( | a π (1) | + ··· + | a π ( i ) | ) , are the usual Koszul signs.Proof. A direct inspection shows that the above formulas are true for n = 0 , , 2. In general,expanding the recursive equations of Theorem 2.2 we getΦ nx ( a , . . . , a n ) = i + j = n X i,j ≥ C i,j i ! j ! X π ∈ Σ n ǫ ( π, i, x ) a π (1) · · · a π ( i ) x a π ( i +1) · · · a π ( n ) , for a suitable sequence of rational numbers C i,j . In order to prove the theorem it is sufficientto prove that C ,n = B n for every n ≥ C i,j + C i +1 ,j + C i,j +1 = 0 for every i, j ≥ 0; tothis end it is not restrictive to assume that A is the free unitary associative algebra generatedby x, a , . . . , a n , | x | = | a i | = 0. The coefficient of a · · · a i x a i +1 · · · a n in Φ n +1 x (1 , a , . . . , a n )is equal to ( i + 1) C i +1 ,j ( i + 1)! j ! + ( j + 1) C i,j +1 i !( j + 1)! = C i +1 ,j + C i,j +1 i ! j ! . According to Lemma 3.1, Φ n +1 x (1 , a , . . . , a n ) = − Φ nx ( a , . . . , a n ) and then the above coef-ficient is equal to − C i,j i ! j .The proof that C ,n = B n is done by induction on n . Assuming C ,i = B i for every i < n , the coefficient of x a · · · a n in [Φ n − hx , µ h ], h > 0, is equal to B n − h ( n − h )! n − h − h + 1)! , and therefore C ,n n ! = 1 n n X h =1 ( − h +1 B n − h ( n − h )! n − h − h + 1)! . Since ( − h +1 B n − h ( n − h − 1) = ( − n +1 B n − h ( n − h − 1) for every h we have( − n +1 C ,n = ( n − n X h =1 B n − h ( n − h )! n − h − h + 1)!= n ! n X h =1 B n − h ( n − h )!( h + 1)! − ( n − n X h =1 B n − h ( n − h )! h != 1 n + 1 n X h =1 (cid:18) n + 1 n − h (cid:19) B n − h − n n X h =1 (cid:18) nn − h (cid:19) B n − h = 1 n + 1 n − X s =0 (cid:18) n + 1 s (cid:19) B s − n n − X s =0 (cid:18) ns (cid:19) B s . Since n ≥ n X s =0 (cid:18) n + 1 s (cid:19) B s = n − X s =0 (cid:18) ns (cid:19) B s = 0and therefore( − n +1 C ,n = − n + 1 (cid:18) n + 1 n (cid:19) B n = − B n , C ,n = ( − n B n = B n . NIQUENESS AND INTRINSIC PROPERTIES OF NON-COMMUTATIVE KOSZUL BRACKETS 13 Again by Theorem 2.2 we haveΦ nf ( a , . . . , a n ) = n X k =1 i + j = n − k X i,j ≥ C i,j,k i ! j ! k ! X π ∈ Σ n ǫ ( π, i, f ) a π (1) · · · a π ( i ) ·· f (1 · a π ( i +1) · · · a π ( i + k ) ) · a π ( i + k +1) · · · a π ( n ) , for a suitable sequence of rational numbers C i,j,k . ThereforeΨ nf ( a , . . . , a n ) = n X k =0 i + j = n − k X i,j ≥ C i,j,k i ! j ! k ! X π ∈ Σ n ǫ ( π, i, f ) a π (1) · · · a π ( i ) ·· f (1 · a π ( i +1) · · · a π ( i + k ) ) · a π ( i + k +1) · · · a π ( n ) , where C i,j, = B i,j for every i, j ≥ 0; in order to prove the theorem it is therefore sufficient toshow by induction on k that C i,j,k = B i,j . By Lemma 3.1 we have Ψ n +1 f (1 , a , . . . , a n ) = 0,whereas the coefficient of a · · · a i · f (1 · a i +1 · · · a i + k ) · a i + k +1 · · · a n in Ψ n +1 f (1 , a , . . . , a n )is equal to 0 = ( i + 1) C i +1 ,j,k ( i + 1)! j ! k ! + ( j + 1) C i,j +1 ,k i !( j + 1)! k ! + ( k + 1) C i,j,k +1 i ! j ! ( k + 1)! , and then C i,j,k +1 = − C i +1 ,j,k − C i,j +1 ,k = − B i +1 ,j − B i,j +1 = B i,j . (cid:3) As already pointed out in [7, 22], although the expression [ ... [[ f, L a ] , L a ] . . . , L a n ](1)makes sense in every unitary graded associative algebra, its symmetrization does not givethe Koszul brackets Ψ nf . For instance, if | a | = | b | = | f | = 0 and f (1) = 0, then[[ f, L a ] , L b ] + ( − | a || b | [[ f, L b ] , L a ]2 − Ψ f ( a, b ) = { f ( a ) , b } + { f ( b ) , a } , where {− , −} denotes the graded commutator in A . Theorem 4.3 (Bandiera’s formulas) . Let A be a graded unitary associative algebra. Forevery x ∈ A and every f ∈ Hom ∗ K ( A, A ) we have: Ψ nx ( a , . . . , a n ) = X π ∈ Σ n ε ( π ) n X k =0 ( − n B n − k k !( n − k )! {{· · · { x a π (1) · · · a π ( k ) , a π ( k +1) } , . . . } , a π ( n ) } , Ψ nf ( a , . . . , a n ) = X π ∈ Σ n ε ( π ) n X k =0 B n − k k !( n − k )! {{· · · { f k ( a π (1) , . . . , a π ( k ) ) , a π ( k +1) } , . . . } , a π ( n ) } , where { a, b } = ab − ( − | a || b | ba , and f k : A ⊗ k → A is the sequence of operators f = f (1) , f n ( a , . . . , a n ) = [ · · · [[ f, L a ] , L a ] , . . . , L a n ](1) . Proof. Denoting momentarily byΘ nx ( a , . . . , a n ) = X π ∈ Σ n ε ( π ) n X k =0 ( − n B n − k k !( n − k )! {{· · · { x a π (1) · · · a π ( k ) , a π ( k +1) } , . . . } , a π ( n ) } , Θ nf ( a , . . . , a n ) = X π ∈ Σ n ε ( π ) n X k =0 B n − k k !( n − k )! {{· · · { f k ( a π (1) , . . . , a π ( k ) ) , a π ( k +1) } , . . . } , a π ( n ) } , by Bering’s formulas it is sufficient to prove that(4.4) Θ nx ( a , . . . , a n ) = i + j = n X i,j ≥ B i,j i ! j ! X π ∈ Σ n ǫ ( π, i, x ) a π (1) · · · a π ( i ) x a π ( i +1) · · · a π ( n ) , Θ nf ( a , . . . , a n ) = i + j + k = n X i,j,k ≥ B i,j i ! j ! k ! X π ∈ Σ n ǫ ( π, i, f ) a π (1) · · · a π ( i ) ·· f (1 · a π ( i +1) · · · a π ( i + k ) ) a π ( i + k +1) · · · a π ( n ) . (4.5)It’s easy to see that, for every a , . . . , a n ∈ A , we have:Θ n +1 f (1 , a , . . . , a n ) = 0 , Θ n +1 x (1 , a , . . . , a n ) = − Θ nx ( a , . . . , a n ) , and then we can prove (4.4) and (4.5) in the same way as in Theorem 4.2. In fact, expandingthe commutator brackets we can writeΘ nx ( a , . . . , a n ) = i + j = n X i,j ≥ C i,j i ! j ! X π ∈ Σ n ǫ ( π, i, x ) a π (1) · · · a π ( i ) x a π ( i +1) · · · a π ( n ) , for some rational coefficients C i,j . Looking at the coefficient of xa · · · a n in Θ nx ( a , . . . , a n )we get C ,n n ! = ( − n n X k =0 B n − k k !( n − k )! = ( − n n ! n X k =0 (cid:18) nk (cid:19) B k = B n n ! = B ,n n ! , and the proof of C i,j = B i,j follows from the equality Θ n +1 x (1 , a , . . . , a n ) = − Θ nx ( a , . . . , a n )exactly as in Theorem 4.2. Similarly, expanding the operators f k and the commutator brack-ets we can writeΘ nf ( a , . . . , a n ) = n X k =0 i + j = n − k X i,j ≥ C i,j,k i ! j ! k ! X π ∈ Σ n ǫ ( π, i, f ) a π (1) · · · a π ( i ) ·· f (1 · a π ( i +1) · · · a π ( i + k ) ) · a π ( i + k +1) · · · a π ( n ) , for certain rational coefficients C i,j,k . Comparing the coefficients of f (1 · a · · · a k ) · a k +1 · · · a n we get C ,n − k,k k ! ( n − k )! = B n − k k ! ( n − k )! , C ,n − k,k = B n − k = B ,n − k . We now prove by induction on i that C i,j,k = B i,j and therefore the proof of the equal-ity Ψ f = Θ f follows by Bering’s formula; the coefficient of a · · · a i · f (1 · a i +1 · · · a i + k ) · a i + k +1 · · · a n in Θ n +1 f (1 , a , . . . , a n ) is equal to0 = ( i + 1) C i +1 ,j,k ( i + 1)! j ! k ! + ( j + 1) C i,j +1 ,k i !( j + 1)! k ! + ( k + 1) C i,j,k +1 i ! j ! ( k + 1)! , and then C i +1 ,j,k = − C i,j,k +1 − C i,j +1 ,k = − B i,j − B i,j +1 = B i +1 ,j . (cid:3) NIQUENESS AND INTRINSIC PROPERTIES OF NON-COMMUTATIVE KOSZUL BRACKETS 15 Additional remarks The uniqueness theorem for non-unitary algebras. It is clear from the proof thatTheorem 2.1 admits several slight modifications, either changing the underlying categoriesor the choice of initial terms and gauge fixing conditions. According to Theorem 2.2, thereduced Koszul brackets Φ n also make sense for graded associative algebras without unit.It is therefore natural to expect an uniqueness theorem also for reduced Koszul brackets inthe setup of non-unitary associative algebras. Theorem 5.1. There exists a unique way to assign to every graded associative algebra A a morphism of graded Lie algebras Φ : End ∗ ( A ) → D ( A ) , Φ = X Φ n , Φ n : End ∗ ( A ) → D n − ( A ) , such that the following conditions are satisfied: (1) naturality: for every morphism of graded associative algebras α : A → B and everypair of linear maps f : A → A , g : B → B such that gα = αf , we have α Φ nf = Φ ng α ⊙ n : A ⊙ n → B . (2) base change : the operators Φ nId are multilinear over the graded centre of A : moreprecisely, if c ∈ A and ac = ( − | a || c | ca for every a ∈ A , then Φ nId ( a , . . . , a n c ) = Φ nId ( a , . . . , a n ) c for every a , . . . , a n . (3) initial terms: for every f : A → A we have Φ f = 0 , Φ f = f . (4) gauge fixing: at least one of the following conditions is satisfied: (a) if A = K then Φ Id = P n ≥ ( − n µ n ; (b) if A = K [ t ] and ∂ is the second derivative operator, then Φ ∂ ( a, b ) = 2( ∂a )( ∂b ) and Φ n∂ = 0 for every n ≥ .Proof. Notice first that when the graded associative algebra A is not unitary, then thedefinition of µ n : A ⊙ n +1 → A makes sense only for n ≥ n, m ≥ f = exp " − , ∞ X n =1 K n µ n f where K n is the sequence of rational numbers defined in the proof of Theorem 2.1: when A is unitary we recover the reduced Koszul brackets. In particular, for A = K [ t ] we have ∂ (1) = 0, Φ n∂ ( a , . . . , a n ) = Ψ n∂ ( a , . . . , a n ) = [ ... [[ ∂ , L a ] , L a ] . . . , L a n ](1)and then Φ n∂ = 0 for every n ≥ 3. The recursive formulas of Theorem 2.2 give [Φ ∂ , µ n − ] =[ ∂ , µ n ] for every n .The proof of the unicity is essentially the same as in the the unitary case and we giveonly a sketch. Assume that for every graded associative algebra A it is given a morphismof graded Lie algebras φ : End ∗ ( A ) → D ( A ) which satisfies the condition of the theorem:we want to prove that φ = Φ.For A = K we have φ Id = µ + P n ≥ s n µ n for a suitable sequence s , s , . . . ∈ K and by base change the same holds for A = K [ t ]. The gauge fixing condition implies that s n = ( − n for every n : this is clear in the first case, while in the second case we have[ φ ∂ , µ n − ] = [Φ ∂ , µ n − ] = [ ∂ , µ n ] for every n and from the equality0 = [ φ ∂ , φ Id ] = [ φ ∂ , s n − µ n − ] + [ ∂ , s n µ n ]we get s n = − s n − for every n .Considering the inclusion t K [ t ] → K [ t ], by naturality the formula φ Id = P n ≥ ( − n µ n holds also for the algebra t K [ t ] and the polarization trick gives φ Id = P n ≥ ( − n µ n for every graded associative algebra A . Finally the formula [ φ f , φ Id ] = φ [ f,Id ] = 0 givesimmediately the recursive equations φ f = 0 , φ f = f, φ n +1 f = 1 n n X h =1 ( − h +1 [ φ n − h +1 f , µ h ] . (cid:3) The quantum antibracket of Vinogradov, Batalin and Marnelius. Given a gradedassociative algebra A , the quantum antibracket associated to a homogeneous element Q ∈ A of odd degree | Q | = k is defined by the formula( − , − ) Q : A × A → A, ( a, b ) Q = 12 ( { a, { Q, b }} − ( − ( | a | + k )( | b | + k ) { b, { Q, a }} ) . This bracket has been introduced by Batalin and Marnelius in [5] as the unique bracket (upto a scalar factor) satisfying certain natural properties arising in the context of quantizationof classical dynamic. An essentially equivalent construction was also given by Vinogradovin the algebra of linear endomorphisms of the space of differential forms on a manifold,with Q = d the de Rham differential [9, 13].The bracket ( − , − ) Q is graded skewsymmetric of degree 0 on the shifted complex A [ − k ]and corresponds, by standard shifting degrees (d´ecalage) formulas, to the graded symmetricoperator B Q : A ⊙ → A of degree k : B Q ( a, b ) = ( − k | a | ( a, b ) Q = − (cid:16) {{ Q, a } , b } + ( − | a || b | {{ Q, b } , a } (cid:17) . Therefore, according to Example 3.3, we haveΨ L Q = Ψ R Q = − B Q . Gauge fixing variation and B¨orjeson’s brackets. Changing the gauge fixing conditionin Theorem 5.1 one can obtain different hierarchies of higher brackets: for instance, settingΦ Id = µ we get Φ n = 0 for every n ≥ 2, while setting Φ Id = µ − µ , i.e., K = 1 and K n = 0 for every n > 1, it is easy to see that the resulting higher brackets are the gradedsymmetrizations of the B¨orjeson’s brackets [8, 18]. References [1] F. Akman: On some generalizations of Batalin-Vilkovisky algebras. J. Pure Appl. Algebra, (1997),105-141; arXiv:q-alg/9506027 . 7[2] M. Aschenbrenner and W. Bergweiler: Julia’s equation and differential transcendence. arXiv:1307.6381 [math.CV] . 5[3] R. Bandiera: Nonabelian higher derived brackets. J. Pure Appl. Algebra (2015), 3292-3313; arXiv:1304.4097 [math.QA] . 1, 2, 10[4] R. Bandiera: Formality of Kapranov’s brackets in K¨ahler geometry via pre-Lie deformation theory. Int.Math. Res. Notices, published online December 23, 2015 doi:10.1093/imrn/rnv362; arXiv:1307.8066v3[math.QA] . 1, 4[5] I. A. Batalin and R. 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