Universal Lie Formulas for Higher Antibrackets

We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets, aka Koszul braces) of a linear operator $\Delta$ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis-Richardson brackets having as arguments $\Delta$ and the multiplication operators. As a byproduct, we can immediately extend higher antibrackets to noncommutative algebras in a way preserving the validity of generalized Jacobi identities.


Introduction
In particle physics, the fundamental interactions of the standard model, the electroweak and the QCD interactions, are described by non-abelian gauge theories. The presence of a gauge symmetry implies the appearance of unphysical degrees of freedom in the Lagrangian, which stand in the way of the usual quantization methods. Typically, the redundant degrees of freedom are removed through a gauge-fixing procedure. Ghosts, i.e., fields with unphysical statistics, are introduced to compensate for effects of the gauge degrees of freedom and preserve unitarity. The gauge-fixed action retains a nilpotent, odd, global symmetry involving transformations of both fields and ghosts, the Becchi-Rouet-Stora-Tyutin (BRST) symmetry [10,11,12,30]. The BRST symmetry has played an important role in quantization, renormalization, unitarity, and other aspects of gauge theories. The Batalin-Vilkovisky formalism of antibrackets and antifields [7,8,9,18] retains BRST symmetry as fundamental principle while dealing with very general gauge theories, included those with open or reducible gauge symmetry algebras. The antibracket formalism covers a broad spectrum of applications, ranging from supergravity to string and topological field theories. From the mathematical point of view, probably the most convenient approach to Batalin-Vilkovisky formalism is based upon a certain hierarchy of (super)symmetric multilinear maps introduced by Koszul [20] in the framework of differential operators, Calabi-Yau manifolds and symplectic geometry.
Higher antibrackets, also known in literature as higher Koszul brackets or Koszul braces, were defined in [1,2] as a slight variation of Koszul construction and give a convenient generalization of the Batalin-Vilkovisky formalism working when the underlying algebra is not unitary and when the square-zero operator fails to be a differential operator of second order. Several interesting mathematical properties of higher antibrackets have been studied in [14,21] and more recently in [25,26]. In particular it is proved therein that higher antibrackets satisfy the generalized Jacobi identities, and then they provide a structure of strong homotopy Lie algebra, for every square-zero linear operator.
More recently, the formalism of higher antibrackets has been conveniently used in [17] in the framework of Poisson geometry; here one of the key points was the existence of certain relations involving the linear, bilinear and trilinear antibrackets, the multiplication maps and the Nijenhuis-Richardson bracket on the space of multilinear operators. The initial motivation of this paper is to extend these relations to any order; more precisely we prove that every higher antibracket of a linear operator ∆ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis-Richardson brackets, having as arguments ∆ and the multiplication operators. In doing this we also get an explicit description of a universal gauge trivialization (Theorem 2.1) and of some representations of the Lie algebra of vector fields of the line, which we think interesting in its own right (Section 6). Moreover, as additional byproduct, some of the proved universal formulas for higher antibrackets, when applied to operators in noncommutative algebras, give hierarchies of higher antibrackets satisfying generalized Jacobi identities, cf. [24].
The importance of higher antibrackets relies essentially on the fact that, up to a degree shifting, they satisfy the generalized Jacobi identities [14,21]: this means that for every pair of operators f , g we have for every n, where [−, −] denotes the Nijenhuis-Richardson bracket (see Section 3) on the Lie superalgebra of symmetric operators The Lie superalgebra D(A) contains as special even elements the multiplication operators µ n : A n+1 → A, µ n (a 0 , . . . , a n ) = a 0 · · · a n , n ≥ 0.
The associated adjoint operators are even derivations. A tedious but straightforward computation shows that: The above expressions for Φ 2 f and Φ 3 f were also pointed out and conveniently used in [17] in the framework of Poisson geometry.
It is natural to ask whether similar formulas hold for every n: more precisely, we ask for a sequence of noncommutative polynomials P n (ρ 1 , . . . , ρ n ) with rational coefficients such that, for every f , we have A first way to construct, at least in principle, the polynomials P n , follows immediately from the following result.
Theorem 2.1. In the above notation, for every linear operator f : where K n is the sequence of rational numbers defined by the recursive equations are the Stirling numbers of the second kind.
The first 12 terms of K n are and we refer to Table 2 for the values of n!K n for n ≤ 16. As kindly reported to the authors by one referee, this sequence appeared in certain computations about Hurwitz numbers contained in the paper [28] by Shadrin and Zvonkine, and especially in Conjecture A.9, proved later by Aschenbrenner [3]; more details about that conjecture and its proof will be given in Remark 4.2.
In practice it is more convenient to calculate the polynomials P n by using the following easy consequence of Theorem 2.1.
Corollary 2.2. In the above notation, for every linear operator f : A → A we have The advantage of Corollary 2.2 with respect to standard formulas (2.1) and (2.2) is that it gives an easy way to define higher antibrackets also for associative noncommutative superalgebras, providing that the symmetric operators µ n ∈ D n (A) are now defined as µ n (a 0 , . . . , a n ) = 1 (n + 1)! σ∈Σ n+1 (σ)a σ(0) a σ(1) · · · a σ(n) .
In fact, also in this more general situation the brackets defined in Corollary 2.2 satisfy the generalized Jacobi identities (see Remark 5.6), while neither the symmetrization of (2.1) nor the symmetrization of (2.2) do. Extensions of higher antibrackets to noncommutative associative algebras were discussed in [2,6], and a complete description has been given, with different approaches, in [4,5,13,24].
The polynomials P n (ρ 1 , . . . , ρ n ) are not uniquely determined by equation (2.3). In fact, according to formula (3.1), we have and therefore, by Poincaré-Birkhoff-Witt theorem, we can choose every P n as a linear combination with rational coefficients of monomials of type The following theorem, which is mathematically nontrivial, tells us in particular that, in the (super) commutative case, we can restrict to monomials such that i 2 + · · · + i n ≤ 1. Table 1. Values of x n i = (−1) n n!c n i for n ≤ 10, see Theorem 2.3.
In Table 1 we list the values of (−1) n n!c n i for every n ≤ 10: as we shall see later, for a fixed n, the n coefficients c n i can be computed as the solution of a suitable system of n + 1 linear equations. Moreover, some numerical computations suggest the validity of the following conjecture (verified for n ≤ 12) which we are not able to prove at the moment in full generality.
where every empty product is intended to be equal to 1. Moreover (−1) n c n i > 0 for every n ≥ i ≥ 1.
by V n its symmetric powers and by D n (V ) = Hom * (V n+1 , V ) the super vector space of multilinear supersymmetric maps on n + 1 variables Recall that supersymmetry means that where |v| = 0, 1 denotes the parity of a homogeneous element v.
Given f ∈ D n (V ) and g ∈ D m (V ), n, m ≥ 0, their Nijenhuis-Richardson bracket [27] is defined as and S(m + 1, n) is the set of shuffles of type (m + 1, n), i.e., the set of permutations σ of 0, . . . , n + m such that σ(0) < · · · < σ(m) and σ(m Observe that when f ∈ D 0 (V ) the product f g is the same as the composition product f • g and therefore the Nijenhuis-Richardson bracket reduces to the usual super commutator on D 0 (V ). We denote by D(V ) = n≥0 D n (V ); the Nijenhuis-Richardson brackets induces on D(V ) a structure of Lie superalgebra: Remark 3.1. Denoting by S c (V ) = ⊕ n≥1 V n the reduced symmetric coalgebra generated by V , the composition with the natural projection p : S c (V ) → V gives a morphism of super vector spaces and it is well known [22] that P induces an isomorphism of Lie superalgebras where the bracket on Coder * (S c (V )) is the super commutator.
Assume now that A is a commutative superalgebra, then there exists a distinguished sequence µ n ∈ D n (A), n ≥ 0, corresponding to multiplication in A: µ n (a 0 , . . . , a n ) = a 0 a 1 · · · a n .
We have In fact, the binomial coefficient in the first formula is equal to the cardinality of the set of shuffles involved in the formula for the product .

Koszul numbers
Let's recall the notion of iterative exponential and iterative logarithm [16]. Given a formal power series a(t) ∈ Q[[t]] with rational coefficients such that a(0) = a (0) = 0, the following derivation is well defined as well as its exponential which is an isomorphism of rings. The power series is called the iterative exponential of a(t). Conversely, for every g(t) ∈ Q[[t]] such that g(0) = 0 and g (0) = 1 there exists a unique formal power series a(t) = itlog(g(t)), called the iterative logarithm, such that In this paper we are interested to a special sequence of rational numbers K i ∈ Q which will appear in the natural description of the infinitesimal generator of the Koszul hierarchy: with a certain lack of imagination we shall refer to this sequence as "Koszul numbers". Definition 4.1. By Koszul numbers we mean the sequence of rational numbers K n , n ≥ 1, determined by the formula or equivalently by the equation Remark 4.2. As already mentioned in the introduction, the first 12 Koszul numbers appear in certain computations of Hurwitz numbers by Shadrin and Zvonkine [28], in particular in Conjecture A.9 of [28], cf. also [29]. This conjecture was subsequently proved by Aschenbrenner [3], who also fixed a couple of misprints in the original formula of Shadrin and Zvonkine.
Here we briefly recall the conjecture together with the translation of the Aschenbrenner's proof in our setup, which avoids the use of infinite matrices and combinatorial properties of the Stirling numbers.
The starting point is the following sequence of formal power series a d (z) ∈ Q[[z]]: In order to show that the multiplicity of a d (z) is d (cf. [ In fact and therefore Thus we can define the rational numbers a d,d+k , for d, k ≥ 0, by setting Consider now the ring R = Q[[t 0 , t 1 , . . .]] of formal power series in the pairwise distinct indeterminates t n , n ∈ N. That ring has the complete decreasing filtration and therefore there exists a (unique) morphism of Q-algebras L : R → R such that a d,d+k t d+k for every d ≥ 0. Then the Conjecture A.9 of [28] is expressed by the equality: In order to prove (4.2) notice that the vector subspace T ⊂ R of power series of type ∞ n=0 a n t n is isomorphic to the maximal ideal of Q[[t]] via the continuous linear isomorphism of complete Q-vector spaces For every n ≥ 0 the operator φ −1 • t n+1 (n+1)! d dt • φ : T → T is the restriction to T of the derivation Therefore, by (4.1) and the definition of Koszul numbers, for every d ≥ 0 we have and this implies formula (4.2).

Remark 4.3.
Koszul numbers, and more generally iterative logarithms, may also be computed by using the pre-Lie Magnus expansion (see [15] and references therein): in the pre-Lie algebra of formal vector fields over the line.
Proof . The exponential exp a(t) d dt is a local isomorphism of complete local rings, therefore exp a(t) d dt (t n ) = g(t) n for every n and then ]. In particular, we have .

Julia's equation shows a clear relation between Koszul numbers and the sequence A180609 [19] in the On-Line Encyclopedia of Integer Sequences, defined by the equation
a n t n n!(n + 1)! , a 1 = 1.
It is immediate to see that K n = an n! for every n > 0. In fact, setting a(t) = tA(t) the equation becomes a(e t −1) = a(t)e t . In particular, the sequences A134242 and A134243 [29] are respectively the numerators and the denominators of the sequence a n /n!.
where n i are the Stirling numbers of the second kind. In particular, n!K n is an integer for every n. Table 2. The first 16 terms of the sequence OEIS-A180609, a n = n!K n . Proof . This is proved in [3, Section 7] by using the theory of iterative logarithms. For reader's convenience we reproduce here the proof of the first equality as a consequence of Julia's equation.
Recall that the Stirling numbers of the second kind n i , 1 ≤ i ≤ n, may be defined by the formulas In particular, for every n ≥ 2 we have ] be any formal power series and denote g(t) = f (e t − 1). Then for every n > 0 the nth derivative of g is equal to The proof is done by induction on n. For n = 1 formula (4.3) is precisely the formula of derivation of composite functions. For general n we have Notice that (4.3) applied to f (t) = t k gives Denoting a(t) = K 1 3! +· · · , according to Definition 4.1 and Lemma 4.4 we have a(e t −1) = a(t)e t ; taking the derivative we get a e t − 1 e t = a (t)e t + a(t)e t , a(t) + a (t) = a e t − 1 , and then, by formula (4.3), Evaluating the above expression at t = 0 we get The symmetric coalgebra generated by the indeterminate x over the field Q is the vector space Q[x] of polynomials with rational coefficients, equipped with the coproduct It is a trivial consequence of well known facts about symmetric coalgebras that every coderivation of Q[x] may be written as ∞ n=−1 a n x∂ n+1 (n + 1)! : for a sequence a n ∈ Q, n ≥ −1.
Lemma 4.7. Let K n be the sequence of Koszul numbers, then for every integer h > 0 we have K n x∂ n+1 (n + 1)! x h = x + higher order terms.
Proof . Consider the algebra Q[[t]] of formal power series as the algebraic dual of the coalgebra Q[x], with the duality pairing given by x n , t s = 0 if n = s and x n , t n = n!. It is immediate to see that the dual of the coproduct ∆ is precisely the Cauchy product, and the dual (transpose) of the coderivation x∂ n+1 (n+1)! is the derivation t n+1 ∂ (n+1)! , since x∂ n+1 (n + 1)!
and then for every h > 0
In this case the coalgebra S c (Q) is isomorphic to the coalgebra xQ[x] equipped with the coproduct and the isomorphism is given by a 1 · · · a n → a 1 a 2 · · · a n x n .
Under this isomorphism, the projection S c (Q) p − → Q corresponds to the evaluation in 0 of the first derivative, the multiplication map µ n : S c (Q) − → Q corresponds to the linear map and then the associated coderivation is By Lemma 4.7 we have P exp K n m n = m n and this clearly implies P (exp( µ)) = n µ n . Given f : A → A, according to inversion formula, for every a 1 , . . . , a n ∈ A we have (1) , . . . , a σ(h) )a σ(h+1) · · · a σ(n) = f (a 1 a 2 · · · a n ) = f p exp( µ)(a 1 · · · a n ) = p f exp( µ) (a 1 · · · a n ), and then we have the equalities in Coder * (S c (A)): It is now sufficient to keep in mind that D(A) − → Coder * (S c (A)) is a Lie isomorphism.
The proof of Theorem 2.1 now follows from the fact that, by definition the adjoint operator [−µ, ·] is exactly − n≥1 K n ρ n .
Remark 5.4. If F : S c (A) → S c (A) denotes the unique isomorphism of coalgebras such that pF = n>0 µ n , the above lemma shows in particular the well known equality F −1 f F = n Φ n f , cf. [5,25,26]. An alternative, and more conceptual, proof of the first item of Lemma 5.3 also follows from the results about pre-Lie exponential and pre-Lie Magnus expansion discussed in [15,Section 4].
Theorem 2.1 gives immediately the equalities, first proved in [14], for every n > 0 and every f, g : A → A. In fact, K n ρ n is an even derivation of D(A), its exponential is an isomorphism of Lie superalgebras and then In particular if ∆ is a square-zero odd operator, then [∆, ∆] = 2∆ 2 = 0 and therefore i.e., Φ n ∆ are the Taylor coefficients of an L ∞ structure. Corollary 5.5. In the above notation, for every linear operator f : A → A we have Proof . The higher antibrackets of the identity Id : A → A are Φ n+1 Id = (−1) n µ n and then The proof now follows from the trivial equality Remark 5.6. There exists a natural way to extend the definition of the operators µ n ∈ D(A) to every noncommutative associative superalgebra A by setting µ n (a 0 , . . . , a n ) = 1 (n + 1)! σ∈Σ n+1 (σ)a σ(0) a σ(1) · · · a σ(n) , and it is easy to see that the equality is an isomorphism of Lie superalgebras. In particular, defining the higher antibrackets by the formula K n ρ n f, the same argument as above shows the validity of the equalities and the same proof of Corollary 5.5 works in this case as well. Further properties of this noncommutative extension of Koszul brackets are studied in [24], where it is proved in particular that they coincide with the brackets defined by Bering and Bandiera, while they are different from the symmetrization of Börjeson's brackets, cf. [26].
Remark 5.7. For reference purpose, we point out that the sequence K n is uniquely determined by the formula K n ρ n µ 0 in the case A = Q. In fact Φ n+1 Id = (−1) n µ n , ρ n (µ 0 ) = nµ n , and therefore ((−1) n + nK n )µ n is In particular, for every commutative superalgebra A over Q, the linear map is a morphism of Lie superalgebras: we denote by the corresponding adjoint representation.
Lemma 6.2. The linear map ρ : g → Hom(a, a): is an injective morphism of Lie algebras.
Proof . We first prove that are morphisms of Lie algebras; it is easier to make the computation in the basis L n = (n + 1)!l n , The injectivity is clear. In other terms, for every Ψ ∈ a and every l ∈ g we have Proof . By linearity is sufficient to check (6.2) when l = l k and Ψ = Φ n,i . The proof follows from the following straightforward identities: Theorem 6.4. In the notation above, for every integer k > 0 let's denote by ρ k = ρ(l k ) : a → a. For every integer n > 0, there exists an unique sequence c n 1 , . . . , c n n of rational numbers such that Φ n+1 = c n 1 ρ n 1 + c n 2 ρ n−2 1 ρ 2 + c n 3 ρ n−3 1 ρ 3 + · · · + c n n ρ n Φ 1 .