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 Δ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis-Richardson brackets having as arguments Δ and the multiplication operators. A...
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| Cite this: | Universal Lie Formulas for Higher Antibrackets / M. Manetti, G. Ricciardi // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 30 назв. — англ. |
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| description | We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets, aka Koszul braces) of a linear operator Δ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis-Richardson brackets having as arguments Δ 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.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 053, 20 pages
Universal Lie Formulas for Higher Antibrackets
Marco MANETTI † and Giulia RICCIARDI ‡§
† Dipartimento di Matematica “Guido Castelnuovo”, Università degli studi di Roma La Sapienza,
P. le Aldo Moro 5, I-00185 Roma, Italy
E-mail: manetti@mat.uniroma1.it
URL: http://www1.mat.uniroma1.it/people/manetti/
‡ Dipartimento di Fisica “E. Pancini”, Università degli studi di Napoli Federico II,
Complesso Universitario di Monte Sant’Angelo, Via Cintia, I-80126 Napoli, Italy
§ INFN, Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo,
Via Cintia, I-80126 Napoli, Italy
E-mail: giulia.ricciardi@na.infn.it
URL: http://people.na.infn.it/~ricciard/
Received November 17, 2015, in final form May 31, 2016; Published online June 06, 2016
http://dx.doi.org/10.3842/SIGMA.2016.053
Abstract. We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets,
aka Koszul braces) of a linear operator ∆ on a commutative superalgebra can be defined
by some universal formulas involving iterated Nijenhuis–Richardson brackets having as ar-
guments ∆ and the multiplication operators. As a byproduct, we can immediately extend
higher antibrackets to noncommutative algebras in a way preserving the validity of genera-
lized Jacobi identities.
Key words: Lie superalgebras; higher brackets
2010 Mathematics Subject Classification: 17B60; 17B70
1 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
mailto:manetti@mat.uniroma1.it
http://www1.mat.uniroma1.it/people/manetti/
mailto:giulia.ricciardi@na.infn.it
http://people.na.infn.it/~ricciard/
http://dx.doi.org/10.3842/SIGMA.2016.053
2 M. Manetti and G. Ricciardi
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].
2 Overview of the main results
Given a commutative superalgebra A = A0 ⊕ A1 over a field of characteristic 0 and a linear
operator f : A → A, the hierarchy of higher antibrackets of f is the sequence of operators
Φn
f : A�n → A, n > 0, defined by the formulas [20, 25]:
Φ1
f (a) = f(a),
Φ2
f (a, b) = f(ab)− f(a)b− (−1)|a||b|f(b)a,
Φ3
f (a, b, c) = f(abc)− f(ab)c− (−1)|a|(|b|+|c|)f(bc)a− (−1)|b||c|f(ac)b
+ f(a)bc+ (−1)|a||b|f(b)ac+ (−1)|c|(|a|+|b|)f(c)ab,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
Φn
f (a1, . . . , an) =
n∑
k=1
(−1)n−k
∑
σ∈S(k,n−k)
ε(σ)f(aσ(1) · · · aσ(k))aσ(k+1) · · · aσ(n), (2.1)
where A�n is the symmetric nth power of A, where |a| = 0, 1 is the parity of a nonzero homo-
geneous element a ∈ A, and S(k, n − k) is the set of shuffles of type (k, n − k), i.e., the set of
permutations σ of 1, . . . , n such that σ(1) < · · · < σ(k) and σ(k + 1) < · · · < σ(n). The Koszul
sign ε(σ) is equal to (−1)α, where α is the number of pairs (i, j) such that i < j, σ(i) > σ(j)
and |ai||aj | = 1. The same operators can be conveniently defined by the recursive formulas [1]:
Φ1
f (a) = f(a),
Φn+1
f (a1, . . . , b, c) = Φn
f (a1, . . . , bc)− Φn
f (a1, . . . , b)c− (−1)|b||c|Φn
f (a1, . . . , c)b, (2.2)
and differ from the definition given in [2, 14] by an extra sign factor appearing because in their
set-up the operator f acts on the right. It is immediate to observe that every higher antibracket
is a supersymmetric operator, i.e., we have
Φn
f (. . . , ai+1, ai, . . .) = (−1)|ai||ai+1|Φn
f (. . . , ai, ai+1, . . .)
Universal Lie Formulas for Higher Antibrackets 3
for every index 0 < i < n and every sequence a1, . . . , an of homogeneous elements. When A has
a unit 1 ∈ A0, it is well known, and in any case easy to prove, that Φn+1
f = 0 if and only if
f(1) = 0 and f is a differential operator of order ≤ n [20].
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
Φn
[f,g] =
n∑
i=1
[
Φi
f ,Φ
n+1−i
g
]
for every n, where [−,−] denotes the Nijenhuis–Richardson bracket (see Section 3) on the Lie
superalgebra of symmetric operators
D(A) = Hom∗
(
⊕n>0 A
�n, A
)
.
The Lie superalgebra D(A) contains as special even elements the multiplication operators
µn : A�n+1 → A, µn(a0, . . . , an) = a0 · · · an, n ≥ 0.
The associated adjoint operators
ρn : D(A)→ D(A), ρn(ω) = [µn, ω]
are even derivations. A tedious but straightforward computation shows that:
• Φ2
f = −ρ1(f) = −[µ1, f ];
• Φ3
f =
1
2
(
ρ2
1 + ρ2
)
(f) =
1
2
([µ1, [µ1, f ]] + [µ2, f ]);
• Φ4
f = −1
6
(
ρ3
1 + 3ρ1ρ2 + 6ρ3
)
f = −1
6
([µ1, [µ1, [µ1, f ]]] + 3[µ1, [µ2, f ]] + 6[µ3, f ]);
• Φ5
f =
(
1
30
ρ4
1 +
1
5
ρ2
1ρ2 + ρ1ρ3 + 3ρ4
)
f ;
• Φ6
f = −1
3
(
1
90
ρ5
1 +
1
9
ρ3
1ρ2 + ρ2
1ρ3 + 7ρ1ρ4 + 28ρ5
)
f.
The above expressions for Φ2
f and Φ3
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 Pn(ρ1, . . . , ρn) with rational coefficients such that,
for every f , we have
Φn+1
f = Pn(ρ1, . . . , ρn)f. (2.3)
A first way to construct, at least in principle, the polynomials Pn, follows immediately from the
following result.
Theorem 2.1. In the above notation, for every linear operator f : A→ A we have
∞∑
i=1
Φi
f = exp
(
−
∞∑
n=1
Knρn
)
f,
where Kn is the sequence of rational numbers defined by the recursive equations
K1 = 1, Kn =
−2
(n+ 2)(n− 1)
n−1∑
i=1
{
n+ 1
i
}
Ki,
4 M. Manetti and G. Ricciardi
where{
n+ 1
i
}
=
1
i!
i∑
j=0
(−1)i−j
(
i
j
)
jn+1
are the Stirling numbers of the second kind.
The first 12 terms of Kn are
K1 = 1, K2 = −1
2
, K3 =
1
2
, K4 = −2
3
, K5 =
11
12
, K6 = −3
4
, K7 = −11
6
,
K8 =
29
4
, K9 =
493
12
, K10 = −2711
6
, K11 = −12406
15
, K12 =
2636317
60
,
and we refer to Table 2 for the values of n!Kn 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 Pn 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
Φ1
f = f, Φn+1
f =
1
n
n∑
h=1
(−1)hρh
(
Φn−h+1
f
)
=
1
n
n∑
h=1
(−1)h
[
µh,Φ
n−h+1
f
]
.
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 superalge-
bras, providing that the symmetric operators µn ∈ Dn(A) are now defined as
µn(a0, . . . , an) =
1
(n+ 1)!
∑
σ∈Σn+1
ε(σ)aσ(0)aσ(1) · · · aσ(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 Pn(ρ1, . . . , ρn) are not uniquely determined by equation (2.3). In fact,
according to formula (3.1), we have
[ρn, ρm] = (n−m)
(n+m+ 1)!
(n+ 1)!(m+ 1)!
ρn+m, (2.4)
and therefore, by Poincaré–Birkhoff–Witt theorem, we can choose every Pn as a linear combi-
nation with rational coefficients of monomials of type
ρi11 ρ
i2
2 · · · ρ
in
n , with
n∑
h=1
hih = n.
The following theorem, which is mathematically nontrivial, tells us in particular that, in the
(super) commutative case, we can restrict to monomials such that i2 + · · ·+ in ≤ 1.
Universal Lie Formulas for Higher Antibrackets 5
Table 1. Values of xni = (−1)nn!cni for n ≤ 10, see Theorem 2.3.
n 2 3 4 5 6 7 8 9 10
xn1 1 1
4
5
4
9
4
21
1
15
8
405
8
1575
4
3465
xn2 1 3
24
5
40
9
20
7
7
5
224
405
32
175
4
77
xn3 6 24 40 40 28
224
15
32
5
16
7
xn4 72 280 480 504
1120
3
1056
5
96
xn5 1120 4320 7560
24640
3
6336 3744
xn6 21600 83160 147840 164736 131040
xn7 498960 1921920 3459456 3931200
xn8 13453440 51891840 94348800
xn9 415134720 1603929600
xn10 14435366400
Theorem 2.3. In the notation above, for every integer n > 0, there exists an unique sequence
cn1 , . . . , c
n
n of rational numbers such that, for every linear operator f : A→ A, we have
Φn+1
f =
(
cn1ρ
n
1 + cn2ρ
n−2
1 ρ2 + cn3ρ
n−3
1 ρ3 + · · ·+ cnnρn
)
f.
In Table 1 we list the values of (−1)nn!cni for every n ≤ 10: as we shall see later, for
a fixed n, the n coefficients cni 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.
Conjecture 2.4. For every n ≥ 2 the coefficients cni of Theorem 2.3 are given by the formula
cni =
(−1)n
i∏
j=2
n(n− 1)− (j − 1)(j − 2)
2
n∑
h=2
h
h∏
j=2
n(n− 1)− (j − 1)(j − 2)
2
n−1∏
j=h
(1− j)(j + 2)
2
,
where every empty product is intended to be equal to 1. Moreover (−1)ncni > 0 for every n ≥
i ≥ 1.
3 The Nijenhuis–Richardson bracket
Unless otherwise specified every (super)vector space, symmetric product, linear map etc. are
considered over the base field Q of rational numbers, although everything works in the same
way over any field of characteristic 0. Given a super vector space V = V 0 ⊕ V 1, we denote
6 M. Manetti and G. Ricciardi
by V �n its symmetric powers and by Dn(V ) = Hom∗(V �n+1, V ) the super vector space of
multilinear supersymmetric maps on n+ 1 variables
f : V × · · · × V︸ ︷︷ ︸
n+1
→ V.
Recall that supersymmetry means that
f(v0, . . . , vi, vi+1, . . . , vn) = (−1)|vi||vi+1|f(v0, . . . , vi+1, vi, . . . , vn),
where |v| = 0, 1 denotes the parity of a homogeneous element v.
Given f ∈ Dn(V ) and g ∈ Dm(V ), n,m ≥ 0, their Nijenhuis–Richardson bracket [27] is
defined as
[f, g] = f Z g − (−1)|f ||g|g Z f ∈ Dn+m(V ),
where
f Z g(v0, . . . , vn+m) =
∑
σ∈S(m+1,n)
ε(σ)f
(
g(vσ(0), . . . , vσ(m)), vσ(m+1), . . . , vσ(m+n)
)
,
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 + 1) < · · · < σ(m + n). The Koszul
sign ε(σ) is equal to (−1)α, where α is the number of pairs (i, j) such that i < j, σ(i) > σ(j)
and |vi||vj | = 1.
Observe that when f ∈ D0(V ) the product f Z g is the same as the composition product
f ◦ g and therefore the Nijenhuis–Richardson bracket reduces to the usual super commuta-
tor on D0(V ). We denote by D(V ) =
∏
n≥0
Dn(V ); the Nijenhuis–Richardson brackets induces
on D(V ) a structure of Lie superalgebra: ∞∑
i=0
fi,
∞∑
j=0
gj
=
∞∑
n=0
n∑
i=0
[fi, gn−i], fi, gi ∈ Di(V ).
Remark 3.1. Denoting by Sc(V ) = ⊕n≥1V
�n the reduced symmetric coalgebra generated by V ,
the composition with the natural projection p : Sc(V ) → V gives a morphism of super vector
spaces
P : Hom∗
(
Sc(V ), Sc(V )
)
→ Hom∗
(
Sc(V ), V
)
= D(V ),
and it is well known [22] that P induces an isomorphism of Lie superalgebras
P : Coder∗
(
Sc(V )
) '−→ D(V ),
where the bracket on Coder∗(Sc(V )) is the super commutator.
Assume now that A is a commutative superalgebra, then there exists a distinguished sequence
µn ∈ Dn(A), n ≥ 0, corresponding to multiplication in A:
µn(a0, . . . , an) = a0a1 · · · an.
We have
µn Z µm =
(
n+m+ 1
m+ 1
)
µn+m, [µn, µm] = (n−m)
(n+m+ 1)!
(n+ 1)!(m+ 1)!
µn+m. (3.1)
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 Z.
Universal Lie Formulas for Higher Antibrackets 7
4 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
a(t)
d
dt
: Q[[t]]→ Q[[t]],
as well as its exponential
exp
(
a(t)
d
dt
)
: Q[[t]]→ Q[[t]],
which is an isomorphism of rings. The power series
itexp(a(t)) = exp
(
a(t)
d
dt
)
(t) = t+ a(t) +
1
2
a(t)a′(t) +
1
6
(a(t)2a′′(t) + a(t)a′(t)2) + · · ·
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
a(0) = a′(0) = 0, g(t) = itexp(a(t)).
In this paper we are interested to a special sequence of rational numbers Ki ∈ 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 Kn, n ≥ 1,
determined by the formula
∞∑
n=1
Kn
tn+1
(n+ 1)!
= itlog
(
et − 1
)
,
or equivalently by the equation
exp
( ∞∑
n=1
Kn
tn+1
(n+ 1)!
d
dt
)
(t) = et − 1.
Remark 4.2. As already mentioned in the introduction, the first 12 Koszul numbers
K1 = 1, K2 = −1
2
, K3 =
1
2
, K4 = −2
3
, K5 =
11
12
, K6 = −3
4
, K7 = −11
6
,
K8 =
29
4
, K9 =
493
12
, K10 = −2711
6
, K11 = −12406
15
, K12 =
2636317
60
, . . .
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 Aschenbren-
ner [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 ad(z) ∈ Q[[z]]:
ad(z) =
d∑
b=0
(−1)d−b
d!
(
d
b
)
1
1− (b+ 1)z
=
∑
n≥0
d∑
b=0
(−1)d−b
d!
(
d
b
)
(b+ 1)nzn, d ≥ 0.
8 M. Manetti and G. Ricciardi
In order to show that the multiplicity of ad(z) is d (cf. [28, Proposition 2.1]), it is convenient to
consider the isomorphism of Q-vector spaces
ψ : Q[[z]]→ tQ[[t]], ψ
(∑
cnz
n
)
=
∑
cn
tn+1
(n+ 1)!
,
and prove that for every d we have
ψ(ad(z)) =
(et − 1)d+1
(d+ 1)!
. (4.1)
In fact
ψ(ad(z)) =
∑
n≥0
d∑
b=0
(−1)d−b
d!
(
d
b
)
(b+ 1)n
tn+1
(n+ 1)!
=
d∑
b=0
(−1)d−b
d!
(
d
b
)
e(b+1)t − 1
b+ 1
,
and therefore
dψ(ad(z))
dt
=
d∑
b=0
(−1)d−b
d!
(
d
b
)
e(b+1)t =
et(et − 1)d
d!
=
d
dt
(et − 1)d+1
(d+ 1)!
.
Thus we can define the rational numbers ad,d+k, for d, k ≥ 0, by setting
ad(z) =
∑
k≥0
ad,d+kz
d+k = zd +
∑
k>0
ad,d+kz
d+k.
Consider now the ring R = Q[[t0, t1, . . .]] of formal power series in the pairwise distinct indeter-
minates tn, n ∈ N. That ring has the complete decreasing filtration
F pR =
∑
p≤i0+2i1+3i2+···<+∞
ai0,i1,i2,...t
i0
0 t
i1
1 t
i2
2 · · ·
,
and therefore there exists a (unique) morphism of Q-algebras L : R→ R such that
L(td) =
∑
k≥0
ad,d+ktd+k = td +
∑
k>0
ad,d+ktd+k
for every d ≥ 0. Then the Conjecture A.9 of [28] is expressed by the equality:
L = exp
∑
n>0
Kn
∑
k≥0
(
k + n+ 1
n+ 1
)
tn+k
∂
∂tk
. (4.2)
In order to prove (4.2) notice that the vector subspace T ⊂ R of power series of type
∞∑
n=0
antn
is isomorphic to the maximal ideal of Q[[t]] via the continuous linear isomorphism of complete
Q-vector spaces
φ : T → tQ[[t]], φ(tn) = ψ(zn) =
tn+1
(n+ 1)!
.
For every n ≥ 0 the operator φ−1 ◦ tn+1
(n+1)!
d
dt ◦ φ : T → T is the restriction to T of the derivation
rn : R→ R, rn =
∑
k≥0
(
k + n+ 1
n+ 1
)
tk+n
∂
∂tk
.
Universal Lie Formulas for Higher Antibrackets 9
Therefore, by (4.1) and the definition of Koszul numbers, for every d ≥ 0 we have
φ
exp
∑
n>0
Kn
∑
k≥0
(
k + n+ 1
n+ 1
)
tn+k
∂
∂tk
td
= exp
( ∞∑
n=1
Kn
tn+1
(n+ 1)!
d
dt
)
td+1
(d+ 1)!
=
(et − 1)d+1
(d+ 1)!
= ψ(ad(z)) = φ(L(td)),
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):
∞∑
n=1
Kn
tn+1
(n+ 1)!
d
dt
= Ω
( ∞∑
n=1
tn+1
(n+ 1)!
d
dt
)
in the pre-Lie algebra of formal vector fields over the line.
Lemma 4.4 (Julia’s equation). Given a(t) ∈ Q[[t]] such that a(0) = a′(0) = 0 and denoting
g(t) = itexp(a(t)), we have a(g(t)) = a(t)g′(t).
Proof. The exponential exp
(
a(t)ddt
)
is a local isomorphism of complete local rings, therefore
exp
(
a(t)
d
dt
)
(tn) = g(t)n
for every n and then
exp
(
a(t)
d
dt
)
(f(t)) = f(g(t))
for every f(t) ∈ Q[[t]]. In particular, we have
a(g(t)) = exp
(
a(t)
d
dt
)
(a(t)) = exp
(
a(t)
d
dt
)(
a(t)
d
dt
)
(t)
=
(
a(t)
d
dt
)
exp
(
a(t)
d
dt
)
(t) =
(
a(t)
d
dt
)
(g(t))
= a(t)g′(t). �
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(t) = A(et − 1)
1− e−t
t
, A(t) =
∞∑
n=1
ant
n
n!(n+ 1)!
, a1 = 1.
It is immediate to see that Kn = an
n! for every n > 0. In fact, setting a(t) = tA(t) the
equation A(t) = A(et−1)1−e−t
t becomes a(et−1) = a(t)et. In particular, the sequences A134242
and A134243 [29] are respectively the numerators and the denominators of the sequence an/n!.
Lemma 4.5 (Aschenbrenner). For every n ≥ 2 we have
Kn =
−2
(n+ 2)(n− 1)
n−1∑
i=1
{
n+ 1
i
}
Ki,
Kn =
n∑
k=1
(−1)k+1
k
∑
1<n1<···<nk−1<nk=n+1
{
n2
n1
}{
n3
n2
}
· · ·
{
nk
nk−1
}
,
where
{
n
i
}
are the Stirling numbers of the second kind. In particular, n!Kn is an integer for
every n.
10 M. Manetti and G. Ricciardi
Table 2. The first 16 terms of the sequence OEIS-A180609, an = n!Kn.
a1 = 1 a9 = 14908320
a2 = −1 a10 = −1639612800
a3 = 3 a11 = −33013854720
a4 = −16 a12 = 21046667685120
a5 = 110 a13 = −549927873855360
a6 = −540 a14 = −637881314775344640
a7 = −9240 a15 = 76198391578224115200
a8 = 292320 a16 = 41404329870413936025600
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{
n
1
}
=
{
n
n
}
= 1,
{
n
i
}
=
{
n− 1
i− 1
}
+ i
{
n− 1
i
}
, 1 < i < n.
In particular, for every n ≥ 2 we have{
n
2
}
= 2n−1 − 1,
{
n
n− 1
}
=
(
n
2
)
.
Let f(t) ∈ Q[[t]] be any formal power series and denote g(t) = f(et − 1). Then for every n > 0
the nth derivative of g is equal to
g(n)(t) =
n∑
k=1
{
n
k
}
f (k)
(
et − 1
)
ekt. (4.3)
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
g(n+1)(t) =
∑
k
{
n
k
}(
f (k+1)
(
et − 1
)
e(k+1)t + if (k)
(
et − 1
)
ekt
)
,
g(n+1)(t) =
∑
k
({
n
k − 1
}
+ k
{
n
k
})
f (k)
(
et − 1
)
ekt.
Notice that (4.3) applied to f(t) = tk gives
g(n)(0) =
{
n
k
}
k!,
(et − 1)k
k!
=
∑
n≥0
{
n
k
}
tn
n!
=
∑
m≥0
{
m+ k
k
}
tm+k
(m+ k)!
.
Denoting a(t) = K1
t2
2! +K2
t3
3! +· · · , according to Definition 4.1 and Lemma 4.4 we have a(et−1) =
a(t)et; taking the derivative we get
a′
(
et − 1
)
et = a′(t)et + a(t)et, a(t) + a′(t) = a′
(
et − 1
)
,
and then, by formula (4.3),
a(h)(t) + a(h+1)(t) =
h∑
i=1
{
h
i
}
a(i+1)
(
et − 1
)
eit.
Universal Lie Formulas for Higher Antibrackets 11
Evaluating the above expression at t = 0 we get
Kh−1 +Kh =
h∑
i=1
{
h
i
}
Ki = Kh +
(
h
2
)
Kh−1 +
h−2∑
i=1
{
h
i
}
Ki,
Kh−1 =
1
1−
(
h
2
) h−2∑
i=1
{
h
i
}
Ki =
−2
(h+ 1)(h− 2)
h−2∑
i=1
{
h
i
}
Ki. �
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
∆: Q[x]→ Q[x]⊗Q Q[x], ∆(xm) =
m∑
k=0
(
m
k
)
xk ⊗ xm−k.
Lemma 4.6. Let ∂ : Q[x]→ Q[x] be the usual derivation operator. Then the operators
dn =
x∂n+1
(n+ 1)!
, n ≥ −1,
are coderivations in the coalgebra (Q[x],∆) such that
[dn, dm] = (n−m)
(n+m+ 1)!
(n+ 1)!(m+ 1)!
dn+m (4.4)
for every n,m ≥ −1.
Proof. We have dn(xm) =
(
m
n+1
)
xm−n, therefore
∆dn(xm) =
m−n∑
k=0
(
m
n+ 1
)(
m− n
k
)
xk ⊗ xm−n−k,
(dn ⊗ Id + Id⊗dn)∆(xm) =
m∑
s=0
(
m
s
)(
s
n+ 1
)
xs−n ⊗ xm−s
+
m∑
k=0
(
m
k
)(
m− k
n+ 1
)
xk ⊗ xm−k−n,
and the conclusion follows from the straightforward equality(
m
n+ 1
)(
m− n
k
)
=
(
m
k + n
)(
k + n
n+ 1
)
+
(
m
k
)(
m− k
n+ 1
)
. �
It is a trivial consequence of well known facts about symmetric coalgebras that every coderiva-
tion of Q[x] may be written as
∞∑
n=−1
an
x∂n+1
(n+ 1)!
: Q[x]→ Q[x]
for a sequence an ∈ Q, n ≥ −1.
Lemma 4.7. Let Kn be the sequence of Koszul numbers, then for every integer h > 0 we have
exp
( ∞∑
n=1
Kn
x∂n+1
(n+ 1)!
)
xh = x+ higher order terms.
12 M. Manetti and G. Ricciardi
Proof. Consider the algebra Q[[t]] of formal power series as the algebraic dual of the coal-
gebra Q[x], with the duality pairing given by 〈xn, ts〉 = 0 if n 6= s and 〈xn, tn〉 = 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 tn+1∂
(n+1)! , since〈
x∂n+1
(n+ 1)!
xm+n, tm
〉
= m!
(
m+ n
n+ 1
)
=
m(m+ n)!
(n+ 1)!
=
〈
xm+n,
tn+1∂
(n+ 1)!
tm
〉
for every n, m.
Therefore exp
( ∞∑
n=1
Kn
tn+1
(n+1)!
d
dt
)
is the dual of exp
( ∞∑
n=1
Kn
x∂n+1
(n+1)!
)
and then for every h > 0〈
exp
( ∞∑
n=1
Kn
x∂n+1
(n+ 1)!
)
xh, t
〉
=
〈
xh, exp
( ∞∑
n=1
Kn
tn+1
(n+ 1)!
d
dt
)
t
〉
= 1. �
5 Proof of Theorem 2.1
As in Section 2 let A = A0⊕A1 be a commutative superalgebra; for every linear endomorphism
f : A→ A consider the hierarchy of higher antibrackets of f :
Φn
f (a1, . . . , an) =
n∑
k=1
(−1)n−k
∑
σ∈S(k,n−k)
ε(σ)f(aσ(1) · · · aσ(k))aσ(k+1) · · · aσ(n).
Lemma 5.1 (inversion formula). For every f ∈ Hom∗(A,A) and every a1, . . . , an ∈ A we have
f(a1a2 · · · an) =
n∑
k=1
∑
σ∈S(k,n−k)
ε(σ)Φk
f (aσ(1), . . . , aσ(k))aσ(k+1) · · · aσ(n).
Proof. The proof follows by a straightforward direct computation. We have
n∑
k=1
∑
σ∈S(k,n−k)
ε(σ)Φk
f (aσ(1), . . . , aσ(k))aσ(k+1) · · · aσ(n)
=
n∑
k=1
∑
σ∈S(k,n−k)
k∑
h=1
(−1)k−h
∑
τ∈S(h,k−h)
ε(σ)f(aστ(1) · · · aστ(h)) · · · aστ(k)aσ(k+1) · · · aσ(n)
=
n∑
h=1
∑
σ∈S(h,n−h)
n−h∑
r=0
(−1)r
∑
η∈Pr(h,σ)
ε(σ)f(aσ(1) · · · aσ(h))aησ(h+1) · · · aησ(n)
= f(a1 · · · an) +
n−1∑
h=1
∑
σ∈S(h,n−h)
n−h∑
r=0
(−1)r
∑
η∈Pr(h,σ)
ε(σ)f(aσ(1) · · · aσ(h))aησ(h+1) · · · aησ(n),
where Pr(h, σ) is the set of permutations η of {σ(h+ 1), . . . , σ(n)} such that
η(σ(h+ 1)) < η(σ(h+ 2)) < · · · < η(σ(h+ r)), η(σ(h+ r + 1)) < · · · < η(σ(n)).
Since the cardinality of Pr(h, σ) is
(
n−h
r
)
, for every h < n and every shuffle σ ∈ S(h, n− h) we
have
n−h∑
r=0
(−1)r
∑
η∈Pr(h,σ)
ε(σ)f(aσ(1) · · · aσ(h))aησ(h+1) · · · aησ(n)
= ε(σ)f(aσ(1) · · · aσ(h))aσ(h+1) · · · aσ(n)
n−h∑
r=0
(−1)r
(
n− h
r
)
= 0. �
Universal Lie Formulas for Higher Antibrackets 13
Definition 5.2. For every α ∈ D(V ) we will denote by α̂ ∈ Coder∗(Sc(V )) the (unique)
coderivation such that P (α̂) = α.
The above definition makes sense in view of the isomorphism of Lie superalgebras
P : Coder∗
(
Sc(A)
) '−→ D(A)
described in Remark 3.1.
Lemma 5.3. In the above setup, let µ =
∞∑
n=1
Knµn ∈ D(A). Then:
1) the map exp(µ̂) : Sc(A)→ Sc(A) is an isomorphism of coalgebras and
P (exp(µ̂)) =
∑
n≥0
µn ∈ D(A);
2) for every f ∈ Hom∗(A,A) we have∑
n>0
Φn
f = exp([−µ, ·])Φ1
f = exp([−µ, ·])f.
Proof. According to [22, 23] we have
µ̂(a0 � · · · � an) =
n∑
h=1
Kh
∑
σ∈S(h+1,n−h)
ε(σ)aσ(0) · · · aσ(h) � aσ(h+1) � · · · � aσ(n),
and then
p(exp(µ̂)(a0 � · · · � an)) = qn(K1, . . . ,Kn)a0a1 · · · an,
where
qn(K1, . . . ,Kn) =
∑
a1+2a2+···+nan=n
qa1,...,anK
a1
1 · · ·K
an
n
is a quasi-homogeneous polynomial of weight n in K1, . . . ,Kn, where Ki has weight i, and with
coefficients qa1,...,an ∈ Q depending only on a1, . . . , an.
Thus, in order to prove that qn(K1, . . . ,Kn) = 1 for every n, it is not restrictive to carry out
the computations for the “simplest” algebra A = Q.
In this case the coalgebra Sc(Q) is isomorphic to the coalgebra xQ[x] equipped with the
coproduct
∆: xQ[x]→ xQ[x]⊗Q xQ[x], ∆(xm) =
m−1∑
k=1
(
m
k
)
xk ⊗ xm−k,
and the isomorphism is given by
a1 � · · · � an 7→ a1a2 · · · anxn.
Under this isomorphism, the projection Sc(Q)
p−→ Q corresponds to the evaluation in 0 of the
first derivative, the multiplication map µn : Sc(Q) −→ Q corresponds to the linear map
mn : Qxn+1 → Qx, mn
(
xn+1
)
= x,
14 M. Manetti and G. Ricciardi
and then the associated coderivation is
m̂n(xh) =
(
h
n+ 1
)
xh−n = x
∂n+1
(n+ 1)!
(xh), m̂n =
x∂n+1
(n+ 1)!
.
By Lemma 4.7 we have P
(
exp
(∑
Knm̂n
))
=
∑
mn and this clearly implies P (exp(µ̂)) =∑
n µn.
Given f : A→ A, according to inversion formula, for every a1, . . . , an ∈ A we have
p exp(µ̂)
(∑
h
Φ̂h
f (a1 � · · · � an)
)
=
n∑
h=1
µn−h
∑
σ∈S(h,n−h)
ε(σ)Φh
f (aσ(1), . . . , aσ(h))� aσ(h+1) � · · · � aσ(n)
=
n∑
h=1
∑
σ∈S(h,n−h)
ε(σ)Φh
f (aσ(1), . . . , aσ(h))aσ(h+1) · · · aσ(n)
= f(a1a2 · · · an) = fp exp(µ̂)(a1 � · · · � an)
= p
(
f̂ exp(µ̂)
)
(a1 � · · · � an),
and then we have the equalities in Coder∗(Sc(A)):∑
n
Φ̂n
f = exp(µ̂)−1f̂ exp(µ̂) = exp(−µ̂)f̂ exp(µ̂) = exp([−µ̂, ·])f̂ .
It is now sufficient to keep in mind that D(A) −̂→ Coder∗(Sc(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
Knρn.
Remark 5.4. If F : Sc(A) → Sc(A) denotes the unique isomorphism of coalgebras such that
pF =
∑
n>0
µn, the above lemma shows in particular the well known equality F−1f̂F =
∑
n Φ̂n
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],
Φn
[f,g] =
n∑
i=1
[
Φi
f ,Φ
n−i+1
g
]
for every n > 0 and every f, g : A → A. In fact,
∑
Knρn is an even derivation of D(A), its
exponential is an isomorphism of Lie superalgebras and then
∞∑
i=1
Φi
[f,g] = exp
(
−
∞∑
n=1
Knρn
)
([f, g])
=
[
exp
(
−
∞∑
n=1
Knρn
)
f, exp
(
−
∞∑
n=1
Knρn
)
g
]
=
[ ∞∑
i=1
Φi
f ,
∞∑
i=1
Φi
g
]
.
Universal Lie Formulas for Higher Antibrackets 15
In particular if ∆ is a square-zero odd operator, then [∆,∆] = 2∆2 = 0 and therefore
n∑
i=1
[
Φi
∆,Φ
n−i+1
∆
]
= 0,
i.e., Φn
∆ are the Taylor coefficients of an L∞ structure.
Corollary 5.5. In the above notation, for every linear operator f : A→ A we have
Φ1
f = f, Φn+1
f =
1
n
n∑
h=1
(−1)h
[
µh,Φ
n−h+1
f
]
.
Proof. The higher antibrackets of the identity Id: A→ A are Φn+1
Id = (−1)nµn and then
0 = Φn+1
[Id,f ] =
n+1∑
i=1
[
Φi
Id,Φ
n−i+2
f
]
=
[
µ0,Φ
n+1
f
]
+
n∑
h=1
(−1)h
[
µh,Φ
n−h+1
f
]
.
The proof now follows from the trivial equality [µ0,Φ
n+1
f ] = −nΦn+1
f . �
Remark 5.6. There exists a natural way to extend the definition of the operators µn ∈ D(A)
to every noncommutative associative superalgebra A by setting
µn(a0, . . . , an) =
1
(n+ 1)!
∑
σ∈Σn+1
ε(σ)aσ(0)aσ(1) · · · aσ(n),
and it is easy to see that the equality
[µn, µm] = (n−m)
(n+m+ 1)!
(n+ 1)!(m+ 1)!
µn+m
still holds. The adjoint operators ρn = [µn,−] : D(A)→ D(A) are derivations and the map
exp
(
−
∑
n≥1
Knρn
)
: D(A)→ D(A)
is an isomorphism of Lie superalgebras. In particular, defining the higher antibrackets by the
formula
∞∑
i=1
Φi
f = exp
(
−
∞∑
n=1
Knρn
)
f,
the same argument as above shows the validity of the equalities
Φn+1
Id = (−1)nµn, Φn
[f,g] =
n∑
i=1
[
Φi
f ,Φ
n−i+1
g
]
,
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 Kn is uniquely determined
by the formula
∞∑
i=1
Φi
Id = exp
(
−
∞∑
n=1
Knρn
)
µ0
in the case A = Q. In fact Φn+1
Id = (−1)nµn, ρn(µ0) = nµn, and therefore ((−1)n + nKn)µn is
the component in Dn(Q) of exp
(
−
n−1∑
i=1
Kiρi
)
µ0.
16 M. Manetti and G. Ricciardi
6 Natural higher brackets
In order to prove the Theorem 2.3 it is useful, and we believe interesting in itself, to consider
the Koszul higher brackets as special natural derived brackets: the notion of naturality has been
discussed extensively in the paper [26]; for our application we simply define a natural derived
bracket of a linear endomorphism f : A→ A as an element of type∑
i+j=n
ψijµi Z (f Z µj) ∈ Dn(A), ψij ∈ Q.
We denote by
a =
{
Ψ ∈ HomQ(Q[x],Q[x]) |Ψ(1) = 0
}
=
{ ∞∑
n=0
pn(x)∂n+1 | pn(x) ∈ Q[x]
}
the associative algebra of linear endomorphisms of the Q-vector space Q[x] vanishing in 1,
equipped with the composition product. Every element Ψ ∈ a is uniquely determined by its
Taylor coefficients, i.e., by the infinite double sequence ψij , i, j ≥ 0, defined by the formulas:
Ψ(xi+1) =
∞∑
j=0
ψij
xj
j!
, i ≥ 0, ψij ∈ Q. (6.1)
Definition 6.1. Let Ψ ∈ a be a fixed element. For a commutative superalgebra A over Q and
a linear endomorphism f : A→ A, the Ψ-bracket of f is
Ψf ∈ D(A), Ψf =
∑
i,j≥0
ψijµj Z (f Z µi),
where the ψij ’s are the Taylor coefficients of Ψ as in (6.1).
For every n ≥ i > 0, denote by Φn,i ∈ a the operator
Φn,i(xi) =
xn−i
(n− i)!
, Φn,i(xs) = 0 for s 6= i.
Then for every f : A→ A we have Φn,i
f = µn−i Z (f Z µi−1):
Φn,i
f (a1, . . . , an) =
∑
σ∈S(i,n−i)
ε(σ)f(aσ(1) · · · aσ(i))aσ(i+1) · · · aσ(n).
In particular, setting
Φn =
n∑
i=1
(−1)n−iΦn,i
we recover the definition of higher Koszul brackets Φn
f . The formulas (2.4), (3.1) and (4.4)
suggest the introduction of the Lie algebra g over the field Q of rational numbers with basis
l0, l1, l2, . . . and bracket
[ln, lm] = (n−m)
(n+m+ 1)!
(n+ 1)!(m+ 1)!
ln+m.
Notice that, setting Ln = (n+ 1)!ln we obtain the more familiar expression
[Ln, Lm] = (n−m)Ln+m.
Universal Lie Formulas for Higher Antibrackets 17
In particular, for every commutative superalgebra A over Q, the linear map
µA : g→ D(A), µA(ln) = µn
is a morphism of Lie superalgebras: we denote by
ρA : g→ Hom(D(A), D(A)), ρA(ln)(φ) = [µn, φ]
the corresponding adjoint representation.
Lemma 6.2. The linear map ρ : g→ Hom(a, a):
ρ(ln)(Ψ) =
(
xn
n!
− xn+1∂
(n+ 1)!
)
◦Ψ−Ψ ◦ x∂n+1
(n+ 1)!
is an injective morphism of Lie algebras.
Proof. We first prove that
φ : g→ Hom(Q[x],Q[x]), φ(ln) =
x∂n+1
(n+ 1)!
,
ψ : g→ Hom(Q[x],Q[x]), ψ(ln) =
xn
n!
− xn+1∂
(n+ 1)!
are morphisms of Lie algebras; it is easier to make the computation in the basis Ln = (n+ 1)!ln,
[φ(Ln), φ(Lm)]xs =
[
x∂n+1, x∂m+1
]
xs
= x∂n+1
(
m∏
i=0
(s− i)
)
xs−m − x∂m+1
(
n∏
i=0
(s− i)
)
xs−n
= (s−m)
m+n∏
i=0
(s− i)xs−m−n − (s− n)
m+n∏
i=0
(s− i)xs−m−n
= (n−m)x∂n+m+1xs = (n−m)φ(Ln+m)xs.
Since ψ(Ln)xs = (n+ 1− s)xn+s we have
[ψ(Ln), ψ(Lm)]xs = ψ(Ln)(m+ 1− s)xm+s − ψ(Lm)(n+ 1− s)xn+s
= ((n+ 1− s−m)(m+ 1− s)− (m+ 1− n− s)(n+ 1− s))xn+m+s
= (n−m)(n+m+ 1− s)xn+m+s = (n−m)ψ(Ln+m)xs.
Finally
[ρ(x), ρ(y)]Ψ = ρ(x)ρ(y)Ψ− ρ(x)ρ(y)Ψ
= ψ(x)ρ(y)Ψ− ρ(y)Ψφ(x)− ψ(y)ρ(x)Ψ + ρ(x)Ψφ(y)
= ψ(x)ψ(y)Ψ + Ψφ(y)φ(x)− ψ(y)ψ(x)Ψ−Ψφ(x)φ(y)
= [ψ(x), ψ(y)]Ψ−Ψ[φ(x), φ(y)]
= ψ([x, y])Ψ−Ψφ([x, y]) = ρ([x, y])Ψ.
The injectivity is clear. �
Lemma 6.3. For every commutative superalgebra A and every f ∈ D0(A) the map
a→ D(A), Ψ 7→ Ψf
is a morphism of g-modules. In other terms, for every Ψ ∈ a and every l ∈ g we have
(ρ(l)Ψ)f = ρA(l)(Ψf ) = [µA(l),Ψf ]. (6.2)
18 M. Manetti and G. Ricciardi
Proof. By linearity is sufficient to check (6.2) when l = lk and Ψ = Φn,i. The proof follows
from the following straightforward identities:
ρ(lk)Φ
n,i =
((
n− i+ k
k
)
−
(
n− i+ k
k + 1
))
Φn+k,i −
(
k + i
k + 1
)
Φn+k,i+k,
[
µk,Φ
n,i
f
]
=
((
n− i+ k
k
)
−
(
n− i+ k
k + 1
))
Φn+k,i
f −
(
k + i
k + 1
)
Φn+k,i+k
f . �
Theorem 6.4. In the notation above, for every integer k > 0 let’s denote by ρk = ρ(lk) : a→ a.
For every integer n > 0, there exists an unique sequence cn1 , . . . , c
n
n of rational numbers such that
Φn+1 =
(
cn1ρ
n
1 + cn2ρ
n−2
1 ρ2 + cn3ρ
n−3
1 ρ3 + · · ·+ cnnρn
)
Φ1.
Proof. Denote by V n ⊂ a the subspace generated by Φn,1, . . . ,Φn,n; notice that dimV n = n
and ρk(V
n) ⊂ V n+k. We claim that, for every n ≥ 2 the n vectors
ρn−2
1
(
Φ2,2
)
, ρa1ρn−a
(
Φ1,1
)
, 0 ≤ a ≤ n− 2,
form a basis of V n. This is clear for n = 2 since ρ1(Φ1,1) = Φ2,1 −Φ2,2. By induction on n, it is
sufficient to prove that for every n ≥ 2 the linear map
V n ⊕ V 1 → V n+1, (x, y) 7→ ρ1(x) + ρn(y)
is an isomorphism. Consider first the case n = 2: in terms of matrix multiplication we have
(
ρ1
(
Φ2,2
)
, ρ1
(
Φ2,1
)
, ρ2
(
Φ1,1
))
=
(
Φ3,3,Φ3,2,Φ3,1
)−3 0 −1
1 −1 0
0 1 1
and the determinant of the matrix is 6= 0. Assume now n ≥ 3: since
ρn
(
Φ1,1
)
= Φn+1,1 − Φn+1,n+1, ρ1
(
Φn,n−2
)
= −
(
n− 1
2
)
Φn+1,n−1,
we can write(
ρ1
(
Φn,n
)
, . . . , ρ1
(
Φn,1
)
, ρn
(
Φ1,1
))
=
(
Φn+1,n+1, . . . ,Φn+1,1
)(A B
0 C
)
,
where A ∈M3,3(Z) and C ∈Mn−2,n−2(Z) are lower triangular matrices with nonzero entries in
the diagonal, and therefore the determinant of the block matrix is 6= 0. Thus, for every n there
exists an unique sequence of n+ 1 rational numbers cn1 , . . . , c
n
n, bn such that
Φn+1 =
(
cn1ρ
n
1 + cn2ρ
n−2
1 ρ2 + · · ·+ cnnρn
)
Φ1 + bnρ
n−1
1
(
Φ2,2
)
and we have to show that bn = 0. To this end it is sufficient to consider the representation
a→ D(Q[x]), Ψ 7→ Ψ∂ ,
where, as usual ∂ = d
dt ∈ D0(Q[x]). Since ∂ is a derivation we have Φn+1
∂ = [µn, ∂] = 0 for every
n ≥ 0, and therefore it is sufficient to prove that ρn1 (Φ2,2
∂ ) 6= 0 for every n. By construction
Φ2,2
∂ (ta, tb) = (a+ b)ta+b−1 and an easy induction on n shows that for every n ≥ 3 we have
ρn−2
1
(
Φ2,2
∂
)(
ta1 , ta2 , . . . , tan
)
=
[
n∏
i=3
(
i− 1
2
)]
(a1 + · · ·+ an)t
∑
ai−1. �
Universal Lie Formulas for Higher Antibrackets 19
Acknowledgments
We thank Bruno Vallette for useful comments and for bringing to our attention the pre-Lie
Magnus expansion. We are indebted with the anonymous referees for several remarks and
especially for letting us into the knowledge of the papers [3, 28]. M.M. acknowledges partial
support by Italian MIUR under PRIN project 2012KNL88Y “Spazi di moduli e teoria di Lie”;
G.R. acknowledges partial support by Italian MIUR under PRIN project 2010YJ2NYW and
INFN under specific initiative QNP.
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1 Introduction
2 Overview of the main results
3 The Nijenhuis–Richardson bracket
4 Koszul numbers
5 Proof of Theorem 2.1
6 Natural higher brackets
References
|
| id | nasplib_isofts_kiev_ua-123456789-147749 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:18:32Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Manetti, M. Ricciardi, G. 2019-02-15T19:08:20Z 2019-02-15T19:08:20Z 2016 Universal Lie Formulas for Higher Antibrackets / M. Manetti, G. Ricciardi // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 30 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B60; 17B70 DOI:10.3842/SIGMA.2016.053 https://nasplib.isofts.kiev.ua/handle/123456789/147749 We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets, aka Koszul braces) of a linear operator Δ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis-Richardson brackets having as arguments Δ 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. We thank Bruno Vallette for useful comments and for bringing to our attention the pre-Lie
 Magnus expansion. We are indebted with the anonymous referees for several remarks and
 especially for letting us into the knowledge of the papers [3, 28]. M.M. acknowledges partial
 support by Italian MIUR under PRIN project 2012KNL88Y “Spazi di moduli e teoria di Lie”;
 G.R. acknowledges partial support by Italian MIUR under PRIN project 2010YJ2NYW and
 INFN under specific initiative QNP. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Universal Lie Formulas for Higher Antibrackets Article published earlier |
| spellingShingle | Universal Lie Formulas for Higher Antibrackets Manetti, M. Ricciardi, G. |
| title | Universal Lie Formulas for Higher Antibrackets |
| title_full | Universal Lie Formulas for Higher Antibrackets |
| title_fullStr | Universal Lie Formulas for Higher Antibrackets |
| title_full_unstemmed | Universal Lie Formulas for Higher Antibrackets |
| title_short | Universal Lie Formulas for Higher Antibrackets |
| title_sort | universal lie formulas for higher antibrackets |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147749 |
| work_keys_str_mv | AT manettim universallieformulasforhigherantibrackets AT ricciardig universallieformulasforhigherantibrackets |