ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

Topics on n-ary Algebraic Structures

J. A. de Azcárraga, J. M. Izquierdo

Abstract

We review the basic definitions and properties of two types of n-ary structures, the Generalized Lie Algebras (GLA) and
the Filippov (≡ n-Lie) algebras (FA), as well as those of their Poisson counterparts, the Generalized Poisson (GPS) and
Nambu-Poisson (N-P) structures. Wedescribe theFilippov algebra cohomology complexes relevant for the central extensions
and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that
they are rigid. This extends Whitehead’s lemma to all n ≥ 2, n =2 being the original Lie algebra case. Some comments on
n-Leibniz algebras are also made.

1 Introduction
TheJacobi identity (JI) forLie algebras g, [X, [Y, Z]]+
[Y, [Z, X]]+ [Z, [X, Y ]] = 0, may be looked at in two
ways. First, onemay see it as a consequence of the as-
sociativity of the composition of generators in the Lie
bracket. Secondly, it may be viewed as the statement
that the adjointmap is a derivation of the Lie algebra,
adX [Y, Z] = [adX Y, Z]+ [Y, adX Z].
A natural problem is to consider n-ary generaliza-

tions, i.e. to look for the possible characteristic iden-
tities that a n-ary bracket,

(X1, . . . , Xn) ∈ G×. . .×G �→ [X1, . . . , Xn] ∈ G, (1.1)

antisymmetric in its arguments (this may be relaxed;
see last section), may satisfy. When n > 2 two gener-
alizations of the JI suggest themselves. These are:
(a) Higher order Lie algebras or generalized Lie alge-
bras (GLA) G, proposed independently in [1, 2, 3]
and [4, 5, 6, 7]. Their bracket is defined by the
full antisymmetrization

[Xi1, . . . , Xin] :=∑
σ∈Sn

(−1)π(σ)Xiσ(1) . . . Xiσ(n) .
(1.2)

For n even, this definition implies the generalized
Jacobi identity (GJI)∑

σ∈S2n−1

(−1)π(σ)
[
[Xiσ(1), . . . , Xiσ(n)],

Xiσ(n+1) . . . , Xiσ(2n−1)
]
=0

(1.3)

which follows from the associtivity of the
products in (1.2) (for n odd, the r · h · s
is n!(n −1)![Xi1, . . . , Xi2n−1] rather than zero).
Chosenabasis ofG, the bracketmaybewrittenas
[Xi1, . . . , Xi2p] = Ωi1...i2p

j Xj, where the Ωi1...i2p
j

are the structure constants of the GLA.
(b) n-Lie or Filippov algebras (FA) G. The charac-
teristic identity that generalizes the n = 2 JI is

the Filippov identity (FI) [8]

[X1, . . . , Xn−1, [Y1, . . . Yn]] =
n∑

a=1

[Y1, . . . Ya−1, [X1, . . . , Xn−1, Ya], Ya+1, . . . Yn] .

(1.4)

If we introduce fundamental objects X =
(X1, . . . , Xn−1) antisymmetric in their (n−1) en-
tries and acting on G as

X · Z ≡ adXZ := [X1, . . . , Xn−1, Z] (1.5)
∀Z ∈ G ,

then the FI just expresses that adX is a deriva-
tion of the bracket,

adX[Y1, . . . , Yn] =
n∑

a=1

[Y1, . . . , adXYa, . . . , Yn] .
(1.6)

Chosen a basis, a FA may be defined through its
structure constants,

[Xa1 . . . Xan] = fa1...an
d Xd , (1.7)

and the FI is written as

fb1...bn
l fa1...an−1l

s =
n∑

k=1

fa1...an−1bk
l fb1...bk−1lbk+1...bn

s .
(1.8)

2 Some definitions and
properties of FA

The definitions of ideals, solvable ideals and semisim-
ple algebras can be extended to the n > 2 case as
follows [9]. A subalgebra I of G is an ideal of G if

[X1, . . . , Xn−1, Z] ⊂ I (2.9)
∀X1, . . . , Xn−1 ∈ G , ∀Z ∈ I .

To appear in the proceedings of the meeting Selected topics in mathematical and particle physics, May 5–7 2009 (Niederlefest),
held in Prague on occasion of the 70th birthday of Professor J. Niederle.

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Acta Polytechnica Vol. 50 No. 3/2010

An ideal I is (n-)solvable if the series

I(0) := I, I(1) := [I(0), . . . , I(0)], . . . ,

I(s) := [I(s−1), . . . , I(s−1)], . . .
(2.10)

ends. AFA is then semisimple if it does not have solv-
able ideals, and simple if [G, . . . , G] �= {0} and does
not contain non-trivial ideals. There is also a Cartan-
like criterion for semisimplicity [10]. Namely, A FA is
semisimple if

k(X, Y)= k(X1, . . . , Xn−1, Y1, . . . , Yn−1) :=

T r(adXadY)
(2.11)

is non-degenerate in the sense that

k(Z, G, n−2. . . , G, G, n−1. . . , G)=0 ⇒ Z =0 . (2.12)

It can also be shown [11] that a semisimple FA is the
sum of simple ideals,

G =
k⊕

s=1

Gs = G(1) ⊕ . . . ⊕ G(k) (2.13)

The derivations of a FA G generate a Lie algebra.
To see it, introduce first the composition of fundamen-
tal objects,

X · Y :=
n−1∑
a=1

(Y1, . . . , Ya−1, [X1, . . . , Xn−1, Ya], Ya+1, . . . , Yn−1)

(2.14)

which reflects that X acts as a derivation. It is then
seen that FI implies that

X · (Y · Z)− Y · (X · Z)= (X · Y) · Z, (2.15)
∀X, Y, Z ∈ ∧n−1G
adXadYZ − adYadXZ = adX·YZ, (2.16)
∀X, Y ∈ ∧n−1G, ∀Z ∈ G ,

which means that adX ∈ EndG satisfies adX·Y =
−adY·X. These two identities show that the inner

derivations adX associated with the fundamental ob-
jects X generate (the ad map is not necessarily injec-
tive) an ordinary Lie algebra, the Lie algebra associ-
ated with the FA G.
An important type of FAs, because of its rele-

vance inphysicalapplicationswherea scalarproduct is
usually needed (as in theBagger-Lambert-Gustavsson
model in M-theory), is the class of metric Filippov al-
gebras. These are endowed with a metric 〈, 〉 on G,
〈Y, Z〉= gabY a Zb, ∀ Y, Z ∈ G that is invariant i.e.,

X · 〈Y, Z〉 = 〈X · Y, Z〉+ 〈Y, X · Z〉
= 〈[X1, . . . , Xn−1, Y ], Z〉+ (2.17)
〈Y, [X1, . . . , Xn−1, Z]〉 =0 .

This means that the structure constants with all in-
dices down fa1...an−1bc are completely antisymmetric
since the invariance of g above implies fa1...an−1b

l glc+
fa1...an−1c

l gbl = 0. The fa1...an+1 define a skewsym-
metric invariant tensor under the action of X, since
the FI implies

n+1∑
i=1

fa1...an−1bi
l fb1...bi−1lbi+1...bn+1 =0

or LX .f =0 .

(2.18)

3 Examples of n-ary structures

3.1 Examples of GLAs

Let n = 2p. We look for structure constants Ωi1...i2p
j

that satisfy the GJI (1.3) i.e., such that

Ω[j1...j2p
lΩj2p+1...j4p−1]l

s =0 . (3.19)

It turns out [3, 2] that given a simple compact Lie
algebra, the coordinates of the (odd) cocyles for the
Lie algebra cohomology satisfy the GJI identity (1.2).
These provide the structure constants of an infinity of
GLAs,withbracketswith n =2(mi−1) entries (where
i =1, . . . , l and l is the rank of the algebra), according
to the table below:

g dimg Orders mi of invariants (and Casimirs) Orders 2mi −1 of g-cocycles

Al (l +1)
2 −1 [l ≥ 1] 2,3, . . . , l +1 3,5, . . . ,2l +1

Bl l(2l +1) [l ≥ 2] 2,4, . . . ,2l 3,7, . . . ,4l −1
Cl l(2l +1) [l ≥ 3] 2,4, . . . ,2l 3,7, . . . ,4l −1
Dl l(2l −1) [l ≥ 4] 2,4, . . . ,2l −2, l 3,7, . . . ,4l −5,2l −1
G2 14 2,6 3,11

F4 52 2,6,8,12 3,11,15,23

E6 78 2,5,6,8,9,12 3,9,11,15,17,23

E7 133 2,6,8,10,12,14,18 3,11,15,19,23,27,35

E8 248 2,8,12,14,18,20,24,30 3,15,23,27,35,39,47,59

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Acta Polytechnica Vol. 50 No. 3/2010

3.2 Examples of FAs

An important example of finite Filippov algebras is
provided by the real euclidean simple n-Lie algebras
An+1 defined on an euclidean (n+1)-dimensional vec-
tor space. Let us fix a basis {ei} (i = 1, . . . , n +1).
The basic commutators are given by

[e1 . . . êi . . . en+1] = (−1)n+1ei

or [ei1 . . . ein] = (−1)
n

n+1∑
i=1

�i1...in
iei .

(3.20)

There are also infinite-dimensional FAs that gen-
eralize the ordinary Poisson algebra by means of the
bracket of n functions fi = fi (x1, x2, . . . , xn) defined
by

[f1, f2, . . . , fn] := �
i1...in
1 ... n ∂i1f

1 . . . ∂in f
n =∣∣∣∣ ∂(f1, f2, . . . , fn)∂(x1, x2, . . . , xn)

∣∣∣∣ , (3.21)
considered by Nambu [12] specially for n = 3. The
conmmutators in (3.20) and the above Jacobian n-
bracket satisfy the FI, which can be checked by using
the Schouten identities technique. All these examples
are also metric FAs.

4 n-ary Poisson generalizations

Both GLAs and FAs have n-ary Poisson structure
counterparts. These satisfy the associated GJI and
FI characteristic identities, to which Leibniz’s rule is
added.

4.1 Generalized Poisson structures
(GPS)

The generalized Poisson structures [2] (GPS, n even)
are defined by brackets {f1, . . . , fn} where the fi,
i = 1, . . . , n, are functions on a manifold. They are
skewsymmtric

{f1, . . . , fi, . . . , fj, . . . , fn} =
−{f1, . . . , fj, . . . , fi, . . . , fn} ,

(4.22)

satisfy the Leibniz identity,

{f1, . . . , fn−1, gh} =
g{f1, . . . , fn−1, h}+{f1, . . . , fn−1, g}h ,

(4.23)

and the characteristic identity of the GLAs, the GJI
(1.3), ∑

σ∈S4s−1

(−1)π(σ){fσ(1), . . . , fσ(2s−1),

{fσ(2s), . . . , fσ(4s−1)}} =0 .
(4.24)

As with ordinary Poisson structures, there are lin-
ear GPS given in terms of coordinates of the odd co-
cyles of the g in the table of Sec. 3.1. They are given
by the multivector

Λ=
1

(2m −2)!
Ωi1...i2m−2

σ
· xσ∂

i1 ∧ . . . ∧ ∂i2m−2 (4.25)

since, as it may be checked [2], Λ has zero Schouten-
Nijenhuis bracket with itself, [Λ,Λ]SN =0, which cor-
responds to the GJI. All GLAs associatedwith a sim-
ple algebra define linear GPS.

4.2 Nambu-Poisson structures (N-P)

These are defined by relations (4.22) and (4.23), but
now the characteristic identity is the FI,

{f1, . . . , fn−1, {g1, . . . , gn}} =
{{f1, . . . , fn−1, g1}, g2, . . . , gn}+
{g1, {f1, . . . , fn−1, g2}, g3, . . . , gn}+ . . . + (4.26)
{g1 . . . , gn−1, {f1, . . . , fn−1, gn}} .

The Filippov identity for the (Nambu) jacobians of
n functions was first written by Filippov [8], and
by Sahoo and Valsakumar [13] and Takhtajan [14]
(who called it fundamental identity) in the context of
Nambu mechanics [12]. Physically, the FI is a consis-
tency condition for the time evolution [13, 14], given
in terms of (n − 1) ‘hamiltonian’ functions that cor-
respond to the adX derivations of a FA. Every even
N-P structure is also aGPS, but the conversedoes not
hold.
Thequestionof thequantizationofNambu-Poisson

mechanics has been the subject of a vast literature; it
is probably fair to say that it remains a problem (for
n > 2!) aggravated by the fact that there are not so
many physical examples ofN-Pmechanical systems to
be quantized. We shall just refer here to [15, 16, 17],
from which the earlier literature can be traced.

5 Lie algebra cohomology,
extensions and deformations

Given a Lie algebra g, the p-cochains of the Lie alge-
bra cohomology are p-antisymmetric, V -valued maps
(where V is a g-module),

Ωp : g ×
p
· · ·×g → V , ΩA =

1
p!
ΩAi1...ip ω

i1 ∧ . . . ∧ ωip ,
(5.27)

where {ωi} is a basis of the coalgebra g∗. The
coboundary operator (for the left action) s : Ωp ∈
Cp(g, V ) �→ (sΩp) ∈ Cp+1(g, V ), s2 =0, is given by

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Acta Polytechnica Vol. 50 No. 3/2010

(sΩp)A (X1, . . . , Xp+1) :=
p+1∑
i=1

(−)i+1ρ(Xi)A.B Ω
pB(X1, . . . , X̂i, . . . , Xp+1) (5.28)

+
p+1∑

j,k=1
j<k

(−)j+kΩpA([Xj , Xk], X1, . . . ,

X̂j , . . . , X̂k, . . . , Xp+1) .

For the trivial action (ρ =0), this simplifies to

(sΩp)(X1, . . . , Xp+1)=
p+1∑
1≤j<k

(−1)jΩp(X1, . . . , X̂j, . . . , adXj Xk, . . . , Xp+1) .

(5.29)

The p-cocycles Ωp ∈ Zpρ(g, V ) are p-cochains such
that sΩp = 0; the p-coboundaries Ωp ∈ Bpρ(g, V )
are such that Ωp = sΩp−1 for some (p − 1)-cochain.
The p-th cohomology groups are then Hpρ(g, V ) :=
Zpρ(g, V )/B

p
ρ(g, V ).

For semisimple Lie algebras, Whitehead’s lemma
states that H20(g)= 0, H

2
ad(g, g)= 0. Hence, semisim-

pleLie algebrasdonotadmit non-trivial central exten-
sions and are moreover rigid (non-deformable) since
central extensions and infinitesimal deformations are
governed, respectively, by H20(g) and H

2
ad(g, g). Let us

now turn to the FA case.

6 Central extensions and
deformations of FAs

6.1 Central extensions of a FA
GivenaFilippov algebra G with n-bracket [. . . ], a cen-
tral extension G̃ has the form

[X̃a1, . . . , X̃an] := f
b
a1...an

X̃b + α
1(X1, . . . , Xn)Ξ ,

[X̃1, . . . , X̃n−1,Ξ]=0 , (6.30)

X̃ ∈ G̃ , α1 ∈ ∧n−1G∗ ∧ G∗ .

If α1 defines a centrally extended FA, it must satisfy
the condition that follows from the FI for the above
bracket. If we now introduce p-cochains as maps

αp ∈ ∧n−1G∗ ⊗ . . . ⊗∧n−1G∗ ∧ G∗ ,
αp : (X1, . . . , Xp, Z) �→ αp(X1, . . . , Xp, Z) ,

(6.31)

the condition imposed by the FI on the one-cochain
in (6.30), written in terms of the fundamental objects
with Yn = Z, reads

α1(X, Y · Z)− α1(X · Y, Z)− α1(Y, X · Z) ≡
(δα1)(X, Y, Z)= 0 .

(6.32)

Note that α1 above would become a two-cochain for
n = 2; we define the order of the p-cochains for FAs

(n ≥ 3) as thenumber p of fundamental objects among
the arguments of the cochain (for aLie algebra X = X
and p counts the number of algebra elements).
A central extension is actually trivial if it is pos-

sible to find new generators X̃′ = X̃ − β(X)Ξ such
that

[X̃′a1, . . . , X̃
′
an
] = f ba1...an X̃

′
b =

f ba1...an X̃b − β([Xa1, . . . , Xan])Ξ .

But this is equivalent to saying (with Xan = Z) that

α1(X1, . . . , Xn−1, Z)= −β([X1, . . . , Xn−1, Z]), (6.33)

which may be rewritten in the form

α1(X, Z)= −β([X1, . . . , Xn−1, Z])≡
(δβ)(X1, . . . , Xn−1, Z)≡ (δβ)(X, Z) ,

(6.34)

where β is the zero-cochain β ∈ G∗ generating the one-
cocycle. Therefore, central extensions of FAs are char-
acterized by one-cocycles modulo one-coboundaries.
The above suffices to define the full FA coho-

mology complex suitable for central extensions. Let
αp ∈ ∧n−1G∗ ⊗ . . . ⊗ ∧n−1G∗ ∧ G∗ be a p-cochain.
Then (C•0(G), δ) is defined by

(δα)(X1, . . . , Xp+1, Z)= (6.35)
p+1∑
1≤i<j
(−1)iα(X1, . . . , X̂i, . . . , Xi · Xj , . . . , Xp+1, Z)+

p+1∑
i=1

(−1)iα(X1, . . . , X̂i, . . . , Xp+1, Xi · Z) .

Defining p-cocycles and p-coboundaries as usual, the
p-th FA cohomology group (for the trivial action) is
H

p
0(G) =Z

p
0(G)/B

p
0(G). Therefore, a FA admits non-

trivial central extensions when H10(G) �=0.

6.2 Infinitesimal deformations of FAs

An infinitesimaldeformationofaFAinGerstenhaber’s
[18] sense is obtained by modifying the n-bracket as

[X1, . . . , Xn]t =

[X1, . . . , Xn]+ tα
1(X1, . . . , Xn) ,

(6.36)

where α1 is now G-valued, so that G will now act on
it. Again, the FI constrains α1 by

[X1, . . . , Xn−1, [Y1, . . . , Yn]t]t = (6.37)
n∑

a=1

[Y1, . . . , Ya−1, [X1, . . . , Xn−1, Ya]t, Ya+1, . . . , Yn]t

which, with Yn = Z, may we rewritten as

[X,(Y · Z)t]t = [(X · Y)t, Z]t +[Y,(X · Z)t]t . (6.38)

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Acta Polytechnica Vol. 50 No. 3/2010

At first order in t, this gives the following condition
on α1:

[X1, . . . , Xn−1, α
1(Y1, . . . , Yn)]+

α
1(X1, . . . , Xn−1, [Y1, . . . , Yn]) = (6.39)
n∑

a=1

[Y1, . . . , Ya−1, α
1(X1, . . . , Xn−1, Ya), Ya+1, . . . , Yn]+

n∑
a=1

α
1(Y1, . . . , Ya−1, [X1, . . . , Xn−1, Ya], Ya+1, . . . , Yn) .

In terms of the fundamental objects andwith Yn = Z,
this may be read as a one-cocycle conditon for α1,

(δα)(X, Y, Z)= adXα(Y, Z)−
adYα(X, Z)− (α(X, ) · Y) · Z−
α(X · Y, Z)− α(Y, X · Z)+ α(X, Y · Z)= 0 ,

(6.40)

where, for instance for n =3,

α1(X, ) · Y :=
(α1(X, ) · Y1, Y2)+(Y1, α1(X, ) · Y2)= (6.41)
(α1(X, Y1), Y2)+(Y1, α

1(X, Y2)) .

To see whether the G-valued cocycle α1 is a
coboundary, we look for the possible triviality of the
infinitesimal deformation. It will be trivial if new
generators can been found in terms of β, β : G →
G , X′i = Xi − tβ(Xi), such that

[X′1, . . . , X
′
n]t = [X1, . . . , Xn]

′ ≡
[X1, . . . , Xn]− tβ([X1, . . . , Xn]) .

(6.42)

At first order in t this implies

[X′1, . . . , X
′
n]t = [X1, . . . , Xn]t −

t

n∑
a=1

[X1, . . . , Xa−1, β(Xa), Xa+1, . . . , Xn]t =

[X1, . . . , Xn]+ tα
1(X1, . . . , Xn)− (6.43)

t

n∑
a=1

[X1, . . . , Xa−1, β(Xa), Xa+1, . . . , Xn] .

Therefore, the deformation is trivial if

(α1)(X1, . . . , Xn) := −β([X1, . . . , Xn])+
n∑

a=1

[X1, . . . , Xa−1, β(Xa), Xa+1, . . . , Xn] ≡

(δβ)(X, Xn)

(6.44)

i.e., when

α1(X, Z)= (δβ)(X, Z)=

−β(X · Z)+(β( ) · X) · Z + X · β(Z) .
(6.45)

If all one-cocycles are trivial, the FA is stable or rigid.
The above allows us to write the full complex

(C•ad(G, G), δ) adapted to the deformations of FA

problem(see [21] for details). The p-cochainsaremaps

αp : ∧(n−1)G ⊗
p
· · ·⊗∧(n−1) G ∧G → G and the action

of the coboundary operator δ is now defined by

(δαp)(X1, . . . , Xp, Xp+1, Z)=
p+1∑
1≤j<k

(−1)jαp(X1, . . . , X̂j , . . . , Xk−1, Xj ·

Xk, Xk+1, . . . , Xp+1, Z)+ (6.46)
p+1∑
j=1

(−1)j αp(X1, . . . , X̂j , . . . , Xp+1, Xj · Z)+

p+1∑
j=1

(−1)j+1Xj · αp(X1, . . . , X̂j . . . , Xp+1, Z)+

(−1)p(αp(X1, . . . , Xp, ) · Xp+1) · Z ,

where in the last term

αp(X1, . . . , Xp, ) · Y =
n−1∑
i=1

(Y1, . . . , α
p(X1, . . . , Xp, Yi), . . . , Yn−1) .

(6.47)

The above cohomology complex [21] is essentially
equivalent to that given by Gautheron [19] and
Rotkiewicz [20].

7 Whitehead lemma for FAs
It follows from the above discussion that an ana-
logue of the Whitehead lemma for FAs would require
H10(G)= 0 and H

1
ad(G, G)=0 for G semisimple. This

may be proven taking advantage of the fact that all
simple FAs have the same general structure [11, 8].
Specifically, in Filippov’s notation, they have the form

[e1 . . . êi . . . en+1] = (−1)n+1εiei

or [ei1 . . . ein] = (−1)
n

n+1∑
i=1

εi�i1...in
iei ,

(7.48)

where εi = ±1. In otherwords, the simple FAs are the
euclidean An+1 and theLorentzian As,t, (s+t = n+1)
generalizations of the n = 2 so(3) and so(1,2) Lie al-

gebras, [ei, ej] =
∑

k

εk�ijkek, for which Whitehead’s

lemma does apply.
Define the Z10(G) and Z

1
ad(G, G) cocycles by its co-

ordinates,

α1i1...in = α
1(ei1, . . . , ein) , (7.49)

α1i1...in
j = α1(ei1, . . . , ein)

j , i, j =1, . . . ,(n +1) .

Using the explicit form of the simple FAs, it is possi-
ble to show [21] that the above cocycles are necessarily

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Acta Polytechnica Vol. 50 No. 3/2010

one-coboundaries respectively generated by the zero-
cochains βk, β

r
k i.e., that

α1i1...in = β([ei1 . . . ein]) = εk�i1...in
kβk ⇒

βk =
εk
n!

n+1∑
i1,...,in=1

�i1...in k α
1
i1...in

;

α1i1...in
r = −(−1)n

n+1∑
s=1

εs�i1...in
sβr s + (7.50)

(−1)n
n∑

a=1

n+1∑
r=1

εs�i1...ia−1aia+1...in
sβr ia ⇒

βrs = −
(−1)n
2

[
εs(α

1)rs −
1

n −1

n+1∑
t=1

εt(α
1)ttδ

rs

]
.

The (α1)rs above is the Poincaré dual (with �11...inr)
of α1i1...in

s; it may be seen to be (rs)-symmetric be-
cause of the cocycle condition. Therefore, H10(G) =
0 , H1ad(G, G) = 0 for a simple FA. Using now that a
semisimple FA is the sum (2.13) of simple ideals the
following result is obtained [21]:
Lemma (Whitehead lemma for n ≥ 2)
Semisimple Filippov (n-Lie) algebras, n ≥ 2, do

not admit non-trivial central extensions and are,more-
over, rigid.

8 A comment on FA and
Leibniz algebra cohomology

Leibniz algebras [22] L are a non-commutative ver-
sion of Lie algebras: their bracket does not need being
anticommutative ([X, Y ] �= −[Y, X]) but still satisfies
the (left, say) ‘Leibniz’ identity

[X, [Y, Z]] = [[X, Y ], Z]+ [Y, [[X, Z]] . (8.51)

Lie algebras are Leibniz algebras the bracket of which
is anticommutative.
Similarly, one may define n-Leibniz algebras L

[23, 24] by dropping the anticommutatitivity of theFA
n-bracketwhile keeping the (left, say) FI. Introducing
also here fundamental objects for L, the identity reads

X · (Y · Z)= (X · Y) · Z + Y · (X · Z)
∀X, Y, Z ∈ ⊗n−1L .

(8.52)

Note that now X ∈ ⊗n−1L since, in contrastwithFAs,
the anticommutativity of the (n −1) arguments of X
is not assumed. The above is still the (left) FI (1.4)
previously defining FAs; n-Lie algebras are n-Leibniz
algebras thebracket ofwhich is fully anticommutative.
As a result, the characteristic FI

X · (Y · Z)− Y · (X · Z)= (X · Y) · Z
∀ X, Y, Z ∈ ⊗n−1L ,

(8.53)

which determined the nilpotency of the coboundary
operator δ and the differentFAcohomologycomplexes

(as the JI for Lie algebras), still holds here. Therefore,
with a suitable definition of p-cochains, the n-Leibniz
and the above FA cohomological complexes have the
same structure. In fact, n-Leibniz cohomology under-
lies n-Lie cohomology. This is why the N-P cohomol-
ogymaybe studied fromthepoint of viewof n-Leibniz
cohomology [23].
For instance, for n = 2 and reverting to the no-

tation that labels the cochains by the number of el-
ements of the algebra it contains, αp ∈ Cp(L, L) =
Hom(⊗pL, L), eq. (6.46) for the n-Lie case reduces to

(δαp)(X1, . . . , Xp, Xp+1)=
p+1∑
1≤j<k

(−1)jαp(X1, . . . , X̂j , . . . , Xk−1, [Xj , Xk],

Xk+1, . . . , Xp+1)+ (8.54)
p∑

j=1

(−1)j+1Xj · αp(X1, . . . , X̂j . . . , Xp+1)+

(−1)p+1(αp(X1, . . . , Xp) · Xp+1 ,

which coincides with the cohomology complex
(C•(L, L), δ) for Leibniz algebras L [25, 24].
Our proof for theWhiteheadLemma forFAs, how-

ever, relied on the antisymmetry of the n-commutator,
and thus it will not hold when the anticommutativity
is relaxed. Thus, one might expect having a richer
deformation structure for Leibniz deformations. This
has been observed already for the n = 2 case [26] by
looking at Leibniz deformations of a Lie algebra, and
a specific Leibniz deformation of the euclidean 3-Lie
algebra has been found [27]. Thus, a natural exten-
sion of our work is to look e.g. at n-Leibniz defor-
mations of simple n-Lie algebras to see whether this
opensmore possibilities. Our results [28] for n-Leibniz
deformations with brackets that keep the antisymme-
try in their first n − 1 arguments show that rigidity
still holds for n > 3.

Acknowledgement

This work has been partially supported by the re-
search grants FIS2008-01980 and FIS2009-09002 from
the Spanish MICINN, and VA013C05 from the Junta
de Castilla y León (Spain).

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J. A. de Azcárraga
Departamento de F́ısica Teórica and IFIC
(CSIC-UVEG)
Univ. de Valencia
46100-Burjassot (Valencia), Spain

J. M. Izquierdo
Departamento de F́ısica Teórica
Universidad de Valladolid
47011-Valladolid, Spain

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