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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 35, Num. 1 (2020), 99 – 126

doi:10.17398/2605-5686.35.1.99

Available online November 12, 2019

Generalized representations of 3-Hom-Lie algebras

S. Mabrouk 1, A. Makhlouf 2, S. Massoud 3

1 University of Gafsa, Faculty of Sciences Gafsa, 2112 Gafsa, Tunisia
2 Université de Haute Alsace, IRIMAS-département de Mathématiques

6, rue des Frères Lumière F-68093 Mulhouse, France
3 Université de Sfax, Faculté des Sciences, Sfax Tunisia

Mabrouksami00@yahoo.fr , Abdenacer.Makhlouf@uha.fr , sonia.massoud2015@gmail.com

Received June 6, 2019 Presented by Consuelo Mart́ınez
Accepted September 30, 2019

Abstract: The propose of this paper is to extend generalized representations of 3-Lie algebras to

Hom-type algebras. We introduce the concept of generalized representation of multiplicative 3-

Hom-Lie algebras, develop the corresponding cohomology theory and study semi-direct products.
We provide a key construction, various examples and computation of 2-cocycles of the new coho-

mology. Also, we give a connection between a split abelian extension of a 3-Hom-Lie algebra and a
generalized semidirect product 3-Hom-Lie algebra.

Key words: 3-Hom-Lie algebra, representation, generalized representation, cohomology, abelian

extension.

AMS Subject Class. (2010): 17A42, 17B10.

Introduction

The first instances of ternary Lie algebras appeared first in Nambu’s gen-
eralization of Hamiltonian mechanics [23], which was formulated algebraically
by Takhtajan [29]. The structure of n-Lie algebras was studied by Filippov
[15] then completed by Kasymov in [21].

The representation theory of n-Lie algebras was first introduced by Kasy-
mov in [21]. The adjoint representation is defined by the ternary bracket in
which two elements are fixed. Through fundamental objects one may also
represent a 3-Lie algebra and more generally an n-Lie algebra by a Leibniz
algebra [11]. The cohomology of n-Lie algebras, generalizing the Chevalley-
Eilenberg Lie algebras cohomology, was introduced by Takhtajan [30] in its
simplest form, later a complex adapted to the study of formal deformations
was introduced by Gautheron [17], then reformulated by Daletskii and Takhta-
jan [11] using the notion of base Leibniz algebra of an n-Lie algebra. In [2, 3],
the structure and cohomology of 3-Lie algebras induced by Lie algebras has
been investigated.

ISSN: 0213-8743 (print), 2605-5686 (online)

https://doi.org/10.17398/2605-5686.35.1.99
mailto:Mabrouksami00@yahoo.fr
mailto:Abdenacer.Makhlouf@uha.fr
mailto:sonia.massoud2015@gmail.com\ 
https://www.eweb.unex.es/eweb/extracta/
https://creativecommons.org/licenses/by-nc/3.0/


100 s. mabrouk, a. makhlouf, s. massoud

The concept of generalized representation of a 3-Lie algebra was introduced
by Liu, Makhlouf and Sheng in [19]. They study the corresponding generalized
semidirect product 3-Lie algebra and cohomology theory. Furthermore, they
describe general abelian extensions of 3-Lie algebras using Maurer-Cartan
elements. Non-abelian extensions were explored in [26].

The aim of this paper is to extend the concept of generalized representation
of 3-Lie algebras to Hom-type algebras. The notion of Hom-Lie algebras was
introduced by Hartwig, Larsson, and Silvestrov in [18] as part of a study of
deformations of the Witt and the Virasoro algebras. The n-Hom-Lie algebras
and various generalizations of n-ary algebras were considered in [4]. In a
Hom-Lie algebra, the Jacobi identity is twisted by a linear map, called the
Hom-Jacobi identity. In particular, representations and cohomologies of Hom-
Lie algebras were studied in [25], while the representations and cohomology
of n-Hom-Lie algebras were first studied in [1].

The paper is organized as follows. In Section 1, we provide some basics
about 3-Hom-Lie algebras, representations and cohomology. The second Sec-
tion includes the new concept of generalized representation of a 3-Hom-Lie
algebra, extending to Hom-type algebras the notion and results obtained in
[19]. We define a corresponding semi-direct product and provide a twist pro-
cedure leading to generalized representations of 3-Hom-Lie algebras starting
from generalized representations of 3-Hom-Lie algebras and algebra maps.
In Section 3, we construct a new cohomology corresponding to generalized
representations and show examples. In the last section we discuss abelian
extensions of multiplicative 3-Hom-Lie algebras. One recovers the results in
[19] when the twist map is the identity.

1. Representations of 3-Hom-Lie algebras

The aim of this section is to recall some basics about 3-Lie algebras and
3-Hom-Lie algebras. We refer mainly to [15] and [4]. In this paper, all vector
spaces are considered over a field K of characteristic 0.

Definition 1.1. A 3-Lie algebra is a pair (g, [·, ·, ·]) consisting of a K-
vector space g and a trilinear skew-symmetric multiplication [·, ·, ·] satisfying
the Filippov-Jacobi identity: for x,y,z,u,v in g

[u,v, [x,y,z]] = [[u,v,x],y,z] + [x, [u,v,y],z] + [x,y, [u,v,z]].

In this paper, we are dealing with 3-Hom-Lie algebras corresponding to
the following definition.



generalized representations of 3-hom-lie algebras 101

Definition 1.2. A 3-Hom-Lie algebra is a triple (g, [·, ·, ·],α) consisting
of a K-vector space g, a trilinear skew-symmetric multiplication [·, ·, ·] and
an algebra map α : g → g satisfying the Hom-Filippov-Jacobi identity: for
x,y,z,u,v in g

[α(u),α(v), [x,y,z]] = [[u,v,x],α(y),α(z)] + [α(x), [u,v,y],α(z)]

+ [α(x),α(y), [u,v,z]].

Remark 1.3. There is more general definition of 3-Hom-Lie algebras which
are given by a quadruple (g, [·, ·, ·],α1,α2) consisting of a K-vector space g,
two linear maps α1,α2 : g → g and a trilinear skew-symmetric multiplication
[·, ·, ·] satisfying the following generalized Hom-Filippov-Jacobi identity: for
x,y,z,u,v in g

[α1(u),α2(v), [x,y,z]] = [[u,v,x],α1(y),α2(z)] + [α1(x), [u,v,y],α2(z)]

+ [α1(x),α2(y), [u,v,z]].

We get our class of 3-Hom-Lie algebras when α1 = α2 = α and where α is
an algebra morphism. This kind of algebras are usually called multiplicative
3-Hom-Lie algebras.

Proposition 1.4. Let (g, [·, ·, ·]) be a 3-Lie algebra and α : g → g be
a 3-Lie algebra morphism. Then (g, [·, ·, ·]α := α ◦ [·, ·, ·],α) is a 3-Hom-Lie
algebra.

Let (g, [·, ·, ·],α) be a 3-Hom-Lie algebra, elements in ∧2g are called fun-
damental objects of the 3-Hom-Lie algebra (g, [·, ·, ·],α). There is a bilinear
operation [·, ·]L on ∧2g, which is given by

[X,Y ]L = [x1,x2,y1] ∧α(y2) + α(y1) ∧ [x1,x2,y2]

for all X = x1 ∧ x2 and Y = y1 ∧ y2, and a linear map α on ∧2g defined
by α(X) = α(x1) ∧ α(x2), for simplicity, we will write α(X) = α(X). It is
well-known that (∧2g, [·, ·]L,α) is a Hom-Leibniz algebra [1, 31].

Definition 1.5. A representation of a 3-Hom-Lie algebra (g, [·, ·, ·],α) on
a vector space V with respect to A ∈ gl(V ) is a skew-symmetric linear map
ρ : ∧2g → End(V ) such that

ρ(α(x1),α(x2)) ◦A = A◦ρ(x1,x2), (1.1)
ρ(α(x1),α(x2))ρ(x3,x4) −ρ(α(x3),α(x4))ρ(x1,x2) (1.2)

=
(
ρ([x1,x2,x3],α(x4)) −ρ([x1,x2,x4],α(x3))

)
◦A,



102 s. mabrouk, a. makhlouf, s. massoud

ρ([x1,x2,x3],α(x4)) ◦A−ρ(α(x2),α(x3))ρ(x1,x4) (1.3)
= ρ(α(x3),α(x1))ρ(x2,x4) + ρ(α(x1),α(x2))ρ(x3,x4),

for x1,x2,x3 and x4 in g.

Theorem 1.6. Let (g, [·, ·, ·]) be a 3-Lie algebra, (V,ρ) be a representa-
tion, α : g → g be a 3-Lie algebra morphism and A : V → V be a linear map
such that A ◦ ρ(x1,x2) = ρ(α(x1),α(x2)) ◦ A. Then (V,ρ̃ := A ◦ ρ,A) is a
representation of the 3-Hom-Lie algebra (g, [·, ·, ·]α := α◦ [·, ·, ·],α).

Proof. Let xi ∈ g, where 1 ≤ i ≤ 5. Then we have

ρ̃([x3,x4,x5]α,α(x1))◦A− ρ̃(α(x3),α(x4))ρ̃(x5,x1) − ρ̃(α(x4),α(x5))ρ̃(x3,x1)
− ρ̃(α(x5),α(x3))ρ̃(x4,x1)

= A2◦(ρ([x3,x4,x5],x1) −ρ(x3,x4)ρ(x5,x1)
−ρ(x4,x5)ρ(x3,x1) −ρ(x5,x3)ρ(x4,x1)) = 0.

The second condition (1.2) is obtained similarly.

The previous result allows to twist along morphisms a 3-Lie algebra with
a representation to a 3-Hom-Lie algebra with a corresponding representation.

Proposition 1.7. Let (g, [·, ·, ·],α) be a 3-Hom-Lie algebra, V be a vector
space, A ∈ gl(V ) and ρ : ∧2g → gl(V ) be a skew-symmetric linear map. Then
(V ; ρ,A) is a representation of 3-Hom-Lie algebra g if and only if there is a
3-Hom-Lie algebra structure (g⊕V, [·, ·, ·]ρ,αg⊕V ) on the direct sum of vector
spaces g⊕V , defined by

[x1 + v1,x2 + v2,x3 + v3]ρ = [x1,x2,x3] + ρ(x1,x2)v3

+ ρ(x3,x1)v2 + ρ(x2,x3)v1,

and αg⊕V = α + A, for all xi ∈ g, vi ∈ V , 1 ≤ i ≤ 3. The obtained 3-Hom-Lie
algebra is denoted by g nρ V and called semidirect product.

Let (g, [·, ·, ·],α) be a 3-Hom-Lie algebra and (V,ρ,A) be a representation
of g. We denote by C

p
α,A(g,V ) the space of all linear maps

ϕ : ∧2g⊗···⊗∧2g︸ ︷︷ ︸
(p−1)

∧g → V



generalized representations of 3-hom-lie algebras 103

satisfying:

A◦ϕ(X1 ⊗···⊗Xp−1,y) = ϕ(α(X1) ⊗···⊗α(Xp−1),α(y)),

for all X1, . . . ,Xp−1 ∈∧2g, y ∈ g. It is called the space of p-cochains.
Let ϕ be a (p − 1)-cochain, the coboundary operator δρ : C

p−1
α,A (g,V ) →

C
p
α,A(g,V ) is given by

(δρϕ)(X1, . . . ,Xp,z)

=
∑

1≤j<k
(−1)jϕ(α(X1), . . . ,X̂j, . . . ,α(Xk−1), [Xj,Xk]L,α(Xk+1),

. . . ,α(Xp),α(z))

+

p∑
j=1

(−1)jϕ(α(X1), . . . ,X̂j, . . . ,α(Xp), [Xj,z]) (1.4)

+

p∑
j=1

(−1)j+1ρ(αp(Xj))ϕ(X1, . . . ,X̂j, . . . ,Xp,z)

+ (−1)p+1
(
ρ(αp(yp),α

p(z))ϕ(X1, . . . ,Xp−1,xp)

+ ρ(αp(z),αp(xp))α(X1, . . . ,Xp−1,yp)
)
,

for all Xi = (xi,yi) ∈ ∧2g, z ∈ g and where [Xi,z] = [xi,yi,z]. An element
ϕ ∈ Cp−1α,A (g,V ) is called a p-cocycle if δρϕ = 0. It is called a p-coboundary if
there exists some f ∈ Cp−2α,A (g,V ) such that ϕ = δρf. Denote by Z

p
3HL(g; V )

and B
p
3HL(g; V ) the sets of p-cocycles and p-coboundaries respectively. Then

the p-th cohomology group is

H
p
3HL(g; V ) = Z

p
3HL(g; V )/B

p
3HL(g; V ). (1.5)

In [28], the author constructed a graded Lie algebra structure by which one
can describe an n-Leibniz algebra structure as a canonical structure. Here,
we give the precise formulas for the 3-Hom-Lie algebra case, generalizing the
result in [19].

Set Cα,α(g,g) = ⊕p≥0C
p
α,α(g,g). Let ϕ ∈ C

q
α,α(g,g), ψ ∈ C

p
α,α(g,g),

p,q ≥ 0, Xi = xi ∧yi ∈∧2g for i = 1, 2, . . . ,p + q and x ∈ g. For each subset
J = {j1, . . . ,jp}j1<···<jp ⊂ N , {1, 2, . . . ,p + q}, let I = {i1, . . . , iq}i1<···<iq
= N/J. Define on the graded vector space Cα,α(g,g) the graded commutator
bracket

[ϕ,ψ]3HL = (−1)pqjαψ(ϕ) − j
α
ϕ(ψ) = (−1)

pqϕ◦α ψ −ψ ◦α ϕ,



104 s. mabrouk, a. makhlouf, s. massoud

where ϕ◦α ψ ∈ C
p+q
α,α (g,g) is defined by

jαψ(ϕ)(X1, . . . ,Xp+q,x) = ϕ◦α ψ(X1, . . . ,Xp+q,x)

=
∑

J,jq<ik+1≤p+q
(−1)(J,I)ϕ

(
αp(Xi1 ), . . . ,α

p(Xik ),

ψ(Xj1, . . . ,Xjp, ) •α Xik+1, . . . ,α
p(Xiq ),α

p(x)
)

+
∑
J

(−1)(J,I)(−1)qϕ
(
αp(Xi1 ), . . . ,α

p(Xiq ),ψ(Xj1, . . . ,Xjp,x)
)
,

with

ψ(Xj1, . . . ,Xjq, ) •α Xik+1 = ψ(Xj1, . . . ,Xjp,xik+1 ) ∧α
p(yik+1 )

+ αp(xik+1 ) ∧ψ(Xj1, . . . ,Xjp,yik+1 )

where k is uniquely determined by the condition jp ≤ ik+1 and if jp ≤ i1
then jp = p, i1 = p + 1 and (−1)(J,I) is the sign of the permutation (J,I) =
(j1, . . . ,jp, i1, . . . , iq) of N.

We need the following lemma to establish a structure of graded Lie algebra
on Cα,α(g,g) .

Lemma 1.8. We have j[ϕ,ψ]3HL = −[jαϕ,jαψ] for all ϕ,ψ ∈ Cα,α(g,g), where
[·, ·] is the graded commutator on End(Cα,α(g,g)).

Proof. Let ϕ ∈ Cqα,α(g,g), ψ ∈ C
p
α,α(g,g), ξ ∈ Crα,α(g,g), X1, . . . ,Xp+q+r ∈

∧2g and x ∈ g

[jαϕ,j
α
ψ](ξ)(X1,X2, . . . ,Xp+q+r,x)

= (jαϕ(j
α
ψξ) − (−1)

pqjαψ(j
α
ϕξ))(X1,X2, . . . ,Xp+q+r,x)

= D1 − (−1)pqD2,

where

D1 = j
α
ϕ(j

α
ψξ)(X1,X2, . . . ,Xp+q+r,x),

D2 = j
α
ψ(j

α
ϕξ))(X1,X2, . . . ,Xp+q+r,x).

For each subset J = {j1, . . . ,jq}j1<···<jq ⊂ N , {1, 2, . . . ,p + q + r}, let
I = {i1, . . . , ip+r}i1<···<ip+r = N \ J and L = {l1, . . . , lp}l1<···<lp ⊂ I ,
{i1, i2, . . . , ip+r}, let H = {h1, . . . ,hr}h1<···<hr = I \L.



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We have for D1 :

jαϕ(j
α
ψξ)(X1,X2, . . . ,Xp+q+r,x) = (j

α
ψ(ξ)) ◦α ϕ(X1,X2, . . . ,Xp+q+r,x)

=
∑

J,jq<ik+1≤p+q+r
(−1)(J,I)(jαψ(ξ))(α

q(Xi1 ), . . . ,α
q(Xik ),ϕ(Xj1, . . . ,Xjq, ) •α Xik+1,α

q(Xik+2 ), . . . ,α
q(Xip+r ),α

q(x))

+
∑
J

(−1)(J,I)(−1)p+r(jαψ(ξ))(α
q(Xi1 ), . . . ,α

q(Xip+r ),ϕ(Xj1, . . . ,Xjq,x))

=
∑

J,jq<ik+1≤p+q+r

∑
L,lp<hm+1≤ip+r,
ik+1=hn,n≤m

(−1)(J,I)(−1)(L,H)ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhn−1 ),α

p(ϕ(Xj1, . . . ,Xjq, ) •α Xik+1 ),

. . . ,αp+q(Xhm ),ψ(α
q(Xl1 ), . . . ,α

q(Xlp ), ) •α α
q(Xhm+1 ),α

p+q(Xhm+2 ), . . . ,α
p+q(Xhr ),α

p+q(x))

+
∑

J,jq<ik+1≤p+q+r

∑
L,lp<hm+1≤ip+r,

ik+1=hm+1

(−1)(J,I)(−1)(L,H)ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhm ),ψ(α

q(Xl1 ), . . . ,α
q(Xlp ), )

•α (ϕ(Xj1, . . . ,Xjq, ) •α Xik+1 ),α
p+q(Xhm+2 ), . . . ,α

p+q(Xhr ),α
p+q(x))

+
∑

J,jq<ik+1≤p+q+r

∑
L,lp<hm+1≤ip+r,
ik+1=hn,n>m+1

(−1)(J,I)(−1)(L,H)ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhm ),ψ(α

q(Xl1 ), . . . ,α
q(Xlp ), )

•α αp+q(Xhm+1 ),α
p+q(Xhm+2 ), . . . ,α

p+q(Xhn−1 ),α
p(ϕ(Xj1, . . . ,Xjq, )

•α Xik+1 ),α
p+q(Xhn+1, . . . ,α

p+q(Xhr ),α
p+q(x))



1
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s
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a
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s
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+
∑

J,jq<ik+1≤p+q+r

∑
L,lp<hm+1≤ip+r,
ik+1=ls,s≤p

(−1)(J,I)(−1)(L,H)ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhm ),ψ(α

q(Xl1 ), . . . ,α
q(Xls−1 ),

ϕ(Xj1, . . . ,Xjq,Xjq, ) •α Xik+1,α
q(Xls+1 ), . . . ,α

q(Xlp ), ) •α α
q(Xhm+1 ),α

p+q(Xhm+2 ), . . . ,α
p+q(Xhr ),α

p+q(x))

+
∑

J,jq<ik+1≤p+q+r

∑
L,ik+1=hn

(−1)(J,I)(−1)(L,H)(−1)rξ(αp+q(Xh1 ), . . . ,α
p+q(Xhn−1 ),α

p(ϕ(Xj1, . . . ,Xjq, ) •α Xik+1 ),

αp+q(Xhn+1 ), . . . ,α
p+q(Xhr ),ψ(α

q(Xl1 ), . . . ,α
q(Xlp ),α

q(x)))

+
∑

J,jq<ik+1≤p+q+r

∑
L,ik+1=ls

(−1)(J,I)(−1)(L,H)(−1)rξ(αp+q(Xh1 ), . . . ,α
p+q(Xhr ),ψ(α

q(Xl1 ), . . . ,α
q(Xls−1 ),

ϕ(Xj1, . . . ,Xjq, ) •α Xik+1,α
q(Xls+1 ), . . . ,α

q(Xlp ),α
q(x)))

+
∑
J

∑
L,lp<hm+1≤ip+r

(−1)(J,I)(−1)p+r(−1)(L,H)ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhm ),ψ(α

q(Xl1 ), . . . ,α
q(Xlp ), ) •α (α

q(Xhm+1 ),

αp+q(Xhm+2 ), . . . ,α
p+q(Xhr ),α

p(ϕ(Xj1, . . . ,Xjq,x)))

+
∑
J

∑
L

(−1)(J,I)(−1)p+r(−1)(L,H)(−1)rξ(αp+q(Xh1 ), . . . ,α
p+q(Xhr ),ψ(α

q(Xl1 ), . . . ,α
p+q(Xlp ),ϕ(Xj1, . . . ,Xjq,x))).

Similarly one can compute D2.



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Let A = {a1,a2, . . . ,ap+q}a1<a2<···<ap+q ⊆ N = {1, . . . ,p + q + r}, H = {h1, . . . ,hr}h1<h2<···<hr = N \A,

and J = {j1, . . . ,jq}j1<···<jq ⊆ A and L = {l1, . . . , lp}l1<···<lp = A\J. We have:

j[ϕ,ψ]3HL (ξ)(X1, . . . ,Xp+q+r,x)

=
∑

A,ap+q<hm+1≤p+q+r
(−1)(A,H)ξ(αp+q(Xh1 ), . . . ,α

p+q(Xhm ), [ϕ,ψ]
3HL
α (Xa1, . . . ,Xap+q, )

•α Xhm+1,α
p+q(Xhm+2 ), . . . ,α

p+q(Xhr ),α
p+q(x))

+
∑
A

(−1)(A,H)(−1)rξ(αp+q(Xh1 ), . . . ,α
p+q(Xhr ), [ϕ,ψ]

3HL
α (Xa1, . . . ,Xap+q,x))

=
∑

A,ap+q<hm+1≤p+q+r

∑
L,lp<jt+1≤ap+q

(−1)pq(−1)(A,H)(−1)(L,J)ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhm ),ϕ(α

p(Xj1 ), . . . ,α
p(Xjt ),

ψ(Xl1, . . . ,Xlp, ) •α Xjt+1,α
p(Xjt+2 ) . . . ,α

p(Xjq ), ) •α α
p(Xhm+1 ),α

p+q(Xhm+2 ), . . . ,α
p+q(Xhr ),α

p+q(x))

−
∑

A,ap+q<hm+1≤p+q+r

∑
J,jq<ls+1≤ap+q

(−1)(A,H)(−1)(J,L)ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhm ),ψ(α

q(Xl1 ), . . . ,α
q(Xls ),

ϕ(Xj1, . . . ,Xjq, ) •α Xls+1,α
q(Xls+2 ) . . . ,α

q(Xlp ), ) •α α
q(Xhm+1 ), . . . ,α

p+q(Xhr ),α
p+q(x))

+
∑

A,ap+q<hm+1≤p+q+r

∑
L

(−1)(A,H)(−1)(L,J)(−1)pq(−1)qξ(αp+q(Xh1 ), . . . ,α
p+q(Xhm ),ϕ(α

p(Xj1 ), . . . ,α
p(Xjq, )

•α (ψ(Xl1, . . . ,Xlp, ) •α Xhm+1 ),α
p+q(Xhm+2 ), . . . ,α

p+q(Xhr ),α
p+q(x))



1
0
8

s
.

m
a
b
r
o
u
k
,

a
.

m
a
k
h
l
o
u
f
,

s
.

m
a
s
s
o
u
d

−
∑

A,ap+q<hm+1≤p+q+r

∑
J

(−1)(A,H)(−1)(J,L)(−1)pq(−1)Pξ(αp+q(Xh1 ), . . . ,α
p+q(Xhm ),ψ(α

q(Xl1 ), . . . ,α
q(Xlp, )

•α (ϕ(Xj1, . . . ,Xjq, ) •α Xhm+1 ),α
p+q(Xhm+2 ), . . . ,α

p+q(Xhr ),α
p+q(x))

+
∑
A

∑
L,lp<jt+1≤ap+q

(−1)r(−1)pq(−1)(A,H)(−1)(L,J)(ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhr ),ϕ(α

p(Xj1 ), . . . ,α
p(Xjt ),

ψ(Xl1, . . . ,Xlp, ) •α Xjt+1,α
p(Xjt+2 ), . . . ,α

p(Xjq ),α
p(x)))

+
∑
A

∑
L

(−1)r(−1)q(−1)pq(−1)(A,H)(−1)(L,J)(ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhr ),ϕ(α

p(Xj1 ), . . . ,α
p(Xjq ),ψ(Xl1, . . . ,Xlp,x))

−
∑
A

∑
J,Jq<ls+1≤ap+q

(−1)r(−1)(A,H)(−1)(J,L)(ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhr ),ψ(α

q(Xl1 ), . . . ,α
q(Xls ),

ϕ(Xj1, . . . ,Xjq, ) •α Xls+1,α
q(Xls+2 ), . . . ,α

q(Xlp ),α
q(x)))

−
∑
A

∑
J

(−1)p(−1)r(−1)(A,H)(−1)(J,L)(ξ(αp+q(Xh1 ), . . . ,α
p+q(Xhr ),ψ(α

q(Xl1 ), . . . ,α
q(Xlp ),ϕ(Xj1, . . . ,Xjq,x)).

By a straightforward verification, we obtain D1 − (−1)pqD2 = jα[ϕ,ψ]3HL (ξ)(X1, . . . ,Xp+q+r,x). Hence

the proof.



generalized representations of 3-hom-lie algebras 109

Theorem 1.9. The pair (Cα,α(g,g), [·, ·]3HL) is a graded Lie algebra.

Proof. Let ϕ ∈ Cqα,α(g,g), ψ ∈ C
p
α,α(g,g) and φ ∈ Crα,α(g,g).

(1) Skew-symmetry:

[ϕ,ψ]3HL = (−1)pqjαϕ(ψ) − j
α
ψ(ϕ)

= (−1)pq+1
(
(−1)pqjαψ(ϕ) − j

α
ϕ(ψ)

)
= −(−1)pq[ψ,ϕ]3HL.

(2) Graded Jacobi identity:

	ϕ,ψ,φ (−1)qr
[
ϕ, [ψ,φ]3HL

]3HL
= (−1)qpj[ψ,φ]3HL (ϕ) − (−1)

qrjαϕ
(
[ψ,φ]3HL

)
+ (−1)prj[φ,ϕ]3HL (ψ) − (−1)

pqjαψ
(
[φ,ϕ]3HL

)
+ (−1)rqj[ϕ,ψ]3HL (φ) − (−1)

rpjαφ
(
[ϕ,ψ]3HL

)
= (−1)qpj[ψ,φ]3HL (ϕ) − (−1)

qrjαϕ
(
(−1)rpjαφ (ψ) − j

α
ψ(φ)

)
+ (−1)prj[φ,ϕ]3HL (ψ) − (−1)

pqjαψ
(
(−1)qrjαϕ(φ) − j

α
φ (ϕ)

)
+ (−1)rqj[ϕ,ψ]3HL (φ) − (−1)

rpjαφ
(
(−1)qpjαψ(ϕ) − j

α
ϕ(ψ)

)
.

Organizing these terms leads to

	ϕ,ψ,φ(−1)qr
[
ϕ, [ψ,φ]3HL

]3HL
= (−1)pqj[ψ,φ]3HL (ϕ) + (−1)

pq
(
jαψ(j

α
φ (ϕ)) − (−1)

rpjαφ (j
α
ψ(ϕ))

)
+ (−1)rpj[φ,ϕ]3HL (ψ) + (−1)

rp
(
jαφ (j

α
ϕ(ψ)) − (−1)

qrjαϕ(j
α
φ (ψ))

)
+ (−1)qrj[ϕ,ψ]3HL (φ) + (−1)

qr
(
jαϕ(j

α
ψ(φ)) − (−1)

qpjαψ(j
α
ϕ(φ))

)
= (−1)pq

(
[jαψ,j

α
φ ] + j[ψ,φ]3HL

)
(ϕ)

+ (−1)rp
(
[jαφ ,j

α
ϕ] + j[φ,ϕ]3HL

)
(ψ)

+ (−1)qr
(
[jαϕ,j

α
ψ] + j[ϕ,ψ]3HL

)
(φ).

Using the previous lemma, we get

	ϕ,ψ,φ (−1)qr
[
ϕ, [ψ,φ]3HL]3HL = 0.



110 s. mabrouk, a. makhlouf, s. massoud

Remark 1.10. The pair (Cα,α(g,g),◦α) is a right symmetric graded
algebra.

The previous structure of graded Lie algebra is useful to describe 3-Hom-
Lie algebra structures as well as coboundary operators.

Corollary 1.11. The maps π : ∧3g → g and α : g → g define a 3-Hom-
Lie structure if and only if [π,π]3HL = 0.

Let g be a 3-Hom-Lie algebra. Given x1,x2 ∈ g, define ad : ∧2g → gl(g)
by

adx1,x2y = [x1,x2,y].

Then, the pair (ad,α) defines a representation of the 3-Hom-Lie algebra g on
itself, which we call adjoint representation of g. The coboundary operator
associated to this representation is denoted by δg.

Corollary 1.12. If π : ∧3g → g is a 3-Hom-Lie bracket, then we have

[π,ϕ]3HL = δg(ϕ) for all ϕ ∈ Cpα,α(g,g), p ≥ 0.

2. Generalized representations of 3-Hom-Lie algebras

In this section, we provide the Hom-type version of generalized represen-
tations of a 3-Lie algebras introduced in [19]. First, we show that a represen-
tation of a 3-Hom-Lie algebra will give rise to a canonical structure.

Let g be a 3-Hom-Lie algebra and V be a vector space. Let ρ : ∧2g → gl(V )
be a linear map. Then, it induces a linear map ρ : ∧3(g⊕V ) → g⊕V defined
by

ρ(x + u,y + v,z + w) = ρ(x,y)(w) + ρ(y,z)(u) + ρ(z,x)(v)

for all x,y,z ∈ g, u,v,w ∈ V . Consider the graded Lie algebra given in
Theorem 1.9 associated to the vector space g⊕V .

Proposition 2.1. A linear map ρ : ∧2g → gl(V ) is a representation on a
vector space V of the 3-Hom-Lie algebra g with respect to A ∈ gl(V ) if and
only if π + ρ is a canonical structure in the graded Lie algebra associated to
g⊕V , i.e.

[π + ρ,π + ρ]3HL = 0.



generalized representations of 3-hom-lie algebras 111

Proof. By Proposition 1.7, ρ : ∧2g → gl(V ) is a representation of g if and
only if g⊕V is a 3-Hom-Lie algebra, where the 3-Hom-Lie structure is exactly
given by

[x + u,y + v,z + w]ρ = [x,y,z] + ρ(x,y)(w) + ρ(y,z)(u) + ρ(z,x)(v)

= (π + ρ̄)(x + u,y + v,z + w),

and αg⊕V = α+A. Thus, by Lemma 1.11, ρ : ∧2g → gl(V ) is a representation
of g if and only if π + ρ̄ is a canonical structure.

The concept of representation of 3-Lie algebras introduced by Liu,
Makhlouf and Sheng [19] is generalized to Hom-type algebras as follows.

Definition 2.2. A generalized representation of a 3-Hom-Lie algebra
(g, [·, ·, ·],α) with respect to A ∈ gl(V ) consists of linear maps ρ : ∧2g → gl(V ),
ν : g → Hom(∧2V,V ), such that

[π + ρ + ν,π + ρ + ν]3HL = 0,

where ν : ∧3(g⊕V ) → (g⊕V ) is induced by ν via

ν(x + u,y + v,z + w) = ν(x)(v ∧w) + ν(y)(w ∧u) + ν(z)(u∧v)

for all x,y,z ∈ g, u,v,w ∈ V . We will refer to a generalized representation
by (V ; ρ,ν,A).

Remark 2.3. If ν = 0, then we recover the usual definition of a represen-
tation of a 3-Hom-Lie algebra on a vector space V . If the dimension of the
vector space V is 1, then ν must be zero. In this case, we only have the usual
representation.

Given linear maps ρ : ∧2g → End(V ), ν : g → Hom(∧2V,V ), and A : V →
V , define a trilinear bracket operation on g⊕V by

[x + u,y + v,z + w](ρ,ν) = [x,y,z] + ρ(x,y)(w) + ρ(y,z)(u)

+ ρ(z,x)(v) + ν(x)(v ∧w)
+ ν(y)(w ∧u) + ν(z)(u∧v).

(2.1)

Theorem 2.4. Let (g, [·, ·, ·],α) be a 3-Hom-Lie algebra and (V ; ρ,ν,A)
a generalized representation of g with respect to A. Then (g ⊕ V, [·, ·, ·](ρ,ν),
αg⊕V = α + A) is 3-Hom-Lie algebra, where [·, ·, ·](ρ,ν)is given by (2.1). We
call the 3-Hom-Lie algebra (g ⊕V, [·, ·, ·](ρ,ν),αg⊕V = α + A) the generalized
semidirect product of g and V .



112 s. mabrouk, a. makhlouf, s. massoud

Proof. It follows from [x+u,y+v,z+w](ρ,ν) = (π+ρ+ν)(x+u,y+v,z+w)
and Lemma 1.11.

In the following, we give a characterization of a generalized representation
of a 3-Hom-Lie algebra.

Proposition 2.5. Let ρ : ∧2g → End(V ), ν : g → Hom(∧2V,V ) and
A : V → V be linear maps. They give rise to a generalized representation of
a 3-Hom-Lie algebra g with respect to A if and only if for all xi ∈ g, vj ∈ V ,
the following equalities hold:

ρ(α(x1),α(x2))ρ(x3,x4) = ρ([x1,x2,x3],α(x4)) ◦A (2.2)
−ρ([x1,x2,x4],α(x3)) ◦A
+ ρ(α(x3),α(x4))ρ(x1,x2),

ρ([x1,x2,x3],α(x4)) ◦A = ρ(α(x2),α(x3))ρ(x1,x4) (2.3)
+ ρ(α(x3),α(x1))ρ(x2,x4)

+ ρ(α(x1),α(x2))ρ(x3,x4),

ρ(α(x1),α(x2))ν(x3)(v1,v2) = ν([x1,x2,x3])(A(v1),A(v2)) (2.4)

+ ν(α(x3))(ρ(x1,x2)v1,A(v2))

+ ν(α(x3))(A(v2),ρ(x1,x2)v1),

ν(α(x1))(A(v1),ρ(x2,x3)v2) = ν(α(x3))(A(v2),ρ(x2,x1)v1) (2.5)

+ ν(α(x2))(ρ(x3,x1)v1,A(v2))

+ ρ(α(x2),α(x3))ν(x1)(v1,v2),

ν(α(x1))(A(v1),ν(x2)(v2,v3)) = ν(α(x2))(ν(x1)(v1,v2),A(v3)) (2.6)

+ ν(α(x2))(A(v2),ν(x1)(v1,v3)),

ν(α(x1))(ν(x2)(v1,v2),A(v3)) = ν(α(x2))(ν(x1)(v1,v2),A(v3)). (2.7)

Proof. The quadruple (V ; ρ,ν,A) is a generalized representation if and
only if [π + ρ + ν,π + ρ + ν]3HL = 0. By straightforward computations,

[π + ρ̄ + ν̄,π + ρ̄ + ν̄]3HL(x1,x2,x3,x4,v) = 0

is equivalent to (2.2); and

[π + ρ̄ + ν̄,π + ρ̄ + ν̄]3HL(x1,v,x2,x3,x4) = 0



generalized representations of 3-hom-lie algebras 113

is equivalent to (2.3). Other identities can be proved similarly. The details
are omitted.

Remark 2.6. By (2.2) and (2.3), the map ρ in a generalized representa-
tion (V ; ρ,ν,A) gives rise to a usual representation in the sense of Definition
1.5. Conversely, for any representation ρ, (V ; ρ,ν = 0,A) is a generalized
representation.

Definition 2.7. Let (V1; ρ1,ν1,A1) and (V2; ρ2,ν2,A2) be two generalized
representations of a 3-Hom-Lie algebra (g, [·, ·, ·],α). They are said to be
equivalent if there exists an isomorphism of vector spaces T : V1 → V2 such
that

Tρ1(x,y)(u) = ρ2(x,y)(Tu), Tν1(x)(u,v) = ν2(x)(Tu,Tv), T ◦A1 = A2 ◦T

for all x,y ∈ g, u,v ∈ V1. In terms of diagrams, we have

∧2g×V1
id×T

��

ρ1 // V1

T

��
∧2g×V2

ρ2 // V2

, g×∧2V1
id×∧2T

��

ν1 // V1

T

��
g×∧2V2

ν2 // V2

, V1

T
��

A1 // V1

T
��

V2
A2 // V2

.

In the following, we provide a series of examples to illustrate the new con-
cept of generalized representation and also a procedure to twist a generalized
representation along linear maps.

Example 2.8. Let g be an abelian 3-Hom-Lie algebra. Define ρ = 0,
and ν = ξ ⊗ π, where ξ ∈ g∗ and π ∈ Hom(∧2V ⊗ V ) is a Hom-Lie algebra
structure on (V,π,A). Then (V ; ρ,ν,A) is a generalized representation. In
fact, since g is abelian and ρ = 0, (2.2)-(2.5) hold naturally. Since π satisfies
the Hom-Jacobi identity, (2.6) and (2.7) also hold.

Proposition 2.9. Let (g, [·, ·, ·],α) be a 3-Hom-Lie algebra, (V,ρ,ν,A)
be a generalized representation, β : g → g be an algebra morphism and
B : V → V a linear map such that

B ◦ρ(x1,x2) = ρ(β(x1),β(x2)) ◦B,

B ◦ν(x) = ν(β(x)) ◦ (B ⊗B),

B ◦A = A◦B.



114 s. mabrouk, a. makhlouf, s. massoud

Then (V,ρ̃, ν̃,B) is a generalized representation of 3-Hom-Lie algebra (g,
[·, ·, ·]β, β ◦α) where

[·, ·, ·]β = [·, ·, ·] ◦β⊗3, ρ̃ = B ◦ρ, ν̃(x) = B ◦ν(x).

Proof. We have to show that ρ̃ and ν̃ satisfy Eqs. (2.2)-(2.7).
Let x1,x2,x3 ∈ g and v1,v2 ∈ V ,

ρ̃(β◦α(x1),β ◦α(x2))ν̃(x3)(v1,v2) − ν̃([x1,x2,x3]β)(B ◦A(v1),B ◦A(v2))
− ν̃(β ◦α(x3))(ρ̃(x1,x2)v1,B ◦A(v2))
− ν̃(β ◦α(x3))(B ◦A(v2), widetildeρ(x1,x2)v1)

= B ◦ρ(β ◦α(x1),β ◦α(x2)) ◦B ◦ν(x3)(v1,v2)
−B ◦ν(β ◦ [x1,x2,x3]) ◦ (B ⊗B)(A(v1),A(v2))
−B ◦ν(β ◦α(x3))(B ◦ρ(x1,x2)v1,B ◦A(v2))
−B ◦ν(βα(x3))(B ◦A(v2),B ◦ρ(x1,x2)v1))

= B2 ◦ (ρ(α(x1),α(x2)) ◦ν(x3)(v1,v2) −ν([x1,x2,x3])(A(v1),A(v2))
−ν(α(x))(ρ(x1,x2)v1,A(v2))−ν(α(x3))[2pt](A(v2),ρ(x1,x2)v1)) = 0,

ν̃(β◦α(x1))(B ◦A(v1), ρ̃(x2,x3)v2) − ν̃(β ◦α(x3))(B ◦A(v2), ρ̃(x2,x1)v1)
− ν̃(β ◦α(x2))(ρ̃(x3,x1)v1,B ◦A(v2))
− ρ̃(β ◦α(x2),β ◦α(x3))ν̃(x1)(v1,v2)

= B ◦ν(β ◦α(x1))(B ◦A(v1),B ◦ρ(x2,x3)v2)
−B ◦ν(β ◦α(x3))(B ◦A(v2),B ◦ρ(x2,x1)v1)
−B ◦ν(β ◦α(x2))(B ◦ρ(x3,x1)v1,B ◦A(v2))
−B ◦ρ(β ◦α(x2),β ◦α(x3)) ◦B ◦ν(x1)(v1,v2)

= B2 ◦ (ν(α(x1))(A(v1),ρ(x2,x3)v2) −ν(α(x3))(v2,ρ(x2,x1)v1)
−ν(α(x2))(ρ(x3,x1)v1,A(v2)) −ρ(α(x2),α(x3)) ◦ν(x1)(v1,v2)) = 0.

Then identities (2.4) and (2.5) are proved. One similarly proves identities
(2.6) and (2.7).

Corollary 2.10. Let (g, [·, ·, ·]) be a 3-Lie algebra, (V,ρ,ν) be a gener-
alized representation, α : g → g be an algebra morphism and A : V → V be a



generalized representations of 3-hom-lie algebras 115

linear map such that for all x1,x2 ∈ g and v1,v2 ∈ V ,

A◦ρ(x1,x2) = ρ(α(x1),α(x2)) ◦A,
A◦ν(x)(v1,v2) = ν(α(x)) ◦ (A⊗A)(v1,v2).

Then (V,ρ̃ := A ◦ ρ, ν̃ := A ◦ ν,A) is a generalized representation of the
3-Hom-Lie algebra (L, [·, ·, ·]α := α◦ [·, ·, ·],α).

Example 2.11. Let g be the 3-dimensional 3-Lie algebra defined with
respect to a basis {e1,e2,e3} by the skew-symmetric bracket [e1,e2,e3] = e1.
Let V be a 2-dimensional vector space and {v1,v2} its basis. We have a
representation defined by the following maps (ρ,µ), given with respect to
previous bases by

ρ(e1,e2)(v1) = 0, ρ(e1,e2)(v2) = v1, ρ(e1,e3)(v1) = 0,

ρ(e1,e3)(v2) = r1v1, ρ(e2,e3)(v1) = v1, ρ(e2,e3)(v2) = r2v1,

ν(e1)(v1,v2) = 0, ν(e2)(v1,v2) = sv1, ν(e3)(v1,v2) = sr1v1,

where r1,r2,s are parameters in K.
Let α : g → g be an algebra morphism and A ∈ gl(V ) defined respectively

by:

α(e1) = λe1, α(e2) = e2, α(e3) = e3,

A(v1) = λv1, A(v2) = r2v1 + v2,

where λ is a parameter in K. They satisfy

A◦ρ(x1,x2) = ρ(α(x1),α(x2)) ◦A,
A◦ν(x)(u1,u2) = ν(α(x)) ◦ (A⊗A)(u1,u2),

where x,x1,x2 are in g and u1,u2 in V .
Then, using the Twist procedure, (V ; ρ̃, ν̃,A) is a generalized representa-

tion of the 3-Hom-Lie algebra (g, [·, ·, ·]α,α). More precisely, we have

[e1,e2,e3]α = [α(e1),α(e2),α(e3)] = λe1,

ρ̃(e1,e2)(v1) = 0, ρ̃(e1,e2)(v2) = λv1,

ρ̃(e1,e3)(v1) = 0, ρ̃(e1,e3)(v2) = r1r2(λ− 1)v1 −r1v2,
ρ̃(e2,e3)(v1) = λv1, ρ̃(e2,e3)(v2) = r2λv1,

ν̃(e1)(v1,v2) = 0, ν̃(e2)(v1,v2) = sλv1, ν̃(e3)(v1,v2) = sr1λv1.



116 s. mabrouk, a. makhlouf, s. massoud

Example 2.12. Let g be the 4-dimensional 3-Lie algebra defined, with
respect to a basis {e1,e2,e3,e4}, by the skew-symmetric brackets

[e1,e2,e4] = e3, [e1,e3,e4] = e2, [e2,e3,e4] = e1.

Every generalized representation (V ; ρ,ν), on a 2-dimensional vector space V
with trivial ρ, of g is given by one of the following maps ν defined, with respect
to a basis {v1,v2} of V , by

ν(e1)(v1,v2) = 0, ν(e2)(v1,v2) = 0,

ν(e3)(v1,v2) = 0, ν(e4)(v1,v2) = s1v1 + s2v2,

where s1,s2 are parameters in K, and s1s2 6= 0.
Let α : g → g be a 3-Lie algebra morphism and A : V → V be a linear

map, defined respectively by

α(e1) = a1e1, α(e2) = −a1e2, α(e3) = a1e3, α(e4) =
−1
a1
e4,

A(v1) = −a1v1, A(v2) = a2v1 +
a2s2 −a1s1

s1
v2,

where a1,a2,s1,s2 are parameters in K such that, a1s1s2 6= 0.
They satisfy A ◦ ν(x) = ν(α(x)) ◦ (A ⊗ A). Therefore, using the Twist

procedure, (V ; ρ̃, ν̃,A) is a generalized representation of the 3-Hom-Lie algebra
(L, [·, ·, ·]α,α) with trivial ρ̃. Namely, we have

[e1,e2,e4]α = a1e3, [e1,e3,e4]α = −a1e2, [e2,e3,e4]α = a1e1,

and

ν̃(e1)(v1,v2) = 0, ν̃(e2)(v1,v2) = 0, ν̃(e3)(v1,v2) = 0,

ν̃(e4)(v1,v2) = (a2s2 −a1s1)v1 +
(
a2s

2
2

s1
−a1s2

)
v2.

3. New cohomology complex of 3-Hom-Lie algebras

Based on the generalized representations defined in the previous section,
we introduce a new type of cohomology for 3-Hom-Lie algebras.

Let (g, [·, ·, ·],α) be a 3-Hom-Lie algebra and (V ; ρ,ν,A) be a generalized
representation of g. We set C̃

p
α+A,A(g ⊕ V,V ) to be the set of (p + 1)-Hom-

cochains, which are defined as a subset of C
p
α+A,A(g⊕V,V ) such that

C
p
α+A,A(g⊕V,V ) = C̃

p
α+A,A(g⊕V,V ) ⊕C

p
A(V,V ).



generalized representations of 3-hom-lie algebras 117

Elements of C
p
α+A,A(g⊕V,V ) are of the form

ϕ : ∧2(g⊕V )⊗ (p times). . . ⊗∧2 (g⊕V ) ∧ (g⊕V ) → V.

By direct calculation, we have

[π + ρ̄ + ν̄, C̃•α+A,A(g⊕V,V )] ⊆ C̃
•+1
α+A,A(g⊕V,V ).

Define d : C̃
p
α+A,A(g⊕V,V ) → C̃

p+1
α+A,A(g⊕V,V ) by

d(ϕ) := [π + ρ̄ + ν̄,ϕ]3HL, ϕ ∈ C̃pα+A,A(g⊕V,V ).

Theorem 3.1. Let (V ; ρ,ν,A) be a generalized representation of a
3-Hom-Lie algebra g. Then d ◦ d = 0. Thus, we obtain a new cohomology
complex, where the space of p-Hom-cochains is given by C̃

p−1
α+A,A(g⊕V,V ).

Proof. By the graded Jacobi identity, for any ϕ ∈ C̃p−1α+A,A(g ⊕V,V ), one
obtains

d◦d(ϕ) : = [π + ρ̄ + ν̄, [π + ρ̄ + ν̄,ϕ]3HL]3HL

=
1

2
[[π + ρ̄ + ν̄,π + ρ̄ + ν̄]3HL,ϕ]3HL = 0.

An element ϕ ∈ C̃p−1α+A,A(g ⊕ V,V ) is called a p-cocycle if d(ϕ) = 0; it is
called a p-coboundary if there exists f ∈ C̃p−2α+A,A(g⊕V,V ) such that ϕ = d(f).

Denote by Zp3HL(g; V ) and B
p
3HL(g; V ) the sets of p-cocycles and p-

coboundaries respectively. By Theorem 3.1, we have Bp3HL(g; V ) ⊂Z
p(g; V ).

We define the p-th cohomolgy group Hp3HL(g; V ) to beZ
p
3HL(g; V )/B

p
3HL(g; V ).

The following proposition provides a relationship between this new coho-
mology and the one given by (1.5).

Proposition 3.2. There is a forgetful map from Hp3HL(g; V ) to
H
p
3HL(g; V ).

Proof. It is obvious that C
p
α,A(g,V ) ⊆ C̃

p
α+A,A(g⊕V,V ). By direct calcu-

lation, for Xi ∈∧2g, z ∈ g, we have

d(ϕ)(X1, . . . ,Xp+1,z) = δρ(ϕ)(X1, . . . ,Xp+1,z), ϕ ∈ C
p
α,A(g,V ),

where δρ is the coboundary operator given by (1.4). Thus, the natural pro-

jection from C̃
p
α+A,A(g ⊕ V,V ) to C

p
α,A(g,V ) induces a forgetful map from

Hp3HL(g; V ) to H
p
3HL(g; V ).



118 s. mabrouk, a. makhlouf, s. massoud

In the sequel, we give some characterization of low dimensional cocycles.

Proposition 3.3. A linear map ϕ ∈ Hom(g,V ) is a 1-cocycle if only if
for all x1,x2,x3 ∈ g, v ∈ V , the following identities hold:

ϕ◦α = A◦ϕ,
ν(x1)(ϕ(x2),v) −ν(x2)(ϕ(x1),v) = 0,

ϕ([x1,x2,x3]) −ρ(α(x1),α(x2))(ϕ(x3))
−ρ(α(x2),α(x3))(ϕ(x1)) −ρ(α(x3),α(x1))(ϕ(x2)) = 0.

Proof. For ϕ ∈ Hom(g,V ) satisfying ϕ◦α = A◦ϕ, we have

d(ϕ)(x1,x2,v) = ν(x1)(ϕ(x2),v) −ν(x2)(ϕ(x1),v),

and

d(ϕ)(x1,x2,x3) = δρ(ϕ)(x1,x2,x3)

= ρ(α(x1),α(x2))(ϕ(x3)) + ρ(α(x2),α(x3))(ϕ(x1))

+ ρ(α(x3),α(x1))(ϕ(x2)) −ϕ([x1,x2,x3]).

Proposition 3.4. A 2-cochain ϕ1 + ϕ2 + ϕ3 ∈ C̃1α,A(g ⊕ V,V ), where
ϕ1 ∈ Hom(∧2V ∧ g,V ), ϕ2 ∈ Hom(∧2g ∧ V,V ), ϕ3 ∈ Hom(∧3g,V ), is a 2-
cocycle if and only if for all xi ∈ g,vj ∈ V and v ∈ V , the following identities
hold:

0 = −ρ(α(x1),α(x2))(ϕ3(x3,x4,x5)) −ϕ3(α(x1),α(x2), [x3,x4,x5]) (3.1)
+ ρ(α(x4),α(x5))(ϕ3(x1,x2,x3)) + ϕ3([x1,x2,x3],α(x4),α(x5))

+ ρ(α(x5),α(x3))(ϕ3(x1,x2,x4)) + ϕ3(α(x3), [x1,x2,x4],α(x5))

+ ρ(α(x3),α(x4))(ϕ3(x1,x2,x5)) + ϕ3(α(x3),α(x4), [x1,x2,x5]),

0 = ν(α(x4))(ϕ3(x1,x2,x3),A(v)) + ν(α(x3))(A(v),ϕ3(x1,x2,x4)) (3.2)

+ ρ(α(x1),α(x2))(ϕ2(x3,x4,v)) −ρ(α(x3),α(x4))(ϕ2(x1,x2,v))
−ϕ2([x1,x2,x3],α(x4),A(v)) −ϕ2(α(x3), [x1,x2,x4],A(v)),

0 = ν(α(x1))(A(v),ϕ3(x2,x3,x4)) + ρ(α(x3),α(x4))(ϕ2(x1,x2,v)) (3.3)

−ρ(α(x2),α(x4))(ϕ2(x1,x3,v)) + ρ(α(x2),α(x3))(ϕ2(x1,x4,v))
+ ϕ2(α(x3),α(x4),ρ(x1,x2)(v)) −ϕ2(α(x2),α(x4),ρ(x1,x3)(v))
+ ϕ2(α(x2),α(x3),ρ(x1,x4)(v)) −ϕ2(α(x1), [x2,x3,x4],A(v)),



generalized representations of 3-hom-lie algebras 119

0 = ν(α(x3))(A(v2),ϕ2(x1,x2,v1)) + ν(α(x3))(ϕ2(x1,x2,A(v2)),v1) (3.4)

+ ϕ2(α(x1),α(x2),ν(x3)(v1,v2)) + ρ(α(x1),α(x2))(ϕ1(v1,v2,x3))

−ϕ1(ρ(x1,x2)(v1),A(v2),α(x3)) −ϕ1(A(v1),ρ(x1,x2)(v2),α(x3))
−ϕ1(A(v1),A(v2), [x1,x2,x3]),

0 = ν(α(x3))(A(v2),ϕ2(x2,x1,v1)) + ν(α(x2))(ϕ2(x3,x1,v1),A(v2)) (3.5)

−ν(α(x1))(A(v1),ϕ2(x2,x3,v2)) + ϕ2(α(x2),α(x3),ν(x1)(v1,v2))
+ ρ(α(x2),α(x3))(ϕ1(v1,v2,x1)) + ϕ1(ρ(x1,x2)(v1),A(v2),α(x3))

−ϕ1(A(v1),ρ(x2,x3)(v2),α(x1)) + ϕ1(A(v2),ρ(x1,x3)(v1),α(x2)),
0 = ϕ2(α(x1),α(x3),ν(x2)(v1,v2)) −ϕ2(α(x2),α(x3),ν(x1)(v1,v2)) (3.6)

−ϕ2(α(x1),α(x2),ν(x3)(v1,v2)) + ρ(α(x1),α(x3))(ϕ1(v1,v2,x2))
−ρ(α(x1),α(x2))(ϕ1(v1,v2,x3)) −ρ(α(x2),α(x3))(ϕ1(v1,v2,x1))
+ ϕ1(A(v1),A(v2), [x1,x2,x3]),

0 = −ν(α(x2))(ϕ1(v1,v2,x1),A(v3)) −ν(α(x2))(A(v2),ϕ1(v1,v3,x1)) (3.7)
+ ν(α(x1))(A(v1),ϕ1(v2,v3,x2)) −ϕ1(ν(x1)(v1,v2),A(v3),α(x2))
−ϕ1(A(v2),ν(x1)(v1,v3),α(x2)) + ϕ1(A(v1),ν(x2)(v2,v3),α(x1)),

0 = ν(α(x2))(ϕ1(v1,v2,x1),A(v3)) −ν(α(x1))(A(v3),ϕ1(v1,v2,x2)) (3.8)
+ ϕ1(ν(x1)(v1,v2),A(v3),α(x2)) −ϕ1(ν(x2)(v1,v2),A(v3),α(x1)).

Proof. For ϕ3 ∈ Hom(∧3g,V ), we have

d(ϕ3)(x1,x2,x3,x4,x5) = ρ(α(x1),α(x2))(ϕ3(x3,x4,x5))

+ ϕ3(α(x1),α(x2), [x3,x4,x5])

−ρ(α(x4),α(x5))(ϕ3(x1,x2,x3))
−ϕ3([x1,x2,x3],α(x4),α(x5))
−ρ(α(x5),α(x3))(ϕ3(x1,x2,x4))
−ϕ3(α(x3), [x1,x2,x4],α(x5))
−ρ(α(x3),α(x4))(ϕ3(x1,x2,x5))
−ϕ3(α(x3),α(x4), [x1,x2,x5]),

d(ϕ3)(x1,x2,x3,x4,v) = ν(α(x4))(ϕ3(x1,x2,x3),A(v))

+ ν(α(x3))(A(v),ϕ3(x1,x2,x4)),

d(ϕ3)(x1,v,x2,x3,x4) = ν(α(x1))(A(v),ϕ3(x2,x3,x4)).



120 s. mabrouk, a. makhlouf, s. massoud

For ϕ2 ∈ Hom(∧2g∧V,V ), we have

d(ϕ2)(x1,x2,x3,x4,v) = ρ(α(x1),α(x2))(ϕ2(x3,x4,v))

−ρ(α(x3),α(x4))(ϕ2(x1,x2,v))
−ϕ2([x1,x2,x3],α(x4),A(v))
−ϕ2(α(x3), [x1,x2,x4],A(v)),

d(ϕ2)(x1,v,x2,x3,x4) = ρ(α(x3),α(x4))(ϕ2(x1,x2,v))

−ρ(α(x2),α(x4))(ϕ2(x1,x3,v))
+ ρ(α(x2),α(x3))(ϕ2(x1,x4,v))

+ ϕ2(α(x3),α(x4),ρ(x1,x2)(v))

−ϕ2(α(x2),α(x4),ρ(x1,x3)(v))
+ ϕ2(α(x2),α(x3),ρ(x1,x4)(v))

−α2(x1, [x2,x3,x4],v),
d(ϕ2)(x1,x2,v1,v2,x3) = ν(α(x3))(A(v2),ϕ2(x1,x2,v1))

+ ν(α(x3))(ϕ2(x1,x2,v2),A(v1))

+ ϕ2(α(x1),α(x2),ν(x3)(v1,v2)),

d(ϕ2)(x1,v1,x2,v2,x3) = ν(α(x3))(A(v2),ϕ2(x2,x1,v1))

+ ν(α(x2))(ϕ2(x3,x1,v1),A(v2))

−ν(α(x1))(A(v1),ϕ2(x2,x3,v2))
+ ϕ2(α(x2),α(x3),ν(x1)(v1,v2)),

d(ϕ2)(v1,v2,x1,x2,x3) = ϕ2(α(x1),α(x3),ν(x2)(v1,v2))

−ϕ2(α(x2),α(x3),ν(x1)(v1,v2))
−ϕ2(α(x1),α(x2),ν(x3)(v1,v2)).

For ϕ1 ∈ Hom(∧2V ∧g,V ), we have

d(ϕ1)(x1,x2,v1,v2,x3) = ρ(α(x1),α(x2))(ϕ1(v1,v2,x3))

−ϕ1(ρ(x1,x2)(v1),A(v2),α(x3))
−ϕ1(A(v1),ρ(x1,x2)(v2),α(x3))
−ϕ1(A(v1),A(v2), [x1,x2,x3]),

d(ϕ1)(x1,v1,x2,v2,x3) = ρ(α(x2),α(x3))(ϕ1(v1,v2,x1))

+ ϕ1(ρ(x1,x2)(v1),A(v2),α(x3))

−ϕ1(A(v1),ρ(x2,x3)(v2),α(x1))
+ ϕ1(A(v2),ρ(x1,x3)(v1),α(x2)),



generalized representations of 3-hom-lie algebras 121

d(ϕ1)(v1,v2,x1,x2,x3) = ρ(α(x1),α(x3))(ϕ1(v1,v2,x2))

−ρ(α(x1),α(x2))(ϕ1(v1,v2,x3))
−ρ(α(x2),α(x3))(ϕ1(v1,v2,x1))
+ ϕ1(A(v1),A(v2), [x1,x2,x3]),

d(ϕ1)(x1,v1,v2,v3,x2) = −ν(α(x2))(ϕ1(v1,v2,x1),A(v3))
−ν(α(x2))(A(v2),ϕ1(v1,v3,x1))
+ ν(α(x1))(A(v1),ϕ1(v2,v3,x2))

−ϕ1(ν(x1)(v1,v2),A(v3),α(x2))
−ϕ1(A(v2),ν(x1)(v1,v3),α(x2))
+ ϕ1(A(v1),ν(x2)(v2,v3),α(x1)),

d(ϕ1)(v1,v2,x1,x2,v3) = ν(α(x2))(ϕ1(v1,v2,x1),A(v3))

−ν(α(x1))(A(v3),ϕ1(v1,v2,x2))
+ ϕ1(ν(x1)(v1,v2),A(v3),α(x2))

−ϕ1(ν(x2)(v1,v2),A(v3),α(x1)).

Thus, d(ϕ1 + ϕ2 + ϕ3) = 0 if and only if Eqs. (3.1)-(3.8) hold.

In the following we provide an example of computation of 2-cocycles of a
3-dimensional 3-Hom-Lie algebra.

Example 3.5. Let (g, [·, ·, ·],α) be the 3-dimensional 3-Hom-Lie algebra
defined, with respect to a basis {e1,e2,e3}, by [e1,e2,e3] = a1e1, α(e1) = a1e1,
α(e2) = a2e2, α(e3) =

1
a2
e3. Let V be a 2-dimensional vector space, {v1,v2}

its basis and A ∈ gl(V ) defined by: A(v1) = a1v1, A(v2) = a2a3a1 v2, where
a1,a2,a3 are parameters in K.

We consider the generalized representation (V,ρ,ν,A), where ρ and ν are
defined with respect to the basis by

ρ(e1,e2)(v1) = 0, ρ(e1,e2)(v2) = 0, ρ(e1,e3)(v1) = 0,

ρ(e1,e3)(v2) = r2a1v1, ρ(e2,e3)(v1) = a1v1, ρ(e2,e3)(v2) =
r1a2a3
a1

v2,

ν(e1)(v1,v2) = 0, ν(e2)(v1,v2) = 0, ν(e3)(v1,v2) = s1a1v1,

with s1,r1,r2 parameters in K and a1a2s1 6= 0.



122 s. mabrouk, a. makhlouf, s. massoud

We have the following 2-cocycles: ϕ1 = 0, ϕ3 = 0 and ϕ2 defined as

ϕ2(e1,e2,v1) = 0, ϕ2(e1,e2,v2) = 0,

ϕ2(e1,e3,v1) = c1v1, ϕ2(e1,e3,v2) = 0,

ϕ2(e2,e3,v1) = c2v1, ϕ2(e2,e3,v2) = c3v2,

where c1,c2,c3 are parameters in K.

4. Abelian extensions of 3-Hom-Lie algebras

In this section, we show that associated to any abelian extension, there is
a generalized representation and a 2-cocycle.

Definition 4.1. Let (g, [·, ·, ·]g,α), (V, [·, ·, ·]V ,A) and (ĝ, [·, ·, ·]ĝ,αĝ) be
3-Hom-Lie algebras and i : V → ĝ, p : ĝ → g be morphisms of 3-Hom-
Lie algebras. The following sequence of 3-Hom-Lie algebras is a short exact
sequence if Im(i) = Ker(p), Ker(i) = 0 and Im(p) = g:

0 → V i−→ ĝ
p
−−→ g → 0 ,

where A(V ) = αĝ(V ). In this case, we call ĝ an extension of g by V , and
denote it by Eĝ. It is called an abelian extension if V is an abelian ideal of ĝ,
i.e., [u,v,w]V = 0 for all u,v,w ∈ V . A section σ of p : ĝ → g consists of a
linear maps σ : g →ĝ such that p◦σ = idg and σ ◦α = αĝ ◦σ.

Definition 4.2. Two extensions of 3-Hom-Lie algebras,

Eĝ : 0 → V
i1−−→ ĝ

p1−−→ g → 0 and Eg̃ : 0 → V
i2−−→ g̃

p2−−→ g → 0,

are equivalent if there exists a morphism of 3-Hom-Lie algebras φ : ĝ → g̃
such that the following diagram commutes:

0−−→V i1−−−→ ĝ
p1−−−→ g −−→0yIdV

yφ
yIdg

0−−→V i2−−−→ g̃
p2−−−→ g −−→0

A linear map σ : g → ĝ is called a splitting of g if it satisfies p◦σ = idg.
If there exists a splitting which is also a homomorphism between 3-Hom-Lie



generalized representations of 3-hom-lie algebras 123

algebras, we say that the abelian extension is split. Let ĝ be a split abelian
extension and σ : g → ĝ the corresponding splitting. Define ρ : ∧2g → gl(V )
and ν : g → Hom(∧2V,V ) by

ρ(x,y)(u) = [σ(x),σ(y),u]ĝ,

ν(x)(u,v) = [σ(x),u,v]ĝ.

Then, we can transfer the 3-Hom-Lie algebra structure on ĝ to that on g⊕V
in terms of ρ and ν.

Note that the Hom-Filippov-Jacobi identity gives the character of
ρ and ν:

[x + u,y + v,z + w](ρ,ν) = [x,y,z] + ρ(x,y)(w) + ρ(y,z)(u) + ρ(z,x)(v)

+ ν(x)(v ∧w) + ν(y)(w ∧u) + ν(z)(u∧v).

However, by Theorem 2.4, it is straightforward to obtain the following propo-
sition.

Proposition 4.3. Any split abelian extension of 3-Hom-Lie algebras is
isomorphic to a generalized semidirect of product 3-Hom-Lie algebra.

Now, for non-split abelian extensions, we can further define
ω : ∧3g → V by

ω(x,y,z) = [σ(x),σ(y),σ(z)]ĝ −σ[x,y,z]g.

Then, we also transfer the 3-Hom-Lie algebra structure on ĝ to that on g⊕V
in terms of ρ,ν and ω:

[x1 + v1,x2 + v2,x3 + v3](ρ,ν,ω) = [x1,x2,x3]g + ρ(x1,x2)(v3)

+ ρ(x3,x1)(v2) + ρ(x2,x3)(v1)

+ ν(x1)(v2,v3) + ν(x2)(v3,v1)

+ ν(x3)(v1,v2) + ω(x1,x2,x3).

Theorem 4.4. With above notations, (g ⊕ V, [·, ·, ·](ρ,ν,ω),αg⊕V ) is a 3-
Hom-Lie algebra if and only if for all x1,x2,x3,x4,x5 ∈ g and v,v1,v2,v3 ∈ V ,
Eqs. (2.4)-(2.7) and the following identities hold:



124 s. mabrouk, a. makhlouf, s. massoud

0 = −ρ(α(x1),α(x2))(ω(x3,x4,x5)) −ω(α(x1),α(x2), [x3,x4,x5]) (4.1)

+ ρ(α(x4),α(x5))(ω(x1,x2,x3)) + ω([x1,x2,x3],α(x4),α(x5))

+ ρ(α(x5),α(x3))(ω(x1,x2,x4)) + ω(α(x3), [x1,x2,x4],α(x5))

+ ρ(α(x3),α(x4))(ω(x1,x2,x5)) + ω(α(x3),α(x4), [x1,x2,x5]),

0 = ν(α(x1))(A(v),ω(x2,x3,x4)) + ρ([x2,x3,x4],α(x1))(A(v)) (4.2)

+ ρ(α(x3),α(x4))ρ(x1,x2)(v) −ρ(α(x2),α(x4))ρ(x1,x3)(v)

+ ρ(α(x2),α(x3))ρ(x1,x4)(v),

0 = ρ(α(x1),α(x2))ρ(x3,x4)(v) −ρ(α(x3),α(x4))ρ(x1,x2)(v) (4.3)

−ρ([x1,x2,x3],α(x4))(A(v)) −ν(α(x4))(A(v),ω(x1,x2,x3))

−ρ(α(x3), [x1,x2,x4])(A(v)) + ν(α(x3))(A(v),ω(x1,x2,x4)).

The Fundamental Identity gives the character of ρ, ν and ω.

Proof. The triple (g ⊕ V, [·, ·, ·](ρ,ν,ω),αg⊕V ) defines a 3-Hom-Lie algebra
if and only if the Hom-Filippov-Jacobi identity holds on all elements of g ⊕
V . Condition (4.1) is obtained using the Hom-Filippov-Jacobi identity on
{x1,x2,x3,x4,x5} elements of g.

Similarly, elements {x1,v,x2,x3,x4} gives Eq. (4.2), {x1,x2,v,x3,x4}
gives Eq. (4.3), {x1,x2,v1,v2,x3} gives Eq. (2.4), {v1,x1,v2,x2,x3}
gives Eq. (2.5), {v1,x1,v2,v3,x2} gives Eq. (2.6) and {v1,v2,v3,x1,x2} gives
Eq. (2.7).

Conversely, if Eqs. (2.4)-(2.7) and Eqs. (4.1)-(4.3) hold, it is straightfor-
ward to see that for all e1, . . . ,e5 ∈ g ⊕V , the Hom-Filippov-Jacobi identity
holds. Thus, (g⊕V, [·, ·, ·](ρ,ν,ω),αg⊕V ) is a 3-Hom-Lie algebra.

Remark 4.5. The triple (ρ,ν,ω) is a cochain with respect to the new co-
homology defined in Section 3 but it is not necessarily a 2-cocycle.

Acknowledgements

The authors would like to thank Yunhe Sheng for his comments and
suggestions.



generalized representations of 3-hom-lie algebras 125

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 http://arxiv.org/abs/1505.08168v1

	Representations of 3-Hom-Lie algebras
	Generalized representations of 3-Hom-Lie algebras
	New cohomology complex of 3-Hom-Lie algebras
	Abelian extensions of 3-Hom-Lie algebras