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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 37, Num. 2 (2022), 153 – 184

doi:10.17398/2605-5686.37.2.153

Available online March 25, 2022

Extensions, crossed modules and pseudo
quadratic Lie type superalgebras

M. Pouye 1, B. Kpamegan 2

1 Institut de Mathématiques et de Sciences Physiques (IMSP), Bénin
2 Département de Mathématiques, FAST, UAC, Bénin

Received October 13, 2021 Presented by Consuelo Mart́ınez
Accepted February 17, 2022

Abstract: Extensions and crossed modules of Lie type superalgebras are introduced and studied.

We construct homology and cohomology theories of Lie-type superalgebras. The notion of left
super-invariance for a bilinear form is defined and we consider Lie type superalgebras endowed with

nondegenerate, supersymmetric and left super-invariant bilinear form. Such Lie type superalgebras

are called pseudo quadratic Lie type superalgebras. We show that any pseudo quadratic Lie type
superalgebra induces a Jacobi-Jordan superalgebra. By using the method of double extension, we

study pseudo quadratic Lie type superalgebras and theirs associated Jacobi-Jordan superalgebras.

Key words: Lie type superalgebras, Jacobi-Jordan superalgebras, extension, crossed module,
homology, cohomology, double extension, pseudo quadratic Lie type superalgebras.

MSC (2020): 17A15, 17A70, 17A60, 20K35, 17A45.

Introduction

Recently, in order to investigate commutative non-associative algebras,
authors in [5] introduce the so-called Jacobi-Jordan algebras that are commu-
tative algebras satisfying the Jacobi identity. Those algebras were first defined
in [12] and since then they have been studied in various papers [3, 4, 6, 8] under
different name such as Jordan algebras of nil rank 3, Mock-Lie algebras, Lie-
Jordan algebras or pathological algebras. It turns out that the commutativity
and Jacobi identity satisfied by the product of an algebra (A,∗) induce two
relations x∗(y∗z) = −(x∗y)∗z−y∗(x∗z) and x∗(y∗z) = −(x∗y)∗z−(x∗z)∗y
for all x,y,z ∈ A, called respectivelly left Lie-type identity and right Lie-type
identity.

This motivated us to introduce and study in [11] a new type of nonasso-
ciative (super)-algebra called left or right Lie-type superalgebra. A left (resp.
right) Lie type superalgebra consists of a Z2-graded vector space U := U0̄⊕U1̄
endowed with an even bilinear map [ , ] : U ⊗U → U such that [Uα,Uβ] ⊆
Uα+β for all α,β ∈ Z2 and [x, [y,z]] = −[[x,y],z] − (−1)|x||y|[y, [x,z]] (resp.

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

© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.37.2.153
https://publicaciones.unex.es/index.php/EM
https://creativecommons.org/licenses/by-nc/3.0/


154 m. pouye, b. kpamegan

[x, [y,z]] = −[[x,y],z] − (−1)|y||z|[[x,z],y]) for all x,y,z ∈U. It is called sym-
metric Lie type superalgebra if it is simultaneously a left and right Lie type
superalgebra. Lie type superalgebras can be seen as generalization of Jacobi-
Jordan (super)-algebras introduced in [5] which are sub class of the class of
Jordan algebras that plays an important role in physics (see [10]). In fact,
unlike Jacobi-Jordan (super)-algebras, Lie-type superalgebras are not neces-
sary (super)-commutative. For more details about Jacobi-Jordan algebras
(see [5, 9, 3]).

In this paper, we introduce and study extension and crossed module of
Lie type superalgebras. We give a characterization of extension in terms of
two bilinear applications and characterize the notion of isomorphism between
two extensions in terms of linear applications satisfying some properties. The
notion of trivial extension is defined and studied. By following [7] where the
authors studied crossed modules of Leibniz algebras, we define crossed module
for Lie type superalgebras that we call Lie type crossed module. The notion of
normalized, linked and bilateral Lie type crossed module are defined and we
also characterize the equivalence of two Lie type crossed modules. A homology
and cohomology theory of Lie-type superalgebras is introduced and the first
degree cohomology group is given in term of equivalent class of the so-called
restricted trivial extensions.

In [11], we studied quadratic Lie-type superalgebras that are Lie-type su-
peralgebras (U, [ , ]) endowed with a nondegenerate, supersymmetic and in-
variant bilinear form B. We notice that the invariant or associative property
of B that is B([x,y],z) = B(x, [y,z]) for all x,y,z ∈ U, plays an important
role in the study of quadratic structure of Lie-type superalgebras. But the
fact that the bracket of Lie-type superalgebras is not necessary supercom-
mutative allows us to define a new type of invariant of B by B([x,y],z) =
(−1)|x||y|B(y, [x,z]) called left super-invariance that is different from the as-
sociative property.

Another purpose of this paper is the study of the so-called pseudo quadratic
Lie type superalgebras. A Lie type superalgebra is said pseudo quadratic if
it is endowed with a nondegenerate, symmetric and left super-invariant bilin-
ear form. We show that any pseudo quadratic Lie type superalgebra (U, [ , ])
induces a Jacobi-Jordan superalgebra (U,∧). By using the double extension
extented to Lie type superalgebras, we study pseudo quadratic Lie type su-
peralgebra (U, [ , ]) and the associated Jacobi-Jordan superalgebra (U,∧).

This paper is organized as follows. The first section is devoted to the defini-
tions and elementary results. In Section 2, we study homology and cohomol-



lie type superalgebras 155

ogy of Lie type superalgebras. In Section 3, we define extension and crossed
module of Lie type superalgebras and characterize these notions in terms of
linear and bilinear applications. We give a characterization of the notion of
isomorphism between two Lie type crossed modules and the notion of equiva-
lence between extensions of Lie type superalgebra. The notion of normalized
and bilateral Lie type crossed module are defined and studied. In Section 4,
we define left super-invariance for a bilinear form and by using the notion of
double extension, we study pseudo quadratic Lie type superalgebras and the
induced Jacobi-Jordan superalgebras.

Throughout this paper, all vector spaces and algebras considered are de-
fined over an algebraically closed field K of characteristic zero.

Notations: In this paper we shall keep the same notation as in [2].

1. Preliminaries

In this section we give basic definitions and elementary results about Lie-
type superalgebras and Jacobi-Jordan superalgebras.

Definition 1.1. Let U := U0̄ ⊕U1̄ be a Z2-graded vector space endowed
with a bilinear map [ , ] : U ⊗ U → U such that [Uα,Uβ] ⊆ Uα+β for all
α,β ∈ Z2. Then (U, [ , ]) is called left Lie type superalgebra if

[x, [y,z]] = −[[x,y],z]−(−1)|x||y|[y, [x,z]] ∀x ∈U|x|, y ∈U|y|, z ∈U, (1.1)

and (U, [ , ]) is called right Lie type superalgebra if

[x, [y,z]] = −[[x,y],z]−(−1)|y||z|[[x,z],y] ∀x ∈U|x|, y ∈U|y|, z ∈U|z|. (1.2)

The superalgebra (U, [ , ]) is called symmetric Lie type superalgebra if it
is simultaneously a left and right Lie type superalgebra.

Remark 1.1. Let (U, [ , ]) be a left Lie type superalgebra. Define the bi-
linear map { , } : U⊗U →U by {x,y} = (−1)|x||y|[y,x], then (U,{ ,}) is a right
Lie type superalgebra. Therefore the category of left Lie-type superalgebras
is isomorphic to the category of right Lie-type superalgebras.

Let (A, ·) be a superalgebra. We define the anti-associator of A by the
trilinear application Aasso : A⊗A⊗A → A by Aasso(x,y,z) := x · (y ·z) +
(x·y)·z for all x,y,z ∈ A. The superalgebra (A, ·) is said to be anti-associative
if Aasso(x,y,z) = 0 for all x,y,z ∈ A.



156 m. pouye, b. kpamegan

Example 1.1. Let (U, ·) be an anti-associative superalgebra. If we define
the bilinear map [ , ] : U ⊗U →U by [x,y] = x ·y + (−1)|x||y|y ·x for x ∈U|x|
and y ∈U|y|, then (U, [ , ]) is a right Lie type superalgebra.

A homomorphism f : U →W between two Z2-graded vector spaces is said
to be homogeneous of degree α ∈ Z2 if f(Uβ) ⊆ Wα+β for all β ∈ Z2. Given
three Z2-graded vector spaces U, W and H, a bilinear map g : U ⊗W → H
is said to be homogeneous of degree α ∈ Z2 if g(Uβ,Wγ) ⊆ Hα+β+γ for all
β,γ ∈ Z2. The degree of a homogeneous linear or bilinear map f is denoted by
| f | and f is said to be an even (resp. odd) map if | f |= 0̄ (resp. | f |= 1̄). For
any left Lie-type superalgebra (U, [ , ]), the left and the right multiplication
L and R defined by Lx(y) := [x,y] and Rx(y) := (−1)|x||y|[y,x] satisfy the
following relations:

Lemma 1.1. (i) L[x,y] = −Lx◦Ly−(−1)|x||y|Ly◦Lx for all x ∈U|x|,y ∈U|y|;

(ii) R[x,y] = −Lx ◦Ry − (−1)|x||y|Ry ◦Rx for all x ∈U|x|,y ∈U|y|;

(iii) R[x,y] = −Lx ◦Ry − (−1)|x||y|Ry ◦Lx for all x ∈U|x|,y ∈U|y|;

(iv) Ry ◦Rx = Ry ◦Lx for all x ∈U|x|,y ∈U|y|.

Proof. Straightforward computation.

The left centre Zl(U) and the right centre Zr(U) are defined by Zl(U) =
{x ∈ U, [x,U] = 0} and Zr(U) = {x ∈ U, [U,x] = 0}. Define by Ker(U)
the subspace generated by elements of the form [x,y] − (−1)|x||y|[y,x] where
x ∈U|x| and y ∈U|y|. For any left Lie-type superalgebra (U, [ , ]), it holds

Lemma 1.2. (i) Ker(U) ⊆ Zl(U);

(ii) Zl(U) is a two sided ideal and Zr(U) is a sub-superalgebra.

Proof. See [11, Lemma 2.6].

The fact that Ker(U) ⊆ Zl(U) implies that [[x,y],z] = (−1)|x||y|[[y,x],z]
for x,y,z ∈U. One can sees that an analogous result of Lemma 1.2 holds for
right Lie-type superalgebras. In fact, if (U, [ , ]) is a right Lie-type superalgebra
then Ker(U) ⊆ Zr(U). Therefore

[x, [y,z]] = (−1)|y||z|[x, [z,y]] ∀x ∈U|x|, y ∈U|y|, z ∈U|z|. (1.3)



lie type superalgebras 157

Definition 1.2. Let U be a left Lie-type superalgebra and V a Z2-graded
vector space. A representation of U over V is a couple of even linear maps
(ϕ,λ) where ϕ,λ : U → End(V ) such that

ϕ[x,y] = −ϕx ·φy − (−1)
|x||y|ϕy ·φx ,

λ[x,y] = −ϕx ·λy − (−1)
|x||y|λy ·λx ,

λ[x,y] = −ϕx ·λy − (−1)
|x||y|λy ·ϕx ,

for all homogeneous elements x,y ∈ U. If ϕ = λ = 0, the representation is
called trivial representation. We denote RepUV the set of all representations of
U over a given Z2-graded vector space V .

Example 1.2. Let U be a left Lie-type superalgebra. Then according to
Lemma 1.1, (L,R) ∈ RepUU and is called the adjoint representation or the
regular representation of U.

Let (U, [ , ]) be a left (resp. right) Lie type superalgebra, V := V0̄ ⊕V1̄ a
Z2-graded vector space and (ϕ,λ) a representation of U in V. Then the even
bilinear application ψ : U ⊗U → V is said to be an even bi-cocycle of left
(resp. right) Lie type superalgebra with respect to (ϕ,λ) if for all x,y,z ∈ U
we have

ψ(x, [y,z]) + ψ([x,y],z) + (−1)|x||y|ψ(y, [x,z]) + ϕx(ψ(y,z))

+ (−1)|x||y|ϕy(ψ(x,z)) + (−1)|z|(|x|+|y|)λz(ψ(x,y)) = 0

(resp.

ψ(x, [y,z]) + ψ([x,y],z) + (−1)|y||z|ψ([x,z],y) + ϕx(ψ(y,z))

+ (−1)|z|(|x|+|y|)λz(ψ(x,y)) + (−1)|x||y|λy(ψ(x,z)) = 0 ) .

Let (U, [ , ]) be a left(resp. right) Lie type superalgebra, V a Z2-graded vector
space and ψ : U ⊗U → V an even bilinear map. Then, the Z2-graded space
U := U ⊕V endowed with the product

[x + u,y + v]ψ = [x,y] + ψ(x,y) ∀x,y ∈U, u,v ∈V

is a left (resp. right) Lie type superalgebra if and only if

ψ(x, [y,z]) + ψ([x,y],z) + (−1)|x||y|ψ(y, [x,z]) = 0



158 m. pouye, b. kpamegan

(resp.

ψ(x, [y,z]) + ψ([x,y],z) + (−1)|y||z|ψ([x,z],y) = 0) .

Moreover, (U, [ , ]ψ) is a symmetric Lie type superalgebra if and only if
(U, [ , ]) is symmetric and ψ is an even bi-cocycle of U with respect the trivial
representation such that

ψ(x, [y,z]) = (−1)|x|(|y|+|z|)ψ([y,z],x) ∀x,y,z ∈U .

In this case ψ is called an even Lie-type bi-cocycle of U on the trivial U-module
V. We denote by (ZLtype(U,V))0̄ the set of even Lie-type bi-cocycles of U on
the trivial U-module V.

Lemma 1.3. Let U be a left Lie-type superalgebra. Then U is a right
Lie-type superalgebra if and only if

[x, [y,z]] = (−1)|x|(|y|+|z|)[[y,z],x] ∀x,y,z ∈U . (1.4)

Proof. See the proof of [11, Lemma 3.1].

According to the above lemma, a Lie-type superalgebra is symmetric if
and only if relation (1.4) holds.

Definition 1.3. A Jacobi-Jordan superalgebra is a Z2-graded vector
space J := J0̄ ⊕J1̄ endowed with an even bilinear map [ , ] : J ⊗J → J
such that [Jα,Jβ] ⊆Jα+β for all α,β ∈ Z2 and

1. [x,y] = (−1)|x||y|[y,x] for all x ∈J|x|, y ∈J|y|;

2. (−1)|x||z|[x, [y,z]] + (−1)|x||y|[y, [z,x]] + (−1)|y||z|[z, [x,y]] = 0 for all
x ∈J|x|, y ∈J|y|, z ∈J|z|.

Example 1.3. ([1]) The (2n + 1)-dimensional Heisenberg Jacobi-Jordan
superalgebra h(2n + 1,K) = (h0̄ ⊕ h1̄, ·) where h0̄ ⊕ h1̄ = {e1, . . . ,en} ⊕
{f1, . . . ,fn,z} and

ei ·fi = fi ·ei := z ∀i = 1, . . . ,n.

Every Jacobi-Jordan superalgebra is a Lie-type superalgebra. A Lie-type
superalgebra (U, [ , ]) is a Jacobi-Jordan superalgebra if and only if
Ker(U) = {0}.



lie type superalgebras 159

2. Homology and cohomology of Lie-type superalgebras

In this section we study homology and cohomology of right Lie-type
superalgebras.

Definition 2.1. A Z2-graded vector space V := V0̄ ⊕ V1̄ is called right
U-module if it endowed with an action [ , ] : V ⊗U → V such that

[v, [x,y]] = −[[v,x],y]−(−1)|x||y|[[v,y],x] ∀x ∈U|x|, y ∈U|y|, v ∈ V. (2.1)

Let us consider the canonical surjection ϕ : U → Uab := U/[U,U] and V
a right U-module. We define Cn(U,V ) := V ⊗ (ϕ(U))⊗n for all n ∈ N. Then
one can easily see that Cn(U,V ) is a U-module through the following action

[v ⊗x1 ⊗x2 ⊗···⊗xn,x] = (−1)|x|
∑

16k6n|xk|[v,x] ⊗x1 ⊗···⊗xn

for all v ⊗x1 ⊗x2 ⊗···⊗xn ∈ Cn(U,V ) and x ∈U|x|.
In the sequel for simplicity, we denote by x0⊗x1⊗x2⊗···⊗xn an element

of Cn(U,V ) with x0 ∈ V .
Let δ : Cn(U,V ) → Cn−1(U,V ) be the application defined by

δ(x0,x1, . . . ,xn) =
n∑
j=1

(−1)|xj|
∑

0<l<j|xl|[x0,xj] ⊗···⊗ x̂j ⊗···⊗ ·· ·

where the sign ˆ over a variable x means that x has to disappear. Let x =
x0 ⊗x1 ⊗···⊗xn ∈ Cn(U,V ) for any integer n > 0 and y,z ∈Uab. Then we
have the following relations:

Proposition 2.1. (i) δ(x⊗y) = δx⊗y + [x,y];

(ii) [x⊗y,z] = (−1)|y||z|[x,z] ⊗y;

(iii) δ[x,y] = −[δx,y];

(iv) δ2 = 0.



160 m. pouye, b. kpamegan

Proof. For relation (i), let us set xn+1 = y. We have

δ(x⊗y) = δ(x0 ⊗x1 ⊗···⊗xn ⊗xn+1)

=

n+1∑
j=1

(−1)|xj|
∑

0<l<j|xl|[x0,xj] ⊗···⊗ x̂j ⊗···

=
n∑
j=1

(−1)|xj|
∑

0<l<j|xl|[x0,xj] ⊗···⊗ x̂j ⊗···⊗xn+1

+ (−1)|xn+1|
∑n

l=1|xl|[x0,xn+1] ⊗x1 ⊗···⊗xn

=


 n∑
j=1

(−1)|xj|
∑

0<l<j|xl|[x0,xj] ⊗···⊗ x̂j ⊗···


⊗y

+ (−1)|y|
∑n

l=1|xl|[x0,x] ⊗x1 ⊗···⊗xn
= δx⊗y + [x,y] .

For the second relation, we set xn+1 = y and with a simple calculation we
obtain

[x⊗y,z] = [x0 ⊗x1 ⊗···⊗xn ⊗xn+1,z]

= (−1)|z|
∑n+1

i=1 |xi|[x0,z] ⊗···⊗xn ⊗xn+1

= (−1)|z||xn+1|+|z|
∑n

i=1|xi|[x0,z] ⊗···⊗xn ⊗xn+1

= (−1)|y||z|+|z|
∑n

i=1|xi|[x0,z] ⊗···⊗xn ⊗y = (−1)|y||z|[x,z] ⊗y .

For relation (iii), we shall proceed by induction over n. For that let us recall
that since Uab := U/[U,U] is abelian, then for all x,y ∈Uab, the relation (2.1)
becomes

[[v,x],y] = −(−1)|x||y|[[v,y],x] . (2.2)

If n = 1 then x = x0 ⊗x1. Hence according to (2.2) and relation 2, we have

δ([x,y]) = δ([x0 ⊗x1,y])

= (−1)|x1||y|δ([x0,y] ⊗x1)
= −[[x0,x1],y] = −[δx,y] .

Let us assume now that the relation is true up to some integer n. Let x′ := x⊗z
an element of Cn+1(U,V ). Hence according to (i), (ii), relation (2.2) and the



lie type superalgebras 161

induction hypothesis, we have

δ[x′,x] = δ([x⊗z,y]) = (−1)|y||z|δ([x,y] ⊗z)

= (−1)|y||z|(δ[x,y] ⊗z + [[x,y],z])

= −(−1)|y||z|[δx,y] ⊗z + (−1)|y||z|[[x,y],z]
= −[δx⊗z,y] − [[x,z],y]
= −[δx⊗z + [x,z],y] = −[δ(x⊗z),y] = −[δx′,y] ;

then the relation yields for n + 1, and this proves the relation 3). Finally in
order to prove that δ2 = 0, we proceed by induction. For n = 2, by using the
Lie-type identity (1.2) and the fact that Uab is abelian, we have

δ2(x0 ⊗x1 ⊗x2) = δ([x0,x1 ⊗x2] + (−1)|x1||x2|[x0,x2] ⊗x1)

= [[x0,x1],x2] + (−1)|x1||x2|[[x0,x2],x1]
= −[x0, [x1,x2]] = 0 .

Let us assume that the result is true up to some integer n. Let x′ = x⊗y ∈
Cn+1(U,V ) where x ∈ Cn(U,V ) and y ∈Uab. With the help of relation 1), 3)
and the induction hypothesis, we have

δ2(x′) = δ2(x⊗y) = δ(δ(x⊗y)) = δ(δx⊗y + [x,y])

= δ(δx⊗y) + δ[x,y] = δ2x⊗y + [δx,y] + δ[x,y]
= [δx,y] + δ[x,y] = [δx,y] − [δx,y] = 0

and this proves that δ2 = 0.

The above result shows that δ defines a differential of degree −1 over
Cn(U,V ). Then we have the complex (C∗(U,V ),δ) which the homological
groups are called Lie type homological groups of U with coefficients in V and
are denoted by H∗(U,V ). We obtain a similar result to the case of Leibniz
homology concerning the homological groups of degree 0 and 1.

Lemma 2.1. H0(U,V ) = V/[V,Uab] and if V is a trivial module then
H1(U,V ) = V ⊗Uab.

Proof. The proof is straightforward.



162 m. pouye, b. kpamegan

Let U be a Lie type superalgebra and V := V0̄ ⊕ V1̄ a Z2-graded vector
space. For n ∈ N, define by Cn(U,V ) := HomK(U ⊗ (Uab)⊗n,V ). Then
Cn(U,V ) admits a structure of U-module through the following action

(f ·x)(y ⊗y1 ⊗···⊗yn) = (−1)|x||y|f([y,x] ⊗y1 ⊗···⊗yn)

for all x ∈U,y⊗y1⊗···⊗yn ∈U⊗(Uab)⊗n. Let Dn : Cn(U,V ) → Cn+1(U,V )
defined by

Dnf(x0 ⊗x1 ⊗···⊗xn+1) =
n+1∑
j=1

(−1)|xj|
∑

0<l<j|xl|f([x0,xj] ⊗···⊗ x̂j ⊗···).

The application D defines a differential over the graded U-module C∗(U,V ).
In fact we have the following result

Proposition 2.2. D2 = 0.

Proof. It’s sufficient to notice that for all integer n, we have Dnf = fδn+1
and according to 4) of Proposition 2.1 we obtain D2 = 0.

Indeed, we obtain the cochain complex (C∗(U,V ),D) and the cohomolog-
ical groups denoted by H∗(U,V ).

3. Extensions and crossed modules of Lie type superalgebras

In this section, we introduce and study the theory of extension and crossed
module for Lie type superalgebras. In particular we characterize these notions
with more explicit objects such as bilinear applications satisfying some condi-
tions. The cohomological group H1(U,V) is investigated through a particular
type of extension.

3.1. Extensions of Lie type superalgebras. Let (U, [ , ]) and
(U1, [ , ]) be two Lie type superalgebras. Let π : U1 → U an epimorphism
and V := Ker(π). Then the ideal V admits a structure of U1-module via the
action (v,x) ∈ V ⊗U1 7→ [v,x] ∈ V. Moreover if V is an abelian Lie type
superalgebra, then V admits a structure of U-module through the following
action

v ·x = [y,v] ∀x ∈U, y ∈U1 such that x = πy . (3.1)

Conversely if V admits a structure of U-module through the action defined in
(3.1), then V is abelian.



lie type superalgebras 163

Definition 3.1.1. Let U be a Lie type superalgebra and V a right U-
module. An extension of U by V is an exact sequence

0 −→V i−−→U1
π−−→U −→ 0 , (3.2)

where U1 is a Lie type superalgebra and V becomes a U1-module that can be
considered as an abelian Lie type superalgebra.

Example 3.1.1. Let (L, [ , ]) be a Lie type superalgebra. Then U is an
extension of U/Ker(U) by Ker(U). In fact we have the following exact sequence

0 −→ Ker(U) i−−→U π−−→U/Ker(U) −→ 0 .

The U-module V is called the kernel of the extension (3.2) and the Lie
type superalgebra U1 can be identified by V⊕U. In what follows, we give an
explicit description of the notion of an extension of U with kernel V.

Theorem 3.1.1. An extension of U by V is equivalent to the set of two
even bilinear applications α : U ⊗U →V and φ : U ⊗V →V such that

(i) α(x, [x′,x′′]) + φ(x,α(x′,x′′)) = −α([x,x′],x′′) − [α(x,x′),x′′]
−(−1)|x

′||x′′|α([x,x′′],x′) − (−1)|x
′||x′′|[α(x,x′′),x′] ,

(ii) φ(x,φ(x′,v)) = −φ([x,x′],v) − [φ(x,v),x′]

for all x ∈U|x|, x′ ∈U|x′|, x′′ ∈U|x′′| and v ∈V.

Proof. Let 0 → V i−→ U1
π−−→ U → 0 be an extension of U by V. The

Lie type superalgebra U1 is identified by V ⊕U. Since V is a right U-module
and an abelian superalgebra, then there exists two even bilinear applications
α : U ⊗U →V and φ : U ⊗V →V such that the bracket of U1 is given by

[(x,v), (x′,v′)] = ([x,x′],α(x,x′) + φ(x,v′) + [v,x′]) ∀(x,v), (x′,v′) ∈U1.

By applying the Lie type identity (1.2) to ((x, 0), (x′, 0), (x′′, 0)) we obtain
the relation 1). For the relation 2), we apply the Lie type identity (1.2) to
((x, 0), (x′, 0), (0,v)).

Definition 3.1.2. Two extensions 0 → V i−→ U1
π−−→ U → 0 and

0 → V i−→ U2
π−−→ U → 0 of U by V are said to be equivalent or isomor-

phic if there exists a homomorphism ψ : U1 → U2 such that the following



164 m. pouye, b. kpamegan

diagram

0 −−−−→ V i−−−−→ U1
π−−−−→ U −−−−→ 0∥∥∥ y ∥∥∥

0 −−−−→ V i
′

−−−−→ U2
π′−−−−→ U −−−−→ 0

(3.3)

is commutative. In other words π′ ◦ψ = π and ψ ◦ i = i′.

One can easily sees that the application ψ of the definition above is bijec-

tive. In the sequel, the extension 0 → V i−→ U1
π−−→ U → 0 will be identified

by the couple (α,φ) of Theorem 3.1.1.

Proposition 3.1.1. Two extensions (α,φ) and (α′,φ′) are equivalent if
and only if, there exists an even linear application λ : U →V such that

λ([x,x′]) + α(x,x′) = α′(x,x′) + φ′(x,λx′) + [λx,x′] ∀x,x′ ∈U .

Proof. Let ψ : U1 ∼= U⊕V →U2 ∼= U⊕V be a bijective application between
the two extensions. The commutativity of the diagram (3.3) and the fact that
i and i′ are injective imply that ψ(V) ⊆ V and ψ(U) ⊆ U + V. Hence there
exists a linear application λ : U → V such that for all (x,v) ∈ U1 we have
ψ((x,v)) = (x,λx + v). Since ψ is compatible with the brackets of U1 and U2,
then we have

ψ([(x, 0), (x′, 0)]) = [ψ(x, 0),ψ(x′, 0)] ∀(x, 0), (x′, 0) ∈U1
⇔ ψ

(
([x,x′],α(x,x′))

)
= [(x,λx), (x′,λx′)]

⇔
(
[x,x′],λ([x,x′]) + α(x,x′)

)
=
(
[x,x′],α′(x,x′) + φ′(x,λx′) + [λx,x′]

)
.

Hence λ([x,x′]) + α(x,x′) = α′(x,x′) + φ′(x,λx′) + [λx,x′] for all x,x′ ∈U.

An extension (α,φ) of U by V is said to be trivial if V is a trivial U-module
and φ = 0. Let (α, 0) be a trivial extension of U and denote αR the restriction
of the bilinear map α over U ⊗ Uab. We call (αR, 0) the restricted trivial
extension of U by V.

Proposition 3.1.2. Let (α, 0) be a trivial extension of U by V. Then
αR ∈ Ker(D1).



lie type superalgebras 165

Proof. Let x ∈ U and x′,x′′ ∈ Uab. According to the first relation of
Theorem 3.1.1 and the fact that Uab is abelian we have

DαR(x,x′,x′′) = αR([x,x′],x′′) + (−1)|x
′||x′′|αR([x,x

′′],x′)

= −αR(x, [x′,x′′]) = 0

which implies that αR ∈ Ker(D1).

Lemma 3.1.1. Two restricted trivial extensions (αR, 0) and (α
′
R, 0) of U

by V are equivalent if and only if α′R −αR ∈ Im(D0).

Proof. According to Proposition (3.1.1), we have (α, 0) and (α′, 0) are
equivalent if and only if there exists an even linear map λ : U →V such that
λ[x,x′] = α′(x,x′)−α(x,x′) for all x,x′ ∈U, that is (α′−α)(x,x′) = λ([x,x′]).
Hence (α′R −αR)(x,x

′) = λ([x,x′]). Therefore α′R −αR ∈ Im(D0)

The following result gives the cohomological group H1(U,V).

Theorem 3.1.2. Let (U, [ , ]) be a Lie-type superalgebra and V a U-
module. The isomorphic class of restricted trivial extensions of U by V is
isomorphic to H1(U,V).

Proof. The result follows from a combination of Proposition 3.1.2 and
Lemma 3.1.1.

3.2. Crossed modules of Lie type superalgebras. Let (U, [ , ])
be a Lie type superalgebra. A homogeneous endomorphism f of U is called
(−1,−1)-superderivation if

f([x,y]) = −[f(x),y] − (−1)|f||y|[x,f(y)] ∀x ∈U, y ∈U|y| .

The set of all (−1,−1)-superderivations of U will be denoted by
(−1,−1) SDer(U).

Definition 3.2.1. A crossed module of (U, [ , ]) is a triplet (W,d,η) where
W is a Lie type superalgebra, d : U → W an even morphism and η : W →
(−1,−1) SDer(U) an even linear map such that

(a) η([w,w′]) = −η(w′) ◦ η(w) − (−1)|w||w
′|η(w) ◦ η(w′) for all w ∈ W|w|,

w′ ∈W|w′|;



166 m. pouye, b. kpamegan

(b) d(η(w)(u)) = [du,w] for all u ∈U, w ∈W;
(c) η(du)(u′) = [u′,u] for all u,u′ ∈U.

In what follows, we characterize crossed modules of Lie type superalgebras.
For that, let us set

P = coker(d) , L = Ker(d) and V = Im(d) .

We have the following properties:

Lemma 3.2.1. (i) L⊆ Zr(U) and d ∈ Aut(V);
(ii) V is a left ideal of W and P is a complement of V in W;
(iii) U and L are P-modules through η.

Proof. For relation (i), it’s clear that d ∈ Aut(V). Moreover from relation
(c) of Definition 3.2.1, we have [U,L] ⊆ η(dL)(U) = 0 because dL = 0. Then
L ⊆ Zr(U). For the second relation, according to relation (b) of Definition
3.2.1 we have

[V,W] = [d(U),W] ⊆ d(η(W)(U)) ⊆ Im(d) = V

then V is a left ideal of W. One can easily see that P is a complement of V
in W. Finally, relation (a) of Definition 3.2.1 and the fact that d(η(P)(L)) ⊆
[dL,W] = 0 imply that U and L are P-modules through η.

Hence a crossed module (W,d,η) of U leads to the following exact sequence

0 −→L i−−→U d−−→W π−−→P −→ 0 (3.4)

where U can be identified by L⊕V and W by P ⊕V. The crossed module
(W,d,η) is called Lie-type crossed module of kernel L and cokernel P or for
simplicity, a (L,P)-Lie-type crossed module.

Let (W,d,η) be a (L,P)-Lie-type crossed module. We have the following
result:

Theorem 3.2.1. The (L,P)-Lie-type crossed module (W,d,η) is equiva-
lent to a set composed by a structure of W-module over L, an automorphism
d of V, two even bilinear maps α : V ⊗V →L and η0 : P ⊗V →L such that

(i) α(v, [v′,v′′]) = −[α(v,v′),v′′] −α([v,v′],v′′) − (−1)|v
′||v′′|α([v,v′′],v′)

−(−1)|v
′||v′′|[α(v,v′′),v′] ,



lie type superalgebras 167

(ii) η0(p⊗ [v,v′]) + [α(v,v′),p] = −α(d−1[dv,p],v′) − [η0(p⊗v),v′]
−(−1)|p||v

′|α(v,d−1[dv,p]) ,

(iii) η0([p,p
′]P) = −η0(p′) ◦η0(p) − (−1)|p||p

′|η0(p) ◦η0(p′)

for all v ∈V|v|, v′ ∈V|v′|, v′′ ∈V|v′′| and p ∈P|p|, p′ ∈P|p′|.

Proof. Since U can be identified by L⊕V, [L,V] ⊆L and L⊆ Zr(U), then
there exist bilinear applications [ , ]V : V ⊗V → V and α : V ⊗V → L such
that the bracket of U is given by

[(x,v), (x′,v′)] = ([x,v′] + α(v,v′), [v,v′]V) ∀(x,v), (x′,v′) ∈L⊕V.

By applying the Lie type identity (1.2) to the triplet ((0,v), (0,v′), (0,v′′)) we
obtain the relation (i).

For (ii), since W = P ⊕ V and η(w) ∈ End(L) for all w ∈ W then
η(W)(V) = η(P)(V) + η(V)(V). Moreover, according to relation (b) of Defi-
nition 3.2.1, we have

η(V)(V) = η(dU)(V) = [V,U] = [V,V] = [V,V]V + α(V,V) .

Since η(P)(V) ⊆L⊕V, then there exists a bilinear application η0 : P⊗V →L
such that

η(p)(v) = ηV(p)(v) + η0(p)(v)

where ηV(p)(v) is the component in V of η(p)(v) for all p ∈ P and v ∈ V.
According to the relation (b) of Definition 3.2.1, we have

d(η(p)(v)) = d(ηV(p)(v)) + d(η0(p⊗v)) = d(ηV(p)(v)) = [dv,p]

and since d ∈ Aut(V) then ηV(p)(v) = d−1[dv,p]. This implies that η(p)(v) =
d−1[dv,p] + η0(p⊗v). From the fact η : W → (−1,−1) SDer(U) we have

η(p)([v,v′]) = −[η(p)(v),v′] − (−1)|p||v
′|[v,η(p)(v′)]

⇐⇒ η(p)([v,v′]V + α(v,v′)) = −[d−1[dv,p] + η0(p⊗v),v′]

− (−1)|p||v
′|[v,d−1[dv′,p] + η0(p⊗v′)]

⇐⇒ d−1[d[v,v′],p] + η0(p⊗ [v,v′]) + [α(v,v′),p] = −[d−1[dv,p],v′]

− [η0(p⊗v),v′] − (−1)|p||v
′|[v,d−1[dv′,p]] − (−1)|p||v

′|[v,η0(p⊗v′)]

⇐⇒ d−1[d[v,v′],p] + η0(p⊗ [v,v′]) + [α(v,v′),p] = −[d−1[dv,p],v′]

−α(d−1[dv,p],v′) − [η0(p⊗v),v′] − (−1)|p||v
′|[v,d−1[dv′,p]]

− (−1)|p||v
′|α(v,d−1[dv′,p]) − (−1)|p||v

′|[v,η0(p⊗v′)]



168 m. pouye, b. kpamegan

a projection over U of the later relation gives us the relation (ii). The relation
(iii) is obtained from relation (a) of the Definition 3.2.1.

In the sequel, a Lie type crossed module (W,d,η) will be denoted by the
exact sequence (3.4) or by the triplet (d,α,η0) of the Theorem 3.2.1.

Definition 3.2.2. Two Lie type crossed modules 0 → L i−→ U d−−→
W π−−→ P → 0 and 0 → L i

′
−−→ U′ d−−→ W π

′
−−→ P → 0 are isomorphic or

equivalent if there exists an isomorphism ψ : U → U′ such that the following
diagrams:

0 −−−−→ L i−−−−→ U d−−−−→ W π−−−−→ U −−−−→ 0∥∥∥ y ∥∥∥ ∥∥∥
0 −−−−→ L i

′
−−−−→ U′ d

′
−−−−→ W π

′
−−−−→ U −−−−→ 0

(3.5)

and

W ⊗U
η

−−−−→ U

IdW⊗ψ
y yψ

W ⊗U′
η′

−−−−→ U′

(3.6)

are commutative.

Proposition 3.2.1. Two Lie type crossed modules (d,α,η0) and (d
′,α′,

η′0) are equivalent if there exists an even linear map θ : V →L such that

(1) α(v,v′) + θ([v,v′]) = [θv,d′−1dv′] + α′(d′−1dv,d′−1dv′) ,

(2) η0(p⊗v) + θ(d−1[dv,p]) = [θv,p] + η′0(p⊗d
′−1v) ,

(3) [l,d−1v] = [l,d′−1v] .

Proof. Let ψ : U ∼= L⊕V → U′ ∼= L⊕V′ be an isomorphism between U
and U′ such that the diagrams (3.5) and (3.6) be commutative. According to
the commutativity of the diagram (3.5), we have ψ ◦ i = i′ ◦ IdL then for all
x ∈L, ψ(x) = x. Moreover, since ψ(v) ∈L⊕V′ for all v ∈V then there exists
a linear application θ : V →L such that

ψ(l,v) = (l + θv,ψV(v)) ∀(l,v) ∈L⊕V



lie type superalgebras 169

where ψV(v) is the component of ψ(v) in V′. We have ψV(v) = d′−1dv. In
fact, according to the diagram (3.5) we have

IdW ◦d(0,v) = d′ ◦ψ(0,v) ⇐⇒ (0,dv) = d′(θv,ψV(v))

⇐⇒ (0,dv) = (d′θv,d′ψV(v))

⇒ dv = d′ψV(v) ,

hence ψV(v) = d
′−1dv. Therefore, we have ψ(l,v) = (l + θv,d′−1dv). The

compatibility of ψ with the brackets implies that

ψ[(0,v), (0,v′)] = [ψ(0,v),ψ(0,v′)]

⇐⇒ ψ
(
α(v,v′), [v,v′]V

)
= [(θv,d′−1dv), (θv′,d′−1dv′)]

⇐⇒
(
α(v,v′) + θ([v,v′]V),d

′−1d([v,v′]V)
)

=
(
[θv,d′−1dv′] + α′(d′−1dv,d′−1dv′), [d′−1dv,d′−1dv′]V

)
which proves the first relation.

For relation 2), the commutativity of the diagram (3.6) implies that for all
p⊗v ∈P ⊗V we have

ψ ◦η(p⊗v) = η′ ◦ (IdW ⊗ψ)(p⊗v)

⇐⇒ ψ(d−1[dv,p] + η0(p⊗v)) = η′(p⊗θv) + η′(p⊗d′−1dv)

⇐⇒ η0(p⊗v) + θ(d−1[dv,p]) + d′−1[dv,p]
= [θv,p] + d′−1[d′d′−1dv,p] + η′0(p⊗d

′−1dv)

η0(p⊗v) + θ(d−1[dv,p]) + d′−1[dv,p]
= [θv,p] + d′−1[dv,p] + η′0(p⊗d

′−1dv)

a projection over L of the later relation gives us η0(p ⊗ v) + θ(d−1[dv,p]) =
[θv,p] + η′0(p⊗d

′−1v).
Let v ∈V and l ∈L. According to the commutativity of the diagram (3.6)

we have ψ ◦ η(v ⊗ l) = η′ ◦ (IdW ⊗ ψ)(v ⊗ l) hence ψ (η(v)(l)) = η′(v ⊗ l),
therefore ψ([l,d−1v]) = [l,d′−1]. Since [l,d−1v] ∈ L then [l,d−1v] = [l,d′−1v].
This proves relation 3).

Definition 3.2.3. A Lie type crossed module (W,d,η) of (U, [ , ]) is said
to be:
• normalized if d = IdU,
• bilateral if L⊆ Z(U).



170 m. pouye, b. kpamegan

By adapting the characterization of isomorphism between two Lie type
crossed modules (see Proposition 3.2.1) to the normalized crossed modules,
we obtain the following result:

Corollary 3.2.1. Two normalized Lie type crossed modules (α,η0) and
(α′,η′0) are isomorphic if and only if there exists an even linear map θ : V →L
such that

(1) α(v,v′) + θ([v,v′]) = [θv,v′] + α′(v,v′) ,

(2) η′0(p⊗v) = η0(p⊗v) + θ[v,p] − [θv,p] .

Define by β(p,p′) = [p,p′]P − [p,p′] where [p,p′]P is the component in P
of the product in W of p and p′. The following result gives a characterization
of bilateral Lie-type crossed module.

Proposition 3.2.2. A bilateral Lie type crossed module (W,d,η) of ker-
nel L and co-kernel P is equivalent the set composed by bilinear applications
α : V ⊗V →L and η0 : P ⊗V →L such that

(1) α(v, [v′,v′′]) = −α([v,v′],v′′) − (−1)|v
′||v′′|α([v,v′′],v′);

(2) α(v, [p,v′]) = (−1)|p||v
′|α(v, [v′,p]);

(3) η0([p,p
′] ⊗ v) − α(v,d−1β(p,p′)) = −η0(p′ ⊗ d−1[dv,p]) − [η0(p ⊗ v),p′]

− (−1)|p||p
′|η0(p⊗d−1[dv,p′]) − (−1)|p||p

′|[η0(p
′ ⊗v),p];

(4) η0(p⊗[v,v′])+[α(v,v′),p] = −α(d−1[dv,p],v′)−(−1)|p||v
′|α(v,d−1[dv′,p]).

Proof. The relations (1) and (4) can be obtained respectively from rela-
tions (i) and (iii) of Theorem 3.2.1 by using the fact that α(v,v′) ∈L⊆ Z(U)
for all v,v′ ∈V.

For (2), let v ∈ V|v|, v′ ∈ V|v′| and p ∈ P|p|. Since W is a right Lie-type
superalgebra then according to relation (1.3) we have

[v, [p,v′]] = (−1)|p||v
′|[v, [v′,p]] ,

hence

α(v, [p,v′]) + [v, [p,v′]]V = (−1)|p||v
′|α(v, [v′,p]) + (−1)|p||v

′|[v, [v′,p]]V

which implies that α(v, [p,v′]) = (−1)|p||v
′|α(v, [v′,p]).



lie type superalgebras 171

Let p ∈P|p|, p′ ∈P|p′| and l + v ∈L⊕V, we have then

η([p,p′])(l + v) = η(p′) ◦η(p)(l) − (−1)|p||p
′|η(p) ◦η(p′)(l) −η(p′) ◦η(p)(v)

− (−1)|p||p
′|η(p) ◦η(p′)(v)

= −η(p′)([l,p]) − (−1)|p||p
′|η(p)([l,p′]) −η(p)(d−1[dv,p])

−η(p′)(η0(p⊗v)) − (−1)|p||p
′|η(p)(d−1[dv,p′])

− (−1)|p||p
′|η(p)η0(p

′ ⊗v))

= −[[l,p],p′] − (−1)|p||p
′|[[l,p′],p] −d−1[[dv,p],p′]

−η0(p′ ⊗d−1[dv,p])

− [η0(p⊗v),p′] − (−1)|p||p
′|d−1[[dv,p′],p]

− (−1)|p||p
′|η0(p⊗d−1[dv,p′]) − (−1)|p||p

′|[η0(p
′ ⊗v),p]

= [l, [p,p′]] + d−1[dv, [p,p′]] −η0(p′ ⊗d−1[dv,p])

− [η0(p⊗v),p′] − (−1)|p||p
′|η(p⊗d−1[dv,p′])

− (−1)|p||p
′|[η0(p

′ ⊗v),p] .

On the other hand

η([p,p′])(l + v) = η([p,p′]P)(l + v) −η(β(p,p′))(l + v)

= η([p,p′]P)(l) + η([p,p
′]P)(v)

−η(β(p,p′))(l) −η(β(p,p′))(v)

= [l, [p,p′]P] + d
−1[dv, [p,p′]P] + η0([p,p

′]P ⊗v)
− [l,d−1β(p,p′)] − [v,d−1β(p,p′)]

= [l, [p,p′]P] + d
−1[dv, [p,p′]P] + η0([p,p

′]P ⊗v)
− [v,d−1β(p,p′)]V −α(v,d−1β(p,p′))

which implies relation (3).

In what follows, we define and study another relationship between two Lie-
type crossed modules of a Lie-type superalgebra. In fact we study the notion
of linked bilateral Lie-type crossed modules and characterize this relation.

Definition 3.2.4. Let (MC1) : 0 → L
i−→ U d−−→ W π−−→ P → 0 and

(MC2) : 0 → L
i−→ U′ d

′
−−→ W′ π−−→ P → 0 be two bilateral Lie type crossed



172 m. pouye, b. kpamegan

modules. (MC1) and (MC2) are said to be linked if there exist Lie type
superalgebras homomorphisms ψ : U → U′ and φ : W → W′ such that the
following diagrams

0 −−−−→ L i−−−−→ U d−−−−→ W π−−−−→ U −−−−→ 0∥∥∥ y y ∥∥∥
0 −−−−→ L i

′
−−−−→ U′ d

′
−−−−→ W′ π

′
−−−−→ U −−−−→ 0

(3.7)

and
W ⊗U

η
−−−−→ U

φ⊗ψ
y yψ

W′ ⊗U′
η′

−−−−→ U′

(3.8)

are commutative.

The following result gives a characterization of two linked bilateral Lie
type crossed modules.

Proposition 3.2.3. Two Lie type crossed modules (MC1) and (MC2)
are linked if and only if, there exist even linear maps f : V →V′ and g : V →L
such that

(1) α(v,v′) + g([v,v′]) = α′(f(v),f(v′)),

(2) η′0(p⊗f(v)) + [g(v),p] = η0(p⊗v) + g(d
−1[dv,p]),

(3) f(β(p,p′)) = β′(p,p′).

Proof. Recall that U ∼= L⊕V, U′ ∼= L⊕V′, W ∼= P⊕V and W′ ∼= P⊕V′.
According to the commutativity of the diagram (3.7), we have ψ(x) = x and
ψ(v) ∈ L⊕V′ for all x ∈ L and v ∈ V. Therefore, there exist even linear
maps f : V →V′ and g : V →L such that ψ(x + v) = x + f(v) + g(v) for all
x + v ∈U.

Let x + v ∈U and x′ + v′ ∈U. Then by using the compatibility of ψ with
the brackets and the fact that L⊆ Z(U), we obtain

ψ([x + v,x′ + v′]) = [ψ(x + v),ψ(x′ + v′)]

⇐⇒ ψ([v,v′]) = [f(v) + g(v),f(v′) + g(v′)]
⇐⇒ ψ(α(v,v′) + [v,v′]V) = [f(v),f(v′)]
⇐⇒ α(v,v′) + f([v,v′]V) + g([v,v′]) = α′(f(v),f(v′)) + [f(v),f(v′)]V



lie type superalgebras 173

which implies that α′(f(v),f(v′)) = α(v,v′) + g([v,v′]). Hence we have
relation (1).

For relation (2), let p ∈P and v ∈V. According to the commutativity of
the diagram (3.8), we have

η′ ◦ (φ⊗ψ)(p⊗v) = ψ ◦η(p⊗v)

⇐⇒ η′(φ(p))(ψ(v)) = ψ(η(p)(v))

⇐⇒ η′(p)(f(v) + g(v)) = ψ
(
d−1[dv,p] + η0(p⊗v)

)
⇐⇒ η′(p)(f(v)) + η′(p)(g(v)) = η0(p⊗v) + f(d−1[dv,p]) + g(d−1[dv,p])

⇐⇒ d′−1[d′f(v),p] + η′0(p⊗f(v)) + [g(v),p]

= η0(p⊗v) + f(d−1[dv,p]) + g(d−1[dv,p]) ,

hence η′0(p⊗f(v)) + [g(v),p] = η0(p⊗v) + g(d
−1[dv,p]).

For the proof of relation 3), let p,p′ ∈P. The compatibility of φ with the
brackets of W and W′ gives us

φ([p,p′]) = [φ(p),φ(p′)] ⇐⇒ φ([p,p′]P −β(p,p′)) = [p,p′]

⇐⇒ [p,p′]P −f(β(p,p′)) = [p,p′]P −β′(p,p′)

⇐⇒ f(β(p,p′)) = β′(p,p′) .

4. Pseudo quadratic Lie type superalgebras and theirs
associated Jacobi-Jordan superalgebras

In [11], we studied quadratic Lie-type superalgebras that are Lie-type
superalgebras (U, [ , ]) endowed with a nondegenerate, supersymmetric and
invariant bilinear form B. We notice that the invariant property of B that is

B([x,y],z) = B(x, [y,z]) ∀x,y,z ∈U (4.1)

plays an important role in the study of quadratic structure of Lie-type su-
peralgebras. But the fact that the bracket of Lie-type superalgebras is not
necessary supercommutative allows us to define a new type of invariance of B
called left super-invariance that is different from the one in relation (4.1).

In this section, we aim to investigate Lie-type superalgebras endowed with
a nondegenerate, supersymmetric and left super-invariant bilinear form Big.



174 m. pouye, b. kpamegan

Definition 4.1. Let (U, [ , ]) be a Lie-type superalgebra and Big : U ⊗
U → K a bilinear form. Then Big is said to be
• supersymmetric if Big(x,y) = (−1)|x||y|Big(y,x) for all x ∈U|x|, y ∈U|y|;
• nondegenerate if Big(x,y) = 0 for all y ∈U implies x = 0.

Definition 4.2. Let (U, [ , ]) be a Lie-type superalgebra. A bilinear form
Big over U is left super-invariant if

Big ([x,y],z) = (−1)|x||y|Big (y, [x,z]) ∀x ∈U|x|, y ∈U|y|, z ∈U|z| .

If (U, [ , ]) is endowed with nondegenerate, supersymmetric and left super-
invariant bilinear form Big, then (U, [ , ],Big) is called pseudo-quadratic Lie
type superalgebra.

We can notice that the definition of left super-invariance is equivalent
to say that for all x ∈ U|x|, we have Lx is Big-supersymmetric that is
Big (Lx(y),z) = (−1)|x||y|Big (y,Lx(z)) for all x ∈U|x|, y ∈U|y| and z ∈U|z|.

Lemma 4.1. Let (U, [ , ],Big) be a pseudo-quadratic Lie type superalge-
bra. If moreover Big is invariant then (U, [ , ]) is a Jacobi-Jordan superalgebra.

Proof. Let x ∈U|x|,y ∈U|y| and z ∈U|z|. Then

Big
(

[x,y] − (−1)|x||y|[y,x],z
)

= Big ([x,y],z) − (−1)|x||y|Big ([y,x],z)

= (−1)|x||y|Big (y, [x,z]) − (−1)|x||y|Big (y, [x,z]) = 0 ,

and since Big is nondegenerate then [x,y] = (−1)|x||y|[y,x]. Hence (U, [ , ]) is
a Jacobi-Jordan superalgebra.

Let (U, [ , ],Big) be a pseudo-quadratic Lie type superalgebra. Define the
even bilinear application ∧ : U ⊗U →U by

Big ([x,y],z) = Big (x,y ∧z) ∀x ∈U|x|, y ∈U|y|, z ∈U|z| ;

(U,∧) is a superalgebra and will be called the induced superalgebra of
(U, [ , ],Big).

In what follows, we shall establish some properties of the induced super-
algebra and show that the induced superalgebra of any pseudo-quadratic Lie
type superalgebra is a Jacobi-Jordan superalgebra.



lie type superalgebras 175

Proposition 4.1. Let (U, [ , ],Big) be a pseudo-quadratic Lie type super-
algebra and (U,∧) the induced superalgebra. Then we have:

(1) (U,∧) is a Jacobi-Jordan superalgebra;
(2) x∧ (y ∧z) = [x,y ∧z] for all x ∈U|x|, y ∈U|y|, z ∈U|z|;
(3) Lx is a (−1,−1)-superderivation of (U,∧) that is

[x,y∧z] = −[x,y] ∧z− (−1)|x||y|y∧ [x,z] ∀x ∈U|x|, y ∈U|y|, z ∈U|z|;

(4) for all n ∈ N\{0} we have Ln∧x = Ln−1x ·L∧x, where L∧x(y) = x∧y.

Proof. Let x ∈ U|x|,y ∈ U|y|,z ∈ U|z| and t ∈ U|t|. Let us show first the
super-commutativity of (U,∧)

Big
(
z,x∧y−(−1)|x||y|y ∧x

)
= Big (z,x∧y) − (−1)|x||y|Big (z,y ∧x)

= Big ([z,x],y) − (−1)|x||y|Big ([z,y],x)

= (−1)|x||z|Big (x, [z,y]) − (−1)|x||y|Big ([z,y],x)

= (−1)|x||z|
(
Big (x, [z,y]) −Big (x, [z,y])

)
= 0 ,

and since Big is nondegenerate then x∧y = (−1)|x||y|y∧x. Hence the product
∧ is supercommutative. Let us show that x∧(y∧z) = −(x∧y)∧z−(−1)|x||y|y∧
(x∧z).

Big
(
t,x∧ (y ∧z) + (x∧y) ∧z + (−1)|x||y|y ∧ (x∧z)

)
= Big

(
[[t,x],y],z

)
+ Big

(
[t,x∧y],z

)
+ (−1)|x||y|Big

(
[[t,y],x],z

)
= (−1)|y|(|x|+|t|)

(
Big
(
y, [[t,x],z] + [t, [x,z]] + (−1)|x||z|[[t,z],x]

))
= (−1)|y|(|x|+|t|)

(
Big
(
y,−(−1)|x||t|[x, [t,z]] + (−1)|x||z|[[t,z],x]

))
= (−1)|y|(|x|+|t|)

(
Big
(
y,−(−1)|x||z|[[t,z],x] + (−1)|x||z|[[t,z],x]

))
= 0 ,

which implies that x∧(y∧z) = −(x∧y)∧z−(−1)|x||y|y∧(x∧z). Therefore
(U,∧) is a Jacobi-Jordan superalgebra. Let us show that x∧(y∧z) = [x,y∧z]

Big (t,x∧ (y ∧z) − [x,y ∧z]) = Big ([t,x],y ∧z) − (−1)|x||t|Big ([x,t],y ∧z)

= Big ([[t,x],y],z) − (−1)|x||t|Big ([[x,t],y],z)

= Big
([

[t,x] − (−1)|x||t|[x,t],y
]
,z
)

= 0 ,



176 m. pouye, b. kpamegan

and the fact that Big is nondegenerate gives us x∧(y∧z) = [x,y∧z]. A similar
computation of (2) gives us relation (3). For relation (4), we will proceed by
induction over n. If n = 2 then according to (2) we have

L2∧x(y) = x∧ (x∧y) = [x,x∧y] = Lx ·L∧x(y)

therefore L2∧x = Lx ·L∧x.
Let us assume that Ln∧x = L

n−1
x ·L∧x and show that L

n+1
∧x = L

n
x ·L∧x. By

using the relation (2) above and the induction hypothesis, we obtain

Ln+1∧x (y) = L
n
∧x(x∧y) = L

n−1
x ·L∧x(x∧y)

= Ln−1x (x∧ (x∧y)) = L
n−1
x ([x,x∧y]) = L

n
x ◦L∧x(y) .

Therefore, we have Ln+1∧x = L
n
x ·L∧x for all n ∈ N\{0}.

The above proposition shows that the superalgebras (U ∧U,∧) and (U ∧
U, [ , ]) are the same and the existence of a nondegenerate, supersymmetric and
left super-invariant bilinear form over Lie-type superalgebra (U, [ , ]) induces
over the underlying Z2-graded vector space U a structure of Jacobi-Jordan
superalgebra. The following result gives the expression of the product ∧ of
the induced Jacobi-Jordan superalgebra.

Proposition 4.2. Let (U, [ , ],Big) be a pseudo quadratic symmetric Lie
type superalgebra. Then the induced product ∧ : U⊗U →U is given as follow

x∧y = x∗y −ψ(x,y) ∀x,y ∈U

where x∗y =
1

2

(
[x,y] + (−1)|x||y|[y,x]

)
and ψ ∈ (ZtLie(U,U))0̄ .

Proof. Set µ(x,y) = [x,y] −x∧ y. Then we have µ(x,y) ∈ Zr(U) for all
x,y ∈U. Indeed, let t,z ∈U. We have

Big([z,µ(x,y)], t)

= Big([z, [x,y]], t) −Big([z,x∧y], t)

= Big([z, [x,y]], t) − (−1)|z|(|x|+|y|)Big(x∧y, [z,t])

= Big([z, [x,y]], t) − (−1)|t|(|x|+|y|)Big([[z,t],x],y)

= (−1)|z|(|x|+|y|)Big([x,y], [z,t]) − (−1)|t|(|x|+|y|)Big([[z,t],x],y)

= (−1)|z|(|x|+|y|)+|x||y|Big(y, [x, [z,t]]) − (−1)|t|(|x|+|y|)Big([[z,t],x],y)

= (−1)|z|(|x|+|y|)+|y|(|z|+|t|)Big([x, [z,t]],y) − (−1)|t|(|x|+|y|)Big([[z,t],x],y)



lie type superalgebras 177

= (−1)|x||z|+|y||t|Big([x, [z,t]],y) − (−1)|t|(|x|+|y|)Big([[z,t],x],y)

= (−1)|t|(|x|+|y|)Big([[z,t],x],y) − (−1)|t|(|x|+|y|)Big([[z,t],x],y) = 0

and the fact that Big is nondegenerate implies that µ(x,y) ∈ Zr(U) for all
x,y ∈ U. According to Proposition 4.1, the product ∧ is supercommutative.
Therefore, for all x ∈U|x| and y ∈U|y| we have

x∧y =
1

2

(
x∧y + (−1)|x||y|y ∧x

)
=

1

2

(
[x,y] −µ(x,y) + (−1)|x||y|[y,x] − (−1)|x||y|µ(y,x)

)
=

1

2

(
[x,y] + (−1)|x||y|[y,x]

)
−

1

2

(
µ(x,y) + (−1)|x||y|µ(y,x)

)
= x∗y −ψ(x,y) ,

where

x∗y :=
1

2

(
[x,y] + (−1)|x||y|[y,x]

)
, ψ(x,y) :=

1

2

(
µ(x,y) + (−1)|x||y|µ(y,x)

)
.

Since ∧ and ∗ are supercommutative and satisfy the Lie-type identity then ψ
is a bi-cocycle of (U,∧) with value in Zr(U).

Lemma 4.2. Let (U, [ , ],Big) be a pseudo quadratic symmetric Lie type
superalgebra. Then the application ∧ : U ⊗U →U is a bi-cocycle of (U, [ , ]).

Proof. Let x ∈ U|x|,y ∈ U|y|,z ∈ U|z| and t ∈ U|t|. We want to show
that x∧ [y,z] + [x,y] ∧ z + (−1)|x||y|y ∧ [x,z] = 0 and [y,z] ∧x + y ∧ [z,x] +
(−1)|x||z|[y,x] ∧z = 0. For the first relation we have

Big
(
t,x∧ [y,z] + [x,y] ∧z + (−1)|x||y|y ∧ [x,z]

)
= Big (t,x∧ [y,z]) + Big (t, [x,y] ∧z) + (−1)|x||y|Big (t,y ∧ [x,z])

= Big ([t,x], [y,z]) + Big ([t, [x,y]],z) + (−1)|x||y|Big ([t,y], [x,z])

= Big ([[t,x],y],z) + Big ([t, [x,y]],z) + (−1)|x||t|Big ([x, [t,y]],z)

= Big
(

[[t,x],y] + [t, [x,y]] + (−1)|x||t|[x, [t,y]],z
)

= 0 ,

the fact that Big is nondegenerate gives us

x∧ [y,z] + [x,y] ∧z + (−1)|x||y|y ∧ [x,z] = 0

and by proceeding in the same way we show the second relation.



178 m. pouye, b. kpamegan

4.1. Inductive description of pseudo-quadratic Lie type super-
algebras. In this subsection, we continue our study of pseudo-quadratic
Lie type superalgebras. We extend the notion of double extension to pseudo-
quadratic Lie type superalgebras by using central extensions and representa-
tions. we give an inductive description of pseudo-quadratic Lie type superal-
gebras.

Lemma 4.1.1. Let (U, [ , ],Big) be a symmetric pseudo-quadratic Lie type
superalgebra and γ ∈ End(U)0̄. Then the bilinear application ψ : U ⊗U → K
defined by ψ(x,y) = Big (γ(x),y) is a bi-cocycle if and only if

γ ([x,y]) = − [x,γ(y)] − (−1)|x||y| [y,γ(x)] ∀x ∈U|x|, y ∈U|y|
and

(γ̄ −γ) ([U,U]) = 0 ,
where γ̄ is the adjoint of γ with respect to Big.

Proof. Let x ∈ U|x|,y ∈ U|y| and z ∈ U|z|. By using the fact that Big is
supersymmetric and left super-invariant, we have

ψ(x, [y,z]) + ψ([x,y],z) + (−1)|x||y|ψ(y, [x,z]) = 0

⇐⇒ Big (γ(x), [y,z]) + Big (γ([x,y]),z) + (−1)|x||y|Big (γ(y), [x,z]) = 0

⇐⇒ (−1)|x||y|Big ([y,γ(x)],z) + Big (γ([x,y]),z) + Big ([x,γ(y)],z) = 0

⇐⇒ Big
(
γ ([x,y]) + [x,γ(y)] + (−1)|x||y| [y,γ(x)] ,z

)
= 0 ,

and since Big is nondegenerate this equivalent to γ ([x,y]) = − [x,γ(y)] −
(−1)|x||y| [y,γ(x)] . On the other hand, we have

ψ (x, [y,z]) = (−1)|x|(|y|+|z|)ψ ([y,z],x)

⇐⇒ Big (γ(x), [y,z]) = (−1)|x|(|y|+|z|)Big (γ([y,z]),x)

⇐⇒ Big (x,γ̄([y,z])) = Big (x,γ([y,z])) ⇔ γ̄([y,z]) = γ([y,z]) .

This proves the lemma.

Lemma 4.1.2. Let (U, [ , ]) be a Lie type superalgebra, V a Z2-graded
vector space and ψ : U ⊗U → V a bi-cocycle. Then the space U := U ⊕ V
endowed with the product

[x + u,y + w] = [x,y] + ψ(x,y) ∀x + u,y + w ∈U

is a Lie type superalgebra.



lie type superalgebras 179

Proof. Straightforward calculation.

The Lie-type superalgebra U constructed in the Lemma 4.1.2 is called
central extension of U by means of ψ.

Lemma 4.1.3. Let (U, [ , ]) be a Lie type superalgebra, H a Z2-graded
vector space and (ϕ,λ) ∈ RepUH. Then the space U1 := U ⊕H endowed with
the product

[x+h,y+h′] = [x,y]+ϕx(h
′)+(−1)|x||y|λy(h′) ∀x+h ∈ (U1)|x| , y+h ∈ (U1)|y|

is a Lie type superalgebra.

Proof. Straightforward calculation.

Given a vector space H, we denote H∗ the dual space. The ground field K
admits a Z2-graduation as follows

K = K0̄ ⊕K1̄ with K0̄ = K and K1̄ = {0} .

The pseudo-quadratic Lie type superalgebra (U, [ , ],Big) is called even (resp.
odd) pseudo-quadratic Lie type superalgebra if | Big |= 0̄ (resp. | Big |= 1̄).
The following result gives the double extension of pseudo-quadratic Lie type
superalgebra.

Theorem 4.1.1. Let (U, [ , ],Big) be a symmetric pseudo-quadratic Lie
type superalgebra, H = Ke an one dimensional Z2-graded vector space, γ ∈
End(U)0̄ and (F,G) ∈ Rep

U
H such that

(1) γ ([x,y]) = − [x,γ(y)] − (−1)|x||y| [y,γ(x)] for all x ∈U|x|, y ∈U|y|;

(2) (γ̄ −γ) ([U,U]) = 0 ;

(3) Fx ·Fy = Gx ·Fy, F[x,y] = G[x,y] and Fx ·Gy = Gx ·Gy .

Then the space Ũ = H⊕U ⊕H∗ endowed with the product

[h + x + f,h′ + y + g] = [x,y]U + B
ig (γ(x),y) e∗ + Fx(h

′) + (−1)|x||y|Gy(h)

is a symmetric Lie type superalgebra.



180 m. pouye, b. kpamegan

Proof. Set ψ : U ⊗U → H∗ defined by ψ(x,y) = Big(γ(x),y)e∗. Then
according to Lemma 4.1.1 and the relations (1) and (2) we have ψ is a bi-
cocycle. Hence by Lemma 4.1.2, the space U := U ⊕ H∗ endowed with the
bracket

[x + f,y + g] = [x,y] + ψ(x,y) = [x,y] + Big(γ(x),y)e∗

is a Lie type superalgebra. Now let us define F,G : U → End(H) by Fx+f =
Fx and Gx+f = Gx. Since (F,G) ∈ RepUH then it is clear that (F,G) ∈ Rep

U
H.

Therefore, according to Lemma 4.1.3, the space Ũ := U ⊕ H endowed with
the bracket

[(x + f) + h, (y + g) + h′] = [x + f,y + g]U + Fx+f (h
′) + (−1)|x||y|Gy+g(h)

= [x,y] + ψ(x,y) + Fx(h
′) + (−1)|x||y|Gy(h)

= [x,y] + Big(γ(x),y)e∗ + Fx(h
′) + (−1)|x||y|Gy(h)

is a Lie type superalgebra. And by using (3), one can easily sees that Ũ is
symmetric.

The triplet (F,G,γ) in the above theorem is called context of double ex-
tension and the Lie-type superalgebra Ũ is the double extension of U by H by
means of (F,G,γ).

In what follows, we will show that odd pseudo-quadratic Lie type super-
algebra can be constructed from a finite number of odd pseudo-quadratic
Jacobi-Jordan superalgebras via the notion of double extension.

Theorem 4.1.2. Let (U, [ , ],Big) be a symmetric odd pseudo-quadratic
Lie type superalgebra such that Ker(U)0̄ 6= {0}. Then U is isomorphic to a
double extension of a Lie type superalgebra.

Proof. Since Ker(U)0̄ 6= {0} then there exists 0 6= e ∈ Ker(U)0̄ and the
fact that Big is a nondegenerate odd bilinear form implies that there exists
0 6= d ∈U1̄ such that Big(e,d) = 1. Set H = Ke,V = Kd and E = (H⊕V)⊥.
Following the same way as in the proof of [11, Theorem 6.3], one can easily see
that U = H⊕E⊕V and H⊥ = H⊕E is an ideal of U. Therefore [E,E] ⊆H⊕E.
Hence

[x,y]E = [x,y] −ψ(x,y)e ∀x,y ∈E , (4.2)
where ψ : E⊗E → K and [ , ]E : E⊗E →E. By using the Lie-type identity (1.1)
of (U, [ , ]) and relation (4.2) we show that (E, [ , ]E) is a symmetric Lie-type
superalgebra and ψ is a bi-cocycle.



lie type superalgebras 181

Since U = H⊕E ⊕V and H⊥ = H⊕E is an ideal of U then the bracket
of the Lie-type superalgebra U is given by

[x,d] = αϕxe + ξ(x) with α
ϕ
x ∈ K,ξ ∈ End(E) ,

[d,x] = αφxe + δ(x) with α
φ
x ∈ K,δ ∈ End(E) ,

[d,d] = αe + x0 + λd where α,λ ∈ K,x0 ∈E .

Define the applications ϕ,φ : E → End(H) by ϕx(e) = α
ϕ
xe and φx(e) = α

φ
xe

for all x ∈ E|x|. By proceeding in a same way as in the proof of
[11, Theorem 6.3], we show that (ϕ,φ,ξ) is a context of double extension
of E by H. Hence Ẽ = H⊕E ⊕H∗ is a double extension of (E, [ , ]E,B

ig
E ) by

H by means of (ϕ,φ,ξ) where BigE is the restriction of B
ig over E ⊗E. It is

easy to see that Ẽ is isomorphic to U = H⊕E ⊕V.

Corollary 4.1.1. Let (U, [ , ]Big) be a symmetric odd pseudo-quadratic
Lie type superalgebra which is not a Jacobi-Jordan superalgebra. Then U
can be obtained from a finite number of Jacobi-Jordan superalgebras via the
double extension.

Proof. The proof is analogous to the one of [11, Corollary 6.4].

4.2. The Jacobi-Jordan superalgebra (U,∧). In the previous sub-
section, we have proved through an inductive description that the study of
odd pseudo-quadratic Lie type superalgebras can be reduced to the study
of pseudo-quadratic Jacobi-Jordan superalgebras. Then it is necessary to
study pseudo-quadratic Jacobi-Jordan superalgebras. For that study, we
will reconsider the double extension defined above with some modifications
because unlike Lie type superalgebras, Jacobi-Jordan superalgebra (U,∧)
are supercommutative . Moreover, since all pseudo-quadratic Lie type su-
peralgebras (U, [ , ],Big) is isomorphic to a double extension of a pseudo-
quadratic Lie type superalgebra (E, [ , ]E,B

ig
E ) (see Theorem 4.1.2), we shall

investigate the relationships between the Jacobi-Jordan superalgebra (U,∧)
induced by (U, [ , ],Big) and the Jacobi-Jordan superalgebra (E,∧E) induced
by (E, [ , ]E,B

ig
E ).

Definition 4.2.1. Let (U,∧) be a Jacobi-Jordan superalgebra and V =
V0̄ ⊕V1̄ a Z2-graded vector space. A representation of U in V is given by an
even linear map ϕ : U → (End(V ))0̄ such that

ϕx∧y = −ϕx ·ϕy − (−1)|x||y|ϕy ·ϕx ∀x ∈U|x| , y ∈U|y| .



182 m. pouye, b. kpamegan

Lemma 4.2.1. Let (U,∧) be a Jacobi-Jordan superalgebra and ϕ ∈ RepUV .
Then the space U ⊕V endowed with the product

(x + u) Z (y + v) = x∧y + ϕx(v) + (−1)|x||y|ϕy(u)

for all x+u ∈ (U⊕V )|x| and y+v ∈ (U⊕V )|y|, is a Jacobi-Jordan superalgebra.

Proof. Straightforward computation.

The following result gives the double extension of a Jacobi-Jordan super-
algebra.

Theorem 4.2.1. Let (U,∧) be a Jacobi-Jordan superalgebra, H a Z2-
graded vector space, ϕ ∈ RepUH and ψ : U ⊗U → H

∗ a bi-cocycle of (U,∧).
Then the space Ũ := H⊕U ⊕H∗ with the product

(h + x + f) [ (h′ + y + g) = x∧y + ψ(x,y) + ϕx(h′) + (−1)|x||y|ϕy(h)

for all h + x + f ∈ Ũ|x| and h′ + y + g ∈ Ũ|y|, is a Jacobi-Jordan superalgebra.

Proof. The proof is analogous to the one of Theorem 4.1.1.

The couple (ϕ,ψ) is called JJ-context of double extension and the Jacobi-
Jordan superalgebra Ũ obtained in the Theorem 4.2.1 is called the JJ-double
extension of (U,∧) by H by means of (ϕ,ψ).

Given a symmetric odd pseudo-quadratic Lie type superalgebra (U, [ , ],
Big) such that Ker(U)0̄ 6= {0}, let (U,∧) be the Jacobi-Jordan superalgebra
induced by (U, [ , ]). According to Theorem 4.1.2, U is isomorphic to a dou-
ble extension H⊕E ⊕H∗. Consider the Jacobi-Jordan superalgebra (E,∧E)
induced by (E, [ , ]E). Then we have the following result:

Theorem 4.2.2. The Jacobi-Jordan superalgebra (U,∧) is the JJ-double
extension of (E,∧E).

Proof. Since Ker(U)0̄ 6= {0} and Big is odd and nondegenerate then, there
exists 0 6= e ∈ Ker(U)0̄ and d ∈ U1̄ such that Big(e,d) = 1. Put H = Ke,
V = Kd and E = (H⊕V)⊥.

By using the fact that Big is odd and nondegenerate we obtain U = H⊕
E ⊕V. Let us show that H⊥ is an ideal of (U,∧). Let a ∈ Ker(U)|a|,x ∈U|x|
and y ∈U|y| then we have Big(y,a∧x) = Big([y,a],x) = 0, therefore a∧x = 0
because Big is nondegenerate. Which implies that Ker(U) ⊆ Z∧(U), and since



lie type superalgebras 183

H⊆ Ker(U) then H is an ideal of (U,∧). Therefore one can easily show that
H⊥ = H⊕E is an ideal of (U,∧).

Hence the product in (U,∧) is given as follow, let x ∈ E|x| and y ∈ E|y|
then

x∧y = (−1)|x||y|y ∧x = x∧E y + ψE(x,y)e,
ψE : E ⊗E → K , ∧E : E ⊗E →E,

x∧d = (−1)|x|d∧x = π(x) + λ(x)e, π ∈ End(E) , λ ∈E∗,

d∧d = 0 because d∧d = (−1)|d|d∧d = −d∧d.

Let us show that (E,∧E) is a Jacobi-Jordan superalgebra induced by (E, [ , ]E).
Let x ∈E|x|,y ∈E|y| and z ∈E|z| by using relation (4.2) in the proof of Theorem
4.1.2, we have

B
ig
E ([x,y]E,z) = B

ig ([x,y]U −ψ(x,y)e,z) = Big ([x,y],z)

= Big (x,y ∧z) = Big (x,y ∧E z + ψE(y,z)e) = B
ig
E (x,y ∧E z)

therefore (E,∧E) is a Jacobi-Jordan superalgebra induced by (E, [ , ]E). Let us
consider now the application ϕ : (E,∧E) → End(H)0̄ defined by ϕx(e) = λ(x)e.
It is clear that if x ∈E1̄ then λ(x) = 0 because ϕx changes the degree of e. If
x ∈E0̄ then since Big is odd and d ∈U1̄, we have

0 = Big (x∧d,d) = Big (π(x) + λ(x)e,d) = λ(x) ,

hence λ(x) = 0. Which implies that ϕ is a trivial representation of (E,∧E) in
H. Moreover, from the relation

x∧y = (−1)|x||y|y ∧x = x∧E y + ψE(x,y)e

we can deduce that ψE is a bi-cocycle of (E,∧E). Therefore, (ϕ,ψE) is a
JJ-context of double extension of (E,∧E). and according to Theorem 4.2.1
we obtain that the JJ-double extension (H⊕E ⊕H∗,∧) of (E,∧E) by H by
means of (ϕ,ψE). One can easily sees that (U = H⊕E ⊕V,∧) is isomor-
phic to (H⊕E ⊕H∗,∧). This implies that (U,∧) is the JJ-double extension
of (E,∧E).

Acknowledgements

The authors are grateful to the referee for his/her valuable comments
and suggestions on the first version of the article. The first author would
like to thank Prof. Said Benayadi.



184 m. pouye, b. kpamegan

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https://arxiv.org/abs/0712.0228v1

	Preliminaries
	Homology and cohomology of Lie-type superalgebras 
	Extensions and crossed modules of Lie type superalgebras
	Extensions of Lie type superalgebras.
	Crossed modules of Lie type superalgebras.

	Pseudo quadratic Lie type superalgebras and theirs associated Jacobi-Jordan superalgebras
	Inductive description of pseudo-quadratic Lie type superalgebras.
	The Jacobi-Jordan superalgebra (U, ).