# Cosymplectic Jacobi-Jordan Algebras

S. El bourkadi and M. W. Mansouri  Department of Mathematics, Faculty of
Sciences, Ibn Tofail University
Analysis, Geometry and Applications Laboratory (LAGA)
Kenitra, Morocco
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

We introduce the notion of cosymplectic structure on Jacobi-Jordan algebras,
and we state that they are related to symplectic Jacobi-Jordan algebras. We
show in particular, that they support a right-skew-symmetric product. We also
study the double extension constructions of cosymplectic Jacobi-Jordan
algebras and give a complete classification in dimension five.

###### keywords:

Cosymplectic structures, Jacobi-Jordan algebras, Double extensions.

###### MSC:

17C10 , 17C50 , 17C55 , 53D15.

##  1 Introduction

A Jacobi-Jordan algebra, denoted by JJ-algebra for short, is a pair
$({\mathfrak{J}},\cdot)$ consisting of a vector space ${\mathfrak{J}}$ and a
product $``\cdot"$, satisfying the following identities

  1. (i)

$x\cdot y=y\cdot x$ (commutativity).

  2. (ii)

$\oint x\cdot(y\cdot z):=x\cdot(y\cdot z)+y\cdot(z\cdot x)+z\cdot(x\cdot y)=0$
(Jacobi identity),

for any $x,y,z\in{\mathfrak{J}}$, where $\oint$ denotes summation over the
cyclic permutation. In other words, JJ-algebras are commutative algebras which
their products satisfy the Jacobi identity [4]. JJ-algebras are introduced by
Burde and Fialowski in [4]. Note that this type of algebra has been known and
appears under different names in the literature (e.g., Jordan algebras of nil
index $3$ in [10], Lie-Jordan algebras in [8], and mock-Lie algebras in [12]).

An important class of non-associative algebras is given by _Jordan algebras_.
Which are commutative non-associative algebras satisfying the identity:
$x^{2}\cdot(y\cdot x)=(x^{2}\cdot y)\cdot x$, it follows that JJ-algebra is a
subclass of Jordan algebra, i.e., Any JJ-algebra is a Jordan algebra (see [2]
and [4]).

JJ-algebra and related topics have attracted significant attention in recent
years. This interest stems from both the pure geometrical and algebraic
viewpoint. In particular, several authors introduce and study additional
structures of geometric origin ( symplectic structure) on JJ-algebras, see [3]
and [1]. Libermann in [9] defined cosymplectic manifolds as
$(2n+1)$-dimensional manifolds equipped with a closed $1$-form $\alpha$ and
closed $2$-form $\omega$ such that $\alpha\wedge\omega^{n}$ is a volume form.
In light of the previous definition and the frameworks of cosymplectic
geometry (see [5], [6] and [7]), and the connection between symplectic and
cosymplectic structures. All these elements motivate our investigation of
cosymplectic JJ-algebras.

Our principal objective is to introduce the notion of cosymplectic structure
on JJ-algebras, explore their basic properties, and characterize the
relationship between symplectic JJ-algebras and cosymplectic JJ-algebras.

This paper goes as follows: Section $2$ lays the background and preliminary
concepts and some constructions of JJ-algebras. This includes a review and a
discussion of well-known results. Additionally, we present established results
concerning symplectic JJ-algebras, which will serve as a foundation for the
subsequent analysis. In section $3$, we define cosymplectic structure on JJ-
algebra. This structure is a pair $(\alpha,\omega)$, where $\alpha$ is a
$1$-form and $\omega$ is a $2$-form, satisfying certain conditions.
Furthermore, we study the relation between symplectic and cosymplectic JJ-
algebras. In particular, we show that there exists a one-to-one correspondence
between cosymplectic and symplectic JJ-algebras equipped with a two-step
nilpotent symplectic anti-derivation. Section $4$, we adapt the techniques of
double extensions as in symplectic JJ-algebras presented in [3] to describe
cosymplectic JJ-algebra, and we prove that any cosymplectic JJ-algebra is
obtained as a double extension of a cosymplectic JJ -algebra. Finally, in
Section $5$, using the results of section $3$, we give a complete
classification, up to isomorphism, of five dimensional cosymplectic JJ-
algebra.

Notation. Let $\\{e_{i}\\}_{1\leq i\leq n}$ stands for a basis of
${\mathfrak{J}}$, and let $\\{e^{i}\\}_{1\leq i\leq n}$ be its dual basis on
${\mathfrak{J}}^{\ast}$ and $e^{ij}$ the 2-form $e^{i}\wedge
e^{j}\in\wedge^{2}{\mathfrak{J}}^{*}$. Set by $\langle
x\rangle:=\mathrm{span}_{\mathbb{K}}\\{x\\}$ the one-dimensional JJ-algebra
spanned by $x$.

##  2 Preliminaries on Jacobi-Jordan algebras

In this section, we recall some basic definitions and properties of JJ-algebra
and symplectic JJ-algebra.

Here and subsequently, all algebras are considered of finite dimension over a
field $\mathbb{K}$ of characteristic different from $2$ or $3$. Let
$({\mathfrak{J}},\odot)$ be an algebra.

######  Definition 2.1.

The algebra $({\mathfrak{J}},\odot)$ is said to be an anti-associative algebra
if the product $\odot$ satisfies the identity

$(x\odot y)\odot z=-x\odot(y\odot z),$

for any $x,y,z\in{\mathfrak{J}}$.

We will use the symbol $(x,y,z)^{\odot}$ to denote the anti-associator of
three elements $x,y,z\in{\mathfrak{J}}$, it is defined by
$(x,y,z)^{\odot}:=x\odot(y\odot z)+(x\odot y)\odot z$.

The algebra $({\mathfrak{J}},\odot)$ is said to be an anti-associative algebra
if $(x,y,z)^{\odot}=0$, for any $x,y,z\in{\mathfrak{J}}$.

######  Definition 2.2.

Let $({\mathfrak{J}},\odot)$ be an algebra, Consider a new product on
${\mathfrak{J}}$ denoted by $``\cdot"$ and defined by

$x\cdot y:=x\odot y+y\odot x,$

for any $x,y\in{\mathfrak{J}}$. Then, $({\mathfrak{J}},\odot)$ is called an
admissible JJ-algebra if $({\mathfrak{J}},\cdot)$ is a JJ-algebra. Moreover,
the product $\odot$ is called an admissible JJ-product.

As an example, a direct calculation shows that any anti-associative algebra is
an admissible JJ-algebra.

######  Lemma 2.1.

An algebra $({\mathfrak{J}},\odot)$ is an admissible JJ-algebra if and only if

$\oint(x,y,z)^{\odot}=-\oint(x,z,y)^{\odot},$

for any $x,y,z\in{\mathfrak{J}}$.

###### Proof.

It is a straightforward calculation. ∎

######  Definition 2.3.

Let $({\mathfrak{J}},\odot)$ be an algebra. Then

$({\mathfrak{J}},\odot)$ is called right-skew-symmetric algebra if

$(x,y,z)^{\odot}+(x,z,y)^{\odot}=0,$

$({\mathfrak{J}},\odot)$ is called left-skew-symmetric algebra if

$(x,y,z)^{\odot}+(y,x,z)^{\odot}=0,$

for all $x,y,z\in{\mathfrak{J}}$.

######  Corollary 2.1.

Any right(left)-skew-symmetric algebra is an admissible JJ-algebra.

###### Proof.

It follows from the previous lemma. ∎

######  Definition 2.4.

Let $({\mathfrak{J}},\cdot)$ be an algebra, and $\mathrm{D}$ be a linear map
$\mathrm{D}:{\mathfrak{J}}\longrightarrow{\mathfrak{J}}$.

  1. $(1)$

$\mathrm{D}$ is called a derivation of $({\mathfrak{J}},\cdot)$ if

$\mathrm{D}(x\cdot y)=\mathrm{D}(x)\cdot y+x\cdot\mathrm{D}(y),$

  2. $(2)$

$\mathrm{D}$ is called an anti-derivation of $({\mathfrak{J}},\cdot)$ if

$\mathrm{D}(x\cdot y)=-\mathrm{D}(x)\cdot y-x\cdot\mathrm{D}(y),$

for any $x,y\in{\mathfrak{J}}$.

Let $\mathrm{Der}({\mathfrak{J}})$ denotes the set of all derivations of the
algebra $({\mathfrak{J}},\cdot)$ and $\mathrm{Ader}({\mathfrak{J}})$ stands
for the set of all anti-derivations of the algebra $({\mathfrak{J}},\cdot)$.

######  Remark 1.

Let $({\mathfrak{J}},\cdot)$ be a JJ-algebra. For all $x\in{\mathfrak{J}}$, we
denote by $\mathrm{L}_{x}$ and $\mathrm{R}_{x}$ the left and right-
multiplication on $({\mathfrak{J}},\cdot)$, respectively. Since
$({\mathfrak{J}},\cdot)$ is commutative, it follows trivially that
$\mathrm{L}_{x}=\mathrm{R}_{x}$. It follows from the Jacobi identity that
$\mathrm{L}_{x}$ and $\mathrm{R}_{x}$ are anti-derivations of
$({\mathfrak{J}},\cdot)$.

JJ-algebras extensions. Motivated by the classic notion of double extensions
of Lie algebras, we adapt the same approach here with JJ-algebras as follows.

Let $({\mathfrak{J}},\cdot)$ be a JJ-algebra and $\theta$ a symmetric bilinear
form on ${\mathfrak{J}}$.

On the vector space ${\mathfrak{J}}_{(\theta,e)}={\mathfrak{J}}\oplus\langle
e\rangle$, we define the product $``\cdot_{\theta}"$ by

$\displaystyle x\cdot_{\theta}y$ $\displaystyle=x\cdot y+\theta(x,y)e,$

for every $x,y\in{\mathfrak{J}}$. It is clear that
$({\mathfrak{J}}_{(\theta,e)},\cdot_{\theta})$ is a JJ-algebra if and only if
$\theta$ satisfies

$\oint\theta(x\cdot y,z)=0,$ (1)

for any $x,y,z\in{\mathfrak{J}}$. We call
$({\mathfrak{J}}_{(\theta,e)},\cdot_{\theta})$ the central extension of
${\mathfrak{J}}$ by $\theta$.

Let $\tilde{{\mathfrak{J}}}=\langle d\rangle\oplus{\mathfrak{J}}_{(\theta,e)}$
be the direct sum of ${\mathfrak{J}}_{(\theta,e)}$ with the one-dimensional
vector space $\langle d\rangle$ and $\mathrm{D}$ be a linear map
$\mathrm{D}:{\mathfrak{J}}_{(\theta,e)}\longrightarrow{\mathfrak{J}}_{(\theta,e)}$
. Consider the following product on $\tilde{{\mathfrak{J}}}$ denoted by
$``\bullet"$ and defined by

$\displaystyle x\bullet y$ $\displaystyle=x\cdot y+\theta(x,y)e,$
$\displaystyle d\bullet x$ $\displaystyle=\mathrm{D}(x),$ (2) $\displaystyle
d\bullet d$ $\displaystyle=a\in{\mathfrak{J}},$

Assume that the product $``\bullet"$ satisfies the Jacobi identity. Then, a
direct calculation ensures the following proposition.

######  Proposition 2.1.

Let $({\mathfrak{J}},\cdot)$ be a JJ-algebra, then the vector space
$\tilde{{\mathfrak{J}}}=\langle d\rangle\oplus{\mathfrak{J}}\oplus\langle
e\rangle\ $ equipped with the product $``\bullet"$ defined by (2). Then
$(\tilde{{\mathfrak{J}}},\bullet)$ is a JJ-algebra if and only if

$\mathrm{D}\in\mathrm{Ader}({\mathfrak{J}}),\quad
a\in\ker(\mathrm{D})\quad\mbox{and}\quad\mathrm{D}^{2}=-\frac{1}{2}\mathrm{L}_{a}.$
(3)

######  Remark 2.

  1. 1.

If a pair $(\mathrm{D},a)$ satisfies (3), then it is called an admissible pair
of $({\mathfrak{J}},\cdot)$ l (cf. [3], Definition $2.5$) .

  2. 2.

Consider $\mathrm{D}(x)=\varphi(x)+\lambda(x)e$, for any $x\in{\mathfrak{J}}$
where
$(\varphi,\lambda)\in\mathrm{End}({\mathfrak{J}})\times{\mathfrak{J}}^{*}$.
Then the condition (3) is equivalent to $(\varphi,a)$ is an admissible pair of
$({\mathfrak{J}},\cdot)$ and

$\left\\{\begin{array}[]{rl}\lambda(a)&=0,\\\ \lambda(x\cdot
y)&=-\theta(\varphi(x),y)-\theta(\varphi(y),x),\\\
\theta(a,x)&=-2\lambda\circ\varphi(x),\end{array}\right.$

for any $x,y\in{\mathfrak{J}}$.

We say that $(\tilde{{\mathfrak{J}}},\bullet)$ is a _double extension_ of JJ-
algebra $({\mathfrak{J}},\cdot)$ by $(\theta,\mathrm{D},a)$ or
$(\theta,\varphi,\lambda,a)$.

Symplectic JJ-algebras. Let $({\mathfrak{h}},\cdot)$ be a JJ-algebra. We
recall (cf. [3], Definition $3.1$) that a symplectic structure on
${\mathfrak{h}}$ is given by a skew-symmetric non-degenerate bilinear form
$\omega$ satisfying the following condition

$\oint\omega(x,y,z):=\omega(x\cdot y,z)+\omega(z\cdot x,y)+\omega(y\cdot
z,x)=0,$

for any $x,y,z\in{\mathfrak{h}}$. If a JJ-algebra ${\mathfrak{h}}$ is equipped
with such symplectic structure $\omega$, then $({\mathfrak{h}},\omega)$ is
called a symplectic JJ-algebra.

######  Remark 3.

It makes sense to reconsider the condition of non-degeneracy of $\omega$. In
other words, saying that $\omega$ is non-degenerate is equivalent to stating
that $\omega^{n}\neq 0$, where
$\omega^{n}:=\omega\wedge\underbrace{\cdots}_{\text{$n$-times}}\wedge\omega$.

Furthermore (see also [3], Proposition $3.1$), there exists a product on the
vector space ${\mathfrak{h}}$ denoted $``\odot"$ and given by

$\omega(x\odot y,z)=\omega(x,y\cdot z),$ (4)

and satisfies

  1. $(1)$

$x\cdot y=x\odot y+y\odot x$ (admissible)

  2. $(2)$

$(x,y,z)^{\odot}+(x,z,y)^{\odot}=0$, (right-skew-symmetric)

for any $x,y,z\in{\mathfrak{h}}$.

The product $``\odot"$ is called the right-skew-symmetric product associated
with the symplectic JJ-algebra $({\mathfrak{h}},\omega)$.

Recall that two symplectic JJ-algebras $({\mathfrak{h}}_{1},\omega_{1})$ and
$({\mathfrak{h}}_{2},\omega_{2})$ are isomorphic if there exists an
isomorphism of JJ-algebras
$\phi:{\mathfrak{h}}_{1}\longrightarrow{\mathfrak{h}}_{2}$ such that

$\omega_{2}(\phi(x),\phi(y))=\omega_{1}(x,y),$

for any $x,y\in{\mathfrak{h}}_{1}$.

######  Example 1.

Let $(H_{4},\cdot)$ be the four-dimensional JJ-algebra defined above by

$e_{1}\cdot e_{1}=e_{2},\;e_{1}\cdot e_{3}=e_{3}\cdot e_{1}=e_{4},$

where $\\{e_{1},e_{2},e_{3},e_{4}\\}$ is a basis of $H_{4}$. The skew-
symmetric bilinear form $\omega=e^{12}+2e^{23}$ is a symplectic form on
$H_{4}$. Moreover, any four-dimensional symplectic JJ-algebra is isomorphic to
the unique symplectic JJ-algebra $(H_{4},\omega)$ (see, [3]).

##  3 Cosymplectic Jacobi-Jordan algebras

In this section we give a definition of cosymplectic structure on JJ-algebras.
In particular, we state that the dimension of cosymplectic JJ-algebra is
always necessarily odd. We establish that every cosymplectic JJ-algebra
induces a corresponding symplectic JJ-algebra. Furthermore, under certain
conditions and with additional data, we show that there is a one-to-one
correspondence between cosymplectic and symplectic JJ-algebras.

######  Definition 3.1.

An almost cosymplectic structure on a JJ-algebra $({\mathfrak{J}},\cdot)$ is a
pair $(\alpha,\omega)$, where $\alpha$ is a $1$-form and $\omega$ is a skew-
symmetric bilinear form such that $\alpha\wedge\omega^{n}\neq 0$, the triplet
$({\mathfrak{J}},\alpha,\omega)$ is called almost cosymplectic JJ-algebra.

The almost cosymplectic JJ-algebra $({\mathfrak{J}},\alpha,\omega)$ is said to
be a cosymplectic JJ-algebra if $\alpha$ and $\omega$ satisfy the following
conditions

$\alpha(x\cdot y)=0,$ (5) $\oint\omega(x\cdot y,z)=0,$ (6)

for any $x,y,z\in{\mathfrak{J}}$.

From the fact that $\alpha\wedge\omega^{n}\neq 0$, any cosymplectic structure
$(\alpha,\omega)$ on $({\mathfrak{J}},\cdot)$ induces an isomorphism $\Psi$
given by

$\begin{array}[]{rcl}\Psi:{\mathfrak{J}}&\longrightarrow&{\mathfrak{J}}^{*}\\\
x&\longmapsto&\omega(x,.)+\alpha(x)\alpha(.),\end{array}$ (7)

for any $x\in{\mathfrak{J}}$. Consequently, the unique vector
$\xi=\Psi^{-1}(\alpha)$ on ${\mathfrak{J}}$ is called Reeb vector of the
cosymplectic JJ-algebra $({\mathfrak{J}},\alpha,\omega)$, and it is completely
described by the following conditions

$\displaystyle\alpha(\xi)$ $\displaystyle=1,$ (8) $\displaystyle\omega(\xi,.)$
$\displaystyle=0.$ (9)

######  Remark 4.

  1. 1.

Substituting $z=\xi$ into $\eqref{omclosed}$, we find that

$\omega(x,y\cdot\xi)+\omega(y,x\cdot\xi)=0,$ (10)

for any $x,y\in{\mathfrak{J}}$.

  2. 2.

In particular, if $y=\xi$, the equation above gives that
$\omega(x,\xi\cdot\xi)=0$, for all $x\in{\mathfrak{J}}$. Therefore,
$\xi^{2}=\xi\cdot\xi=0$.

  3. 3.

As in the case with symplectic JJ-algebras being even-dimensional, the
condition $\alpha\wedge\omega^{n}\neq 0$, ensures that any cosymplectic JJ-
algebra must have odd dimension.

######  Lemma 3.1.

Let $({\mathfrak{J}},\alpha,\omega)$ be a cosymplectic JJ-algebra. Then,
${\mathfrak{h}}=\mathrm{ker(\alpha)}$ is an ideal of ${\mathfrak{J}}$ and
$({\mathfrak{h}},\omega_{{\mathfrak{h}}})$ is a symplectic JJ-algebra.

###### Proof.

By (5), we have ${\mathfrak{J}}\cdot{\mathfrak{J}}\subset{\mathfrak{h}}$. In
particular, ${\mathfrak{h}}$ is an ideal of ${\mathfrak{J}}$. Set by
$\omega_{{\mathfrak{h}}}:=\omega_{|{\mathfrak{h}}\times{\mathfrak{h}}}$ the
restriction of $\omega$ on ${\mathfrak{h}}$. Let us prove that
$({\mathfrak{h}},\omega_{{\mathfrak{h}}})$ is a symplectic JJ-algebra. It is
clear that $\omega_{\mathfrak{h}}$ is a skew-symmetric bilinear form on
${\mathfrak{h}}$ and satisfies (6). Finally, let $\\{\xi,e_{1},...,e_{2n}\\}$
be a basis of ${\mathfrak{J}}$, and $\\{e_{1},...,e_{2n}\\}$ be a basis of
${\mathfrak{h}}$. We have

$0\not=\alpha\wedge\omega^{n}(\xi,e_{1},...,e_{2n})=\omega_{{\mathfrak{h}}}^{n}(e_{1},...,e_{2n}),$

i.e., $\omega_{{\mathfrak{h}}}$ is non-degenerate. Therefore,
$({\mathfrak{h}},\omega_{\mathfrak{h}})$ is a symplectic JJ-algebra. ∎

Let $({\mathfrak{J}},\alpha,\omega)$ be a cosymplectic JJ-algebra with Reeb
vector $\xi$, and the product $\odot$ be the right-skew-symmetric product
associated with the symplectic JJ-algebra
$({\mathfrak{h}},\omega_{{\mathfrak{h}}})$ as given in (4).

Set $\mathrm{D}_{\xi}$ to be a linear map
$\mathrm{D}_{\xi}:{\mathfrak{J}}\longrightarrow{\mathfrak{J}}$ such that
$\mathrm{D}_{\xi}(x):=\xi\cdot x$, for any $x\in{\mathfrak{J}}$. It follows
from (10) that $\mathrm{D}_{\xi}$ is an anti-derivation of
$({\mathfrak{J}},\cdot)$.

######  Lemma 3.2.

The linear map $\mathrm{D}_{\xi}$, as defined above, is an anti-derivation of
the algebra $({\mathfrak{h}},\odot)$, i.e.,

$\mathrm{D}_{\xi}(x\odot y)=-\mathrm{D}_{\xi}(x)\odot
y-x\odot\mathrm{D}_{\xi}(y),$

for any $x,y\in h$.

###### Proof.

Let $x,y,z\in{\mathfrak{h}}$, we have

$\displaystyle\omega_{{\mathfrak{h}}}(\mathrm{D}_{\xi}(x\odot
y)+\mathrm{D}_{\xi}(x)\odot y,z)$
$\displaystyle=\omega_{{\mathfrak{h}}}(\xi\cdot(x\odot
y),z)+\omega_{{\mathfrak{h}}}((x\cdot\xi)\odot y,z)$
$\displaystyle=\omega_{{\mathfrak{h}}}(x\odot y,\xi\cdot
z)+\omega_{{\mathfrak{h}}}(x\cdot\xi,y\cdot z)$
$\displaystyle=\omega_{{\mathfrak{h}}}(x,y\cdot(\xi\cdot
z))+\omega_{{\mathfrak{h}}}(x,\xi\cdot(y\cdot z))$
$\displaystyle=\omega_{{\mathfrak{h}}}(x,-z\cdot(y\cdot\xi))$
$\displaystyle=\omega_{{\mathfrak{h}}}(-x\odot(y\cdot\xi),z)$
$\displaystyle=\omega_{{\mathfrak{h}}}(-x\odot\mathrm{D}_{\xi}(y),z).$

The non-degeneracy of $\omega_{{\mathfrak{h}}}$ ensures the result. ∎

The following proposition shows that any cosymplectic JJ-algebra give rise to
a right-skew-symmetric algebra.

######  Proposition 3.1.

Let $({\mathfrak{J}},\alpha,\omega)$ be a cosymplectic JJ-algebra. There
exists a product $``*"$ on the underlying vector space ${\mathfrak{J}}$
defined by

$\Psi(x*y)(z)=\Psi(x)(y\cdot z),$ (11)

for any $x,y,z\in{\mathfrak{J}}$, and it satisfies

  1. 1.

For all $x,\,y\in{\mathfrak{h}}$

$x*y=x\odot y+\omega(x,y\cdot\xi)\xi.$ (12)

  2. 2.

For all $x\in{\mathfrak{J}}$

$x*\xi=x\cdot\xi\quad\mbox{and}\quad\xi*x=0.$ (13)

  3. 3.

The algebra $({\mathfrak{J}},*)$ is a right-skew-symmetric algebra.

###### Proof.

The fact that $\Psi$ is an isomorphism induces a well-defined product on
${\mathfrak{J}}$ denoted by $``*"$ and given by (11) , in other terms for all
$x,y\in{\mathfrak{J}}$, and $z\in{\mathfrak{J}}$, the equation (11) is
equivalent to

$\omega(x*y,z)+\alpha(x*y)\alpha(z)=\omega(x,y\cdot z)+\alpha(x)\alpha(y\cdot
z).$ (14)

  1. 1.

Consequently, for all $x,y\in{\mathfrak{h}}$, and $z\in{\mathfrak{J}}$, the
equation (14) becomes

$\displaystyle\omega(x*y,z)+\alpha(x*y)\alpha(z)$
$\displaystyle=\omega(x,y\cdot z)$

Substituting $z=\xi$ in the equation above, it follows that

$\displaystyle\alpha(x*y)$ $\displaystyle=\omega(x,y\cdot\xi).$

In particular, for all $z\in{\mathfrak{h}}$, we have that
$\omega_{{\mathfrak{h}}}(x*y,z)=\omega_{{\mathfrak{h}}}(x,y\cdot z)$, which
means that $``*"$ reduces to $``\odot"$ on the vector space ${\mathfrak{h}}$,
and This shows 1.

  2. 2.

If $x=\xi$ and $y\in{\mathfrak{J}}$, the relation $\eqref{prod}$ becomes

$\displaystyle\omega(\xi*y,z)+\alpha(\xi*y)\alpha(z)=0.$

On the one hand, replacing $z$ by $\xi$ we obtain $\alpha(\xi*y)=0,$ this
gives $\xi*y\in{\mathfrak{h}}$. On the other hand for all
$z\in{\mathfrak{h}}$, we have

$\displaystyle\omega_{{\mathfrak{h}}}(\xi*y,z)$ $\displaystyle=0,$

the non-degeneracy of $\omega_{{\mathfrak{h}}}$ shows that $\xi*y=0$ for all
$y\in{\mathfrak{J}}$.

For all $x\in{\mathfrak{J}}$, if $y=\xi$, we also have

$\displaystyle\omega(x*\xi,z)+\alpha(x*\xi)\alpha(z)$
$\displaystyle=\omega(x,\xi\cdot z).$

Let $z=\xi$, we have $\alpha(x*\xi)=0,$ this implies that
$x*\xi\in{\mathfrak{h}}$. Also for all $z\in{\mathfrak{h}}$

$\displaystyle\omega_{{\mathfrak{h}}}(x*\xi,z)$
$\displaystyle=\omega_{{\mathfrak{h}}}(x\cdot\xi,z).$

As $\omega_{{\mathfrak{h}}}$ is non-degenerate, it follows that
$x*\xi=x\cdot\xi,$ for all $x\in{\mathfrak{J}}.$

  3. 3.

On the one hand, for all $x,y$ and $z\in{\mathfrak{J}}$, by (12) and (13) we
have

$\displaystyle(x,y,z)^{*}$ $\displaystyle=\big{(}x\odot
y+\omega(x,y\cdot\xi)\xi\big{)}*z+x*\big{(}y\odot
z+\omega(y,z\cdot\xi)\xi\big{)}$ $\displaystyle=(x\odot y)\odot
z+\omega(x\odot y,z\cdot\xi)\xi+x\odot(y\odot z)+\omega(x,(y\odot
z)\cdot\xi)\xi+\omega(y,z\cdot\xi)x\cdot\xi$ $\displaystyle=(x\odot y)\odot
z+x\odot(y\odot z)+\omega(x\odot y,z\cdot\xi)\xi+\omega(x,(y\odot
z)\cdot\xi)\xi+\omega(y,z\cdot\xi)x\cdot\xi$
$\displaystyle=(x,y,z)^{\odot}+\omega(x\odot y,z\cdot\xi)\xi+\omega(x,(y\odot
z)\cdot\xi)\xi+\omega(y,z\cdot\xi)x\cdot\xi.$

Thus,

$\displaystyle(x,y,z)^{*}+(x,z,y)^{*}$
$\displaystyle=(x,y,z)^{\odot}+(x,z,y)^{\odot}+\omega(x\odot
y,z\cdot\xi)\xi+\omega(x,(y\odot z)\cdot\xi)\xi$
$\displaystyle\;+\omega(x\odot z,y\cdot\xi)\xi+\omega(x,(z\odot
y)\cdot\xi)\xi+\big{(}\omega(y,z\cdot\xi)+\omega(z,y\cdot\xi)\big{)}x\cdot\xi.$

Since $(h,\odot)$ is right-skew-symmetric (JJ-admissible) algebra, it follows
that

$\displaystyle(x,y,z)^{*}+(x,z,y)^{*}$
$\displaystyle=\big{(}\omega(x,(z\cdot\xi)\cdot y)+\omega(x,(y\cdot
z)\cdot\xi)+\omega(x,(\xi\cdot y)\cdot z\big{)}\xi-\omega(\xi,y\cdot
z)x\cdot\xi$ $\displaystyle=\omega\big{(}x,(z\cdot\xi)\cdot y+(y\cdot
z)\cdot\xi+(\xi\cdot y)\cdot z\big{)}\xi$
$\displaystyle=\omega(x,\oint(z\cdot\xi)\cdot y)$ $\displaystyle=0.$

On the other hand, for all $x,y\in{\mathfrak{h}}$.

$\displaystyle(x,\xi,y)^{*}+(x,y,\xi)^{*}$
$\displaystyle=(x\cdot\xi)*y+\big{(}x\odot
y+\omega(x,(y\cdot\xi)\big{)}\cdot\xi+x\odot(y\cdot\xi)+\omega\big{(}x,(x\cdot\xi)\big{)}\xi$
$\displaystyle=\mathrm{D}_{\xi}(x)\odot
y+\omega\big{(}x\cdot\xi,y\cdot\xi\big{)}\xi+\mathrm{D}_{\xi}(x\odot
y)+\omega\big{(}x,(y\cdot\xi)\cdot\xi\big{)}\xi+x\odot\mathrm{D}_{\xi}(y)$
$\displaystyle=\omega\big{(}\xi,(y\cdot\xi)\cdot x\big{)}\xi$
$\displaystyle=0.$

Finally, we conclude that $({\mathfrak{J}},*)$ is a right-skew-symmetric JJ-
algebra. ∎

######  Remark 5.

Since $({\mathfrak{J}},*)$ is a right-skew-symmetric JJ-algebra, it follows
from Corollary 2.1 that $({\mathfrak{J}},*)$ is an admissible JJ-algebra.

######  Theorem 3.1.

Any $(2n+1)$-dimensional cosymplectic JJ-algebra
$({\mathfrak{J}},\alpha,\omega)$ is in one-to-one correspondence with a
symplectic JJ-algebra $({\mathfrak{h}},\omega_{\mathfrak{h}},\mathrm{D})$ of
dimension $2n$ equipped with an anti-derivation $\mathrm{D}$ satisfying

$D^{2}(x)=0\quad\mbox{and}\quad\omega_{\mathfrak{h}}(\mathrm{D}(x),y)=\omega_{\mathfrak{h}}(x,\mathrm{D}(y)),$
(15)

for any $x,y\in{\mathfrak{h}}$.

###### Proof.

Let $({\mathfrak{J}},\alpha,\omega)$ be a $(2n+1)$-dimensional cosymplectic
JJ-algebra and $``\bullet"$ its product. Set
${\mathfrak{h}}=\mathrm{ker}(\alpha)$ and
$\omega_{{\mathfrak{h}}}:=\omega_{|{\mathfrak{h}}\times{\mathfrak{h}}}$, it
follows from Lemma 3.1 that $({\mathfrak{h}},\omega_{{\mathfrak{h}}})$ is a
symplectic JJ-algebra.

For all $x\in{\mathfrak{h}}$, we have $\xi\bullet
x=\mathrm{D}_{\xi}(x)\in{\mathfrak{h}}$, one can take
$\mathrm{D}=\mathrm{D}_{\xi}\in\mathrm{Ader}({\mathfrak{h}})$. By (10) we have
$\omega_{\mathfrak{h}}(\mathrm{D}_{\xi}(x),y)=\omega_{\mathfrak{h}}(x,\mathrm{D}_{\xi}(y))$,
for any $x,y\in{\mathfrak{h}}$.

Conversely, let $({\mathfrak{h}},\omega_{\mathfrak{h}},\mathrm{D})$ be a
symplectic JJ-algebra of dimension $2n$ endowed with an anti-derivation
$\mathrm{D}$ satisfying the conditions above and $``\cdot"$ its product. It is
clear to see that
$({\mathfrak{J}}=\langle\xi\rangle\oplus{\mathfrak{h}},\bullet)$ is a well
defined JJ-algebra, such that

$\displaystyle x\bullet y$ $\displaystyle=x\cdot y,$ $\displaystyle\xi\bullet
x$ $\displaystyle=\mathrm{D}(x),$ $\displaystyle\xi\bullet\xi$
$\displaystyle=0,$

for any $x,y\in{\mathfrak{h}}$. We extend $\omega_{{\mathfrak{h}}}$ on
${\mathfrak{J}}$ by defining $\omega(\xi,.)=0$ and
$\omega(x,y)=\omega_{{\mathfrak{h}}}(x,y)$ for any $x,y\in{\mathfrak{h}}$.
Since $\omega_{{\mathfrak{h}}}$ is symplectic together with (15), this implies
that $\omega$ satisfies (6).

Let $\alpha$ be a $1$-form on ${\mathfrak{J}}$ defined by
$\alpha_{|{\mathfrak{h}}}=0$ and $\alpha(\xi)=1$. The fact that
$\mathrm{D}(x)=x\bullet\xi\in{\mathfrak{h}}$, implies that $\alpha(\xi\bullet
x)=0$. Hence $\alpha(x\bullet y)=0$, for any $x,y\in{\mathfrak{J}}$.

Finally, let $\\{\xi,e_{1},...,e_{2n}\\}$ be a basis of ${\mathfrak{J}}$, and
$\\{e_{1},...,e_{2n}\\}$ be a basis of ${\mathfrak{h}}$. We have

$\alpha\wedge\omega^{n}(\xi,e_{1},...,e_{2n})=\omega_{{\mathfrak{h}}}^{n}(e_{1},...,e_{2n})\not=0,$

Thus, the skew-symmetric bilinear form $\omega_{{\mathfrak{h}}}$ is non-
degenerate. We deduce that $({\mathfrak{J}},\alpha,\omega)$ is a
$(2n+1)$-dimensional cosymplectic JJ-algebra. ∎

######  Definition 3.2.

Two cosymplectic JJ-algebras $({\mathfrak{J}}_{1},\alpha_{1},\omega_{1})$ and
$({\mathfrak{J}}_{2},\alpha_{2},\omega_{2})$ are isomorphic if there exists an
isomorphism of JJ-algebras
$\Phi:{\mathfrak{J}}_{1}\longrightarrow{\mathfrak{J}}_{2}$ such that

$\omega_{2}(\Phi(x),\Phi(y))=\omega_{1}(x,y)\quad\mbox{and}\quad\alpha_{2}\circ\Phi(x)=\alpha_{1}(x),$

for any $x,y\in{\mathfrak{J}}_{1}$.

######  Remark 6.

In addition to the conditions mentioned above, for two cosymplectic JJ-
algebras $({\mathfrak{J}}_{1},\alpha_{1},\omega_{1})$ and
$({\mathfrak{J}}_{2},\alpha_{2},\omega_{2})$ to be isomorphic, we present an
extra condition, which is $\alpha_{1}(\xi_{1})=\xi_{2}$. Indeed, let $\xi_{i}$
be the Reeb vector of ${\mathfrak{J}}_{i}$, for $i=1,2$. Note that
$\alpha_{2}\big{(}\phi(\xi_{1})\big{)}=\alpha_{1}(\xi_{1})=1$ and
$\omega_{2}(\phi(\xi_{1}),.)=\omega_{1}(\xi_{1},.)=0$, Therefore
$\phi(\xi_{1})=\xi_{2}$.

######  Proposition 3.2.

Let $({\mathfrak{J}}_{1},\alpha_{1},\omega_{1})$ and
$({\mathfrak{J}}_{2},\alpha_{2},\omega_{2})$ two cosymplectic JJ-algebras,
with $({\mathfrak{h}}_{1},\omega_{{\mathfrak{h}}_{1}},\mathrm{D}_{1})$ and
$({\mathfrak{h}}_{2},\omega_{{\mathfrak{h}}_{2}},\mathrm{D}_{2})$ the
corresponding symplectic JJ-algebras and anti-derivations as in Theorem 3.1.
Then, $({\mathfrak{J}}_{1},\alpha_{1},\omega_{1})$ and
$({\mathfrak{J}}_{2},\alpha_{2},\omega_{2})$ are isomorphic if and only if
there exists an isomorphism
$\phi:{\mathfrak{h}}_{1}\longrightarrow{\mathfrak{h}}_{2}$ of symplectic JJ-
algebras, such that $\phi\circ\mathrm{D}_{1}=\mathrm{D}_{2}\circ\phi$.

###### Proof.

Let $({\mathfrak{J}}_{1},\alpha_{1},\omega_{1})$ and
$({\mathfrak{J}}_{2},\alpha_{2},\omega_{2})$ be two $(2n+1)$-dimensional
cosymplectic JJ-algebras, assume that they are isomorphic by
$\phi:{\mathfrak{J}}_{1}\longmapsto{\mathfrak{J}}_{2}$, It follows from Lemma
3.1 that $({\mathfrak{h}}_{1},\omega_{{\mathfrak{h}}_{1}})$ and
$({\mathfrak{h}}_{2},\omega_{{\mathfrak{h}}_{2}})$ are symplectic JJ-algebras
of dimension $2n$, where ${\mathfrak{h}}_{1}=\mathrm{ker}(\alpha_{1})$ and
${\mathfrak{h}}_{2}=\mathrm{ker}(\alpha_{2})$. In particular,
$\phi:{\mathfrak{h}}_{1}\longmapsto\phi({\mathfrak{h}}_{1})$ is an isomorphism
of JJ-algebras and
$\omega_{{\mathfrak{h}}_{2}}(\phi({\mathfrak{h}}_{1}),\phi({\mathfrak{h}}_{1}))=\omega_{{\mathfrak{h}}_{1}}({\mathfrak{h}}_{1},{\mathfrak{h}}_{1})$
.

For all $x\in{\mathfrak{h}}_{1}$, we have
$\alpha_{1}(x)=\alpha_{2}\big{(}\phi(x)\big{)}=0$, then
$\phi({\mathfrak{h}}_{1})\subset{\mathfrak{h}}_{2}$, since
$\mathrm{dim}(\phi({\mathfrak{h}}_{1}))=\mathrm{dim}({\mathfrak{h}}_{2})$,
this implies that $\phi({\mathfrak{h}}_{1})={\mathfrak{h}}_{2}$, we conclude
that $\phi:{\mathfrak{h}}_{1}\longmapsto{\mathfrak{h}}_{2}$ is an isomorphism
of symplectic JJ-algebras. By Theorem 3.1, we state that
$\mathrm{D}_{i}(x)=\xi_{i}\cdot x$, for $i=1,2$ (without loss of generality,
we fix $``\cdot"$ as product for all JJ-algebras). Then, for any
$x\in{\mathfrak{h}}_{1}$, we have
$\phi\circ\mathrm{D}_{1}(x)=\phi(\xi_{1}\cdot
x)=\xi_{2}\cdot\phi(x)=\mathrm{D}_{2}\circ\phi(x)$.

On the other hand, let
$({\mathfrak{h}}_{1},\omega_{{\mathfrak{h}}_{1}},\mathrm{D}_{1})$ and
$({\mathfrak{h}}_{2},\omega_{{\mathfrak{h}}_{2}},\mathrm{D}_{2})$ be two
isomorphic symplectic JJ-algebras with
$\phi:{\mathfrak{h}}_{1}\longrightarrow{\mathfrak{h}}_{2}$ such that
$\phi\circ\mathrm{D}_{1}=\mathrm{D}_{2}\circ\phi$, where $\mathrm{D}_{1}$ and
$\mathrm{D}_{2}$ two anti-derivations as in Theorem 3.1. Similarly to the
second part of the proof of Theorem 3.1, we consider the vector spaces
$({\mathfrak{J}}_{i}={\mathfrak{h}}_{i}\oplus\langle\xi_{i}\rangle,,\alpha_{i},\Omega_{i})$,
for $i=1,2$. Then, we can extend $\phi$ to an isomorphism of JJ-algebras
$\Phi:{\mathfrak{J}}_{1}\longmapsto{\mathfrak{J}}_{2}$ by
$\Phi|_{{\mathfrak{h}}_{1}}=\phi$, and $\Phi(\xi_{1})=\xi_{2}$. Indeed,
$\Phi(\xi_{1}\cdot{\mathfrak{h}}_{1})=\Phi\circ\mathrm{D}_{1}({\mathfrak{h}}_{1})=\phi\circ\mathrm{D}_{1}({\mathfrak{h}}_{1})=\mathrm{D}_{2}\circ\phi({\mathfrak{h}}_{1})=\xi_{2}\cdot{\mathfrak{h}}_{2}=\Phi(\xi_{1})\cdot\Phi({\mathfrak{h}}_{1})$.
Its reminds to show that $\Phi$ is a cosymplectic isomorphism of JJ-algebras.
On the one hand, it is clear that
$\alpha_{2}\circ\Phi({\mathfrak{h}}_{1})=\alpha_{2}({\mathfrak{h}}_{2})=0=\alpha_{1}({\mathfrak{h}}_{1})$,
and $\alpha_{2}\circ\Phi(\xi_{1})=\alpha_{2}(\xi_{2})=1=\alpha_{1}(\xi_{1})$.
On the other hand, it is straightforward to see that
$\omega_{2}\big{(}\Phi({\mathfrak{h}}_{1}),\Phi({\mathfrak{h}}_{1})\big{)}=\omega_{{\mathfrak{h}}_{2}}\big{(}{\mathfrak{h}}_{2},{\mathfrak{h}}_{2}\big{)}=\omega_{{\mathfrak{h}}_{2}}\big{(}\phi({\mathfrak{h}}_{1}),\phi({\mathfrak{h}}_{1})\big{)}=\omega_{1}({\mathfrak{h}}_{1},{\mathfrak{h}}_{1})$.
Finally,
$\omega_{2}\big{(}\Phi(\xi_{1}),\Phi({\mathfrak{h}}_{1})\big{)}=\omega_{{\mathfrak{h}}_{2}}\big{(}\xi_{2},{\mathfrak{h}}_{2}\big{)}=0=\omega_{1}(\xi_{1},{\mathfrak{h}}_{1})$,
which completes the proof. ∎

If Theorem 3.1 establishes a connection between $(2n+1)$-dimensional
cosymplectic JJ-algebras and $2n$-dimensional symplectic JJ-algebras, then the
following proposition presents the opposite effect. It establishes a
relationship between a $(2n+1)$-dimensional cosymplectic JJ-algebra and a
$(2n+2)$-dimensional symplectic JJ-algebras under certain conditions.

######  Proposition 3.3.

Let $({\mathfrak{J}},\cdot)$ be a JJ-algebra, and $\alpha$, $\omega$ be a
$1$-form and $2$-form on ${\mathfrak{J}}$, respectively. On the direct sum JJ-
algebra $({\mathfrak{J}}\oplus\langle e\rangle,\cdot)$, we consider the
$2$-form $\Omega=\omega+\alpha\wedge e^{*}$. Then,
$({\mathfrak{J}},\alpha,\omega)$ is a cosymplectic JJ-algebra if and only if
$({\mathfrak{J}}\oplus\langle e\rangle,\Omega)$ is a symplectic JJ-algebra.

###### Proof.

Suppose that $({\mathfrak{J}},\alpha,\omega)$ is a cosymplectic JJ-algebra. In
view of Theorem 3.1, the vector space ${\mathfrak{J}}$ always can be written
as ${\mathfrak{J}}={\mathfrak{h}}\oplus\langle\xi\rangle$, where
$({\mathfrak{h}},\omega_{{\mathfrak{h}}})$ is a symplectic JJ-algebra.

Note that the vector space ${\mathfrak{J}}\oplus\langle e\rangle$ is a direct
sum of ${\mathfrak{J}}$ with the one-dimensional vector space $\langle
e\rangle$. Therefore, $({\mathfrak{J}}\oplus\langle e\rangle,\cdot)$ is a JJ-
algebra.

Let $\Omega$ be a skew-symmetric $2$-form on ${\mathfrak{J}}\oplus\langle
e\rangle$ defined as follows

$\Omega(x,y)=\omega_{{\mathfrak{h}}}(x,y),\;\;\;\;\;\Omega(\xi,e)=1\quad\mbox{and}\quad\Omega(\xi,x)=\Omega(e,x)=0,$

for every $x,y\in{\mathfrak{h}}$, it is clear that $\Omega$ is non-degenerate
on ${\mathfrak{J}}\oplus\langle e\rangle$. Hence,
$({\mathfrak{J}}\oplus\langle e\rangle,\Omega)$ is a symplectic JJ-algebra.

Conversely. Let ${\mathfrak{J}}$ be a $(2n+1)$-dimensional JJ-algebra.
Consider a $1$-form $\alpha$ and $2$-form $\omega$ on ${\mathfrak{J}}$. Assume
that $({\mathfrak{J}}\oplus\langle e\rangle,\Omega)$ is a $(2n+2)$-dimensional
symplectic JJ-algebra, where $\Omega=\omega+\alpha\wedge e^{*}$.

Since $\Omega$ is a symplectic structure, we have $\Omega^{n+1}\neq 0$. A
straightforward calculation ensures that $\omega^{n}\wedge\alpha\wedge
e^{*}\neq 0$, which implies $\omega^{n}\wedge\alpha\neq 0$ on
${\mathfrak{J}}$.

On the one hand, we have $\oint\Omega(x,y,z):=\omega(x\cdot y,z)+\omega(z\cdot
x,y)+\omega(y\cdot z,x)=0$ for all $x,y,z\in{\mathfrak{J}}$, which makes
$\omega$ a skew-symmetric bilinear form on ${\mathfrak{J}}$ and satisfies the
identity (6).

On the other hand, $\oint\Omega(x,y,e):=\alpha(x\cdot y)=0$, so $\alpha$
satisfies (5). We conclude that $({\mathfrak{J}},\alpha,\omega)$ is a
cosymplectic JJ-algebra. ∎

##  4 Cosymplectic JJ-algebras double extensions

Let $({\mathfrak{J}},\alpha,\omega)$ be a $(2n+1)$-dimensional cosymplectic
JJ-algebra and let $(\tilde{{\mathfrak{J}}}=\langle
d\rangle\oplus{\mathfrak{J}}\oplus\langle e\rangle,\bullet)$ be a double
extension of $({\mathfrak{J}},\cdot)$ by $(\theta,\varphi,\lambda,a)$ and let
$\tilde{\omega}$ be a skew-symmetric $2$-form and $\tilde{\alpha}$ a $1$-form
on $\tilde{{\mathfrak{J}}}$ defined as follows

$\tilde{\omega}=\omega(x,y),\;\forall
x,y\in{\mathfrak{J}},\;\tilde{\omega}(d,e)=1,\;\tilde{\omega}(d,.)=\tilde{\omega}(e,.)=0,$

and

$\tilde{\alpha}(x)=\alpha(x),\;\forall
x\in{\mathfrak{J}},\;\tilde{\alpha}(d)\in\mathbb{K},\;\tilde{\alpha}(e)\in\mathbb{K}.$

On the one hand, suppose that $\tilde{\alpha}$ satisfies (5). It follows that

$\left\\{\begin{array}[]{l}\theta(x,y)\tilde{\alpha}(e)=0,\\\
\alpha\circ\varphi(x)+\lambda(x)\tilde{\alpha}(e)=0,\\\
\alpha(a)=0.\end{array}\right.$ (16)

On the other hand, suppose that $\tilde{\omega}$ satisfies (6). Then, for any
$x,y\in{\mathfrak{J}}$, It follows that

$\left\\{\begin{array}[]{l}\theta(x,y)=\omega_{\varphi}(x,y)\\\
2\lambda(x)=\omega(a,x)\\\ \omega(\varphi(x),a)=0.\end{array}\right.$ (17)

Where $\omega_{\varphi}(x,y)=\omega(\varphi(x),y)+\omega(\varphi(y),x)$ for
any $x$, $y\in{\mathfrak{J}}$. From the first row of the system (16), one can
distinguish two cases.

The first case: $\tilde{\alpha}(e)=0$.

The conditions described above, together with Proposition 2.1, yield the
following result.

######  Theorem 4.1.

Let $({\mathfrak{J}},\alpha,\omega)$ be a cosymplectic JJ-algebra and
$(\varphi,a)$ an admissible pair of $({\mathfrak{J}},\cdot)$. Then
$\tilde{{\mathfrak{J}}}=\langle d\rangle\oplus{\mathfrak{J}}\oplus\langle
e\rangle\ $ equipped with the product

$\displaystyle x\bullet y$ $\displaystyle=x\cdot y+\omega_{\varphi}(x,y)e,$
$\displaystyle d\bullet x$ $\displaystyle=\varphi(x)+\frac{1}{2}\omega(a,x)e,$
$\displaystyle d\bullet d$ $\displaystyle=a\in{\mathfrak{J}},$

for any $x$ and $y\in{\mathfrak{J}}$. Then,

$\tilde{\alpha}_{|{\mathfrak{J}}}=\alpha+td^{*}\quad\mbox{and}\quad\tilde{\omega}=\omega+d^{*}\wedge
e^{*},\quad t\in\mathbb{R}$

is a cosymplectic structure on $\tilde{{\mathfrak{J}}}$ if
$\alpha\circ\varphi=0$ and $\omega(\varphi(x),a)=0$, for any
$x\in{\mathfrak{J}}$.

###### Proof.

To complete the proof, it suffices to show that
$\tilde{\alpha}\wedge\tilde{\omega}^{(n+1)}\neq 0$. This is equivalent to
proving that
$\Psi:\tilde{{\mathfrak{J}}}\longrightarrow\tilde{{\mathfrak{J}}}^{*}$,
$\tilde{x}\longmapsto\tilde{\omega}(\tilde{x},.)+\tilde{\alpha}(\tilde{x})\tilde{\alpha}(.)$
is an isomorphism. Let $\tilde{x}\in\langle
d\rangle\oplus{\mathfrak{J}}\oplus\langle e\rangle$ and write
$\tilde{x}=x_{1}d+x+x_{2}\xi+x_{3}e$, where
$x\in{\mathfrak{h}}=\mathrm{ker(\alpha)}$ and $x_{1}$, $x_{2}$,
$x_{3}\in\mathbb{R}$. One can directly verify that

$\left\\{\begin{array}[]{ll}\Psi(\tilde{x},y)&=\omega_{{\mathfrak{h}}}(x,y),\\\
\Psi(\tilde{x},e)&=x_{1},\\\
\Psi(\tilde{x},\xi)&=x_{1}\tilde{\alpha}(d)+x_{2},\\\
\Psi(\tilde{x},d)&=-x_{3}+(x_{1}\tilde{\alpha}(d)+x_{2})\tilde{\alpha}(d).\end{array}\right.$

for any $y\in{\mathfrak{h}}$. Since $\omega_{{\mathfrak{h}}}$ is a non-
degenerate bilinear form, it follows that if $\Psi(\tilde{x},.)=0$, then
$\tilde{x}=0$. Therefore $\Psi$ is injective. ∎

The second case: $\tilde{\alpha}(e)\not=0$.

For simplicity, one can take $\tilde{\alpha}(e)=1$. So, by (16) and (17) we
obtain that

$\theta(x,y)=0,\quad\alpha(a)=0\quad\mbox{and}\quad\lambda(x)=-\alpha\circ\varphi(x)=\frac{1}{2}\omega(a,x),$
$\omega(\varphi(x),y)=\omega(x,\varphi(y)),$

for any $x,y\in{\mathfrak{J}}$. Under the above assumptions and Proposition
2.1, one conclude that

######  Theorem 4.2.

Let $({\mathfrak{J}},\alpha,\omega)$ be a cosymplectic JJ-algebra and
$(\varphi,a)$ an admissible pair of $({\mathfrak{J}},\cdot)$. Then
$\tilde{{\mathfrak{J}}}=\langle d\rangle\oplus{\mathfrak{J}}\oplus\langle
e\rangle\ $ equipped with the product

$\displaystyle x\bullet y$ $\displaystyle=x\cdot y,$ $\displaystyle d\bullet
x$ $\displaystyle=\varphi(x)+\frac{1}{2}\omega(a,x)e,$ $\displaystyle d\bullet
d$ $\displaystyle=a\in{\mathfrak{J}},$

for any $x$ and $y\in{\mathfrak{J}}$. Then,

$\tilde{\alpha}_{|{\mathfrak{J}}}=\alpha+e^{*}+td^{*},\;\;\tilde{\omega}=\omega+d^{*}\wedge
e^{*},\quad t\in\mathbb{K}$

defines a cosymplectic structure on $\tilde{{\mathfrak{J}}}$ if
$\omega_{\varphi}=0$ and $\lambda=\frac{1}{2}\omega(a,.)=-\alpha\circ\varphi$.

###### Proof.

As in the proof of Theorem 4.1, we only need to show that
$\Psi:\tilde{{\mathfrak{J}}}\longrightarrow\tilde{{\mathfrak{J}}}^{*}$ is an
isomorphism. Let $\tilde{x}\in\langle
d\rangle\oplus{\mathfrak{J}}\oplus\langle e\rangle$ and write
$\tilde{x}=x_{1}d+x+x_{2}\xi+x_{3}e$, where
$x\in{\mathfrak{h}}=\mathrm{ker(\alpha)}$ and
$x_{1},\,x_{2},\,x_{3}\in\mathbb{K}$. Note that

$\left\\{\begin{array}[]{ll}\Psi(\tilde{x},y)&=\omega_{{\mathfrak{h}}}(x,y),\;\forall
y\in{\mathfrak{h}},\\\
\Psi(\tilde{x},\xi)&=x_{1}\tilde{\alpha}(d)+x_{2}-x_{3},\\\
\Psi(\tilde{x},e)&=x_{1}-(x_{1}\tilde{\alpha}(d)+x_{2}-x_{3}),\\\
\Psi(\tilde{x},d)&=-x_{3}+(x_{1}\tilde{\alpha}(d)+x_{2}-x_{3})\tilde{\alpha}(d).\end{array}\right.$

It is clear that if $\Psi(\tilde{x},.)=0$, then $\tilde{x}=0$. ∎

######  Theorem 4.3.

Let $(\tilde{{\mathfrak{J}}},\tilde{\alpha},\tilde{\omega})$ be a cosymplectic
JJ-algebra such that $\tilde{{\mathfrak{J}}}\not=\\{0\\}$. Then,
$(\tilde{{\mathfrak{J}}},\tilde{\alpha},\tilde{\omega})$ is a double extension
of cosymplectic JJ-algebra $({\mathfrak{J}},\alpha,\omega)$.

###### Proof.

Let $(\tilde{{\mathfrak{J}}},\tilde{\alpha},\tilde{\omega})$ be a cosymplectic
JJ-algebra and $``\bullet"$ be its product. It follows from Lemma 3.1 that
$\tilde{{\mathfrak{J}}}=\langle\xi\rangle\oplus\tilde{\mathfrak{h}}$ where
$\tilde{\mathfrak{h}}=\ker(\tilde{\alpha})$ and
$(\tilde{\mathfrak{h}},\tilde{\omega}_{\tilde{\mathfrak{h}}})$ is a symplectic
JJ-algebra. Consequently, Theorem 3.3 [3] implies that
$(\tilde{\mathfrak{h}},\tilde{\omega}_{\tilde{\mathfrak{h}}})$ is a symplectic
double extension of a symplectic JJ-algebra $({\mathfrak{h}},\omega)$. i.e.,
there exists an admissible pair $(\varphi,a)$ of $({\mathfrak{h}},\cdot)$,
such that $\tilde{\mathfrak{h}}=\langle
d\rangle\oplus{\mathfrak{h}}\oplus\langle e\rangle$

$\displaystyle x\bullet y$ $\displaystyle=x\cdot y+\omega_{\varphi}(x,y)e,$
$\displaystyle d\bullet x$ $\displaystyle=\varphi(x)+\frac{1}{2}\omega(a,x)e,$
$\displaystyle d\bullet d$ $\displaystyle=a,$

$\tilde{\omega}_{\tilde{\mathfrak{h}}}=\omega+d^{*}\wedge e^{*}$ and
$\omega(\varphi(x),a)=0$, for any $x,y\in{\mathfrak{h}}$. Note that
$\omega=\tilde{\omega}_{{\mathfrak{h}}}$ and $\tilde{\alpha}\circ\varphi=0$.

In $\tilde{\mathfrak{J}}=\xi\oplus\tilde{\mathfrak{h}}$, for all
$x\in\tilde{\mathfrak{h}}$, we have

$\displaystyle\xi\bullet x$ $\displaystyle=f(x)d+D(x)+l(x)e,\quad$
$\displaystyle d\bullet\xi$ $\displaystyle=kd+v+te,\quad
v\in{\mathfrak{h}},\;k,t\in\mathbb{K},$

where $D$ is an endomorphism of ${\mathfrak{h}}$, $f$ and $l$ are linear forms
of ${\mathfrak{h}}$. Since $\tilde{\omega}$ satisfies (6), then $f\equiv 0$,
$k=0$, $t=0$, $l(x)=\omega(v,x)$ and
$\tilde{\omega}_{\mathfrak{h}}(D(x),y)=\tilde{\omega}_{\mathfrak{h}}(x,D(y))$
for any $x,y\in{\mathfrak{h}}$.

We extend $\varphi$ on ${\mathfrak{J}}=\xi\oplus{\mathfrak{h}}$ by
$\varphi(\xi)=v$.

The product becomes

$\displaystyle x\bullet y$ $\displaystyle=x\cdot
y+\tilde{\omega}_{\varphi}(x,y)e,$ $\displaystyle\xi\bullet x$
$\displaystyle=D(x)+\tilde{\omega}_{\varphi}(\xi,x)e,$ $\displaystyle d\bullet
x$ $\displaystyle=\varphi(x)+\frac{1}{2}\tilde{\omega}(a,x)e,$ $\displaystyle
d\bullet\xi$ $\displaystyle=\varphi(\xi),$ $\displaystyle d\bullet d$
$\displaystyle=a.$

Since the product $``\bullet"$ satisfies Jacobi identity, then
$\mathrm{D}\in\mathrm{Ader}({\mathfrak{J}})$ such that $\mathrm{D}^{2}=0$. By
Theorem 3.1 it follows that
$({\mathfrak{J}},,\tilde{\alpha}_{{\mathfrak{J}}},\tilde{\omega}_{\mathfrak{J}})$
is a cosymplectic JJ-algebra and $(\varphi,a)$ is an admissible pair of
$({\mathfrak{J}},\cdot)$. In addition, $\oint\omega(\xi\bullet d,a)=0$ implies
that $\tilde{\omega}_{\mathfrak{J}}(\varphi(\xi),a)=0$. Finally,
$(\tilde{{\mathfrak{J}}},\tilde{\alpha},\tilde{\omega})$ is cosymplectic JJ-
algebra double extension of
$({\mathfrak{J}},\tilde{\alpha}_{{\mathfrak{J}}},\tilde{\omega}_{\mathfrak{J}})$.
∎

Non-trivial examples in every odd dimension

Using the previous constructions, we give examples of non-trivial cosymplectic
JJ-algebra in every odd dimension.

Let $(\mathbb{K}^{2n+1},\omega,\alpha)$, with
$\omega=\sum_{i=1}^{n}e^{i}\wedge e^{i+n}$ and $\alpha=e^{2n+1}$ be the
trivial $(2n+1)$-dimensional cosymplectic JJ-algebra. According to Theorem
4.1, it suffices to search for an admissible pair $(\varphi,a)$ of
$\mathbb{K}^{2n+1}$ that satisfies $\alpha\circ\varphi=0$ and
$\omega(\varphi(x),a)=0$, for any $x\in{\mathfrak{J}}$. Consider for example

$\left\\{\begin{array}[]{rlr}\varphi(e_{2n+1})&=\sum_{i=1}^{2n}\varphi_{i}e_{i}\quad\mbox{and}\quad\varphi(e_{i})=0,&1\leqslant
i\leqslant 2n,\\\ a&=\sum_{i=1}^{2n}a_{i}e_{i},&\\\
\varphi_{n+i}a_{i}&=\varphi_{i}a_{n+i},&1\leqslant i\leqslant
n.\end{array}\right.$

In this case, a direct calculation yields to

$\left\\{\begin{array}[]{ll}\lambda(e_{i})=-\frac{1}{2}a_{n+i},&1\leqslant
i\leqslant n\\\ \lambda(e_{i})=\frac{1}{2}a_{i},&n+1\leqslant i\leqslant 2n\\\
\end{array}\right.\quad\mbox{and}\quad\left\\{\begin{array}[]{ll}\theta(e_{i},e_{j})=\theta(e_{2n+1},e_{2n+1})=0,&1\leqslant
i,j\leqslant 2n\\\ \theta(e_{2n+1},e_{i})=-\varphi_{i+n},&1\leqslant
i\leqslant n\\\ \theta(e_{2n+1},e_{i})=\varphi_{i},&n+1\leqslant i\leqslant
2n.\end{array}\right.$

Finally, the vector space $\tilde{\mathfrak{J}}=\langle
e_{2n+3}\rangle\oplus\mathbb{K}^{2n+1}\oplus\langle e_{2n+2}\rangle$, endowed
with the product

$\left\\{\begin{array}[]{ll}e_{2n+1}\cdot
e_{i}=-\varphi_{n+i}e_{2n+2},&1\leqslant i\leqslant n\\\ e_{2n+1}\cdot
e_{i}=-\varphi_{i}e_{2n+2},&n+1\leqslant i\leqslant 2n\\\ e_{2n+3}\cdot
e_{i}=-\frac{1}{2}a_{n+i}e_{2n+2},&1\leqslant i\leqslant n\\\ e_{2n+3}\cdot
e_{i}=\frac{1}{2}a_{i}e_{2n+2},&n+1\leqslant i\leqslant 2n\\\ e_{2n+3}\cdot
e_{2n+3}=a,&\end{array}\right.$

and the following structure

$\tilde{\alpha}=e^{2n+1}\quad\mbox{and}\quad\tilde{\omega}=\omega+e^{2n+2}\wedge
e^{2n+3},$

is a cosymplectic double extension of the trivial cosymplectic JJ-algebra
$(\mathbb{K}^{2n+1},\omega,\alpha)$.

##  5 Five-dimensional cosymplectic JJ-algebras

We use Theorem 3.1 and Proposition 3.2 to give a complete classification of
five-dimensional cosymplectic JJ-algebras.

######  Theorem 5.1.

Any five-dimensional cosymplectic non trivial JJ-algebra is isomorphic to one
of the following JJ-algebras:

  1. ${\mathfrak{J}}_{5,1}:$

$e_{1}\cdot e_{1}=e_{2}$, $e_{1}\cdot e_{3}=e_{4}$, $e_{1}\cdot e_{5}=e_{3}$,
$e_{2}\cdot e_{5}=-2e_{4}$,

  2. ${\mathfrak{J}}_{5,2}:$

$e_{1}\cdot e_{1}=e_{2}$, $e_{1}\cdot e_{3}=e_{4}$, $e_{1}\cdot e_{5}=e_{2}$,
$e_{3}\cdot e_{5}=2e_{4}$,

  3. ${\mathfrak{J}}_{5,3}:$

$e_{1}\cdot e_{1}=e_{2}$, $e_{1}\cdot e_{3}=e_{4}$,

endowed with $\alpha=e^{5},\;\omega=e^{14}+2e^{23}.$

###### Proof.

On the one hand, let $({\mathfrak{h}},\omega_{\mathfrak{h}})$ be the unique
four-dimensional symplectic non-trivial JJ-algebra defined by $e_{1}\cdot
e_{1}=e_{2}$, $e_{1}\cdot e_{3}=e_{3}\cdot e_{1}=e_{4}$, where
$\mathfrak{B}:=\\{e_{1},e_{2},e_{3},e_{4}\\}$ is a basis of the vector space
${\mathfrak{h}}$ and $\omega_{h}=te^{14}+2te^{23}$ for
$t\in\mathbb{K}-\\{0\\}$. Let
$\mathrm{D}:{\mathfrak{h}}\longrightarrow{\mathfrak{h}}$ be a linear map
represented in the base $\mathfrak{B}$ by the matrix
$\mathrm{D}=(a_{i,j})_{1\leqslant i,j\leqslant 4}$. A direct calculation shows
that $\mathrm{D}$ is an anti-derivation if and only if $a_{i,j}$ satisfies

$\left\\{\begin{array}[]{l @{=} c}a_{1,1}+a_{4,4}+a_{3,3}&0\\\
a_{2,2}-2a_{4,4}-2a_{3,3}&0\\\ a_{4,2}+2a_{3,1}&0\\\
a_{1,2}=a_{1,3}=a_{1,4}&0\\\ a_{2,4}=a_{3,2}=a_{3,4}&0.\end{array}\right.$

In addition, if
$\omega_{\mathfrak{h}}(\mathrm{D}(e_{i}),e_{j})=\omega_{\mathfrak{h}}(e_{i},\mathrm{D}(e_{j}))$
holds for $1\leqslant i,j\leqslant 4$, then we have

$\left\\{\begin{array}[]{l @{=} c}2a_{2,1}-a_{4,3}&0\\\ 2a_{4,4}+a_{3,3}&0\\\
a_{4,1}=a_{2,3}&0.\end{array}\right.$

It follows from the condition $\mathrm{D}^{2}=0$ that $a_{4,4}=0$.

Therefore, any anti-derivation $\mathrm{D}$ has the following form

$\mathrm{D}=\begin{pmatrix}0&0&0&0\\\ a_{2,1}&0&0&0\\\ a_{3,1}&0&0&0\\\
0&-2a_{3,1}&2a_{2,1}&0\end{pmatrix},$

Furthermore, one can distinguish three cases.

  1. 1.

If $a_{3,1}\not=0$, one can take $t=a_{3,1}$ the following automorphism $T$ of
${\mathfrak{h}}$, satisfies

$T=\begin{pmatrix}1&0&0&0\\\ 0&1&-\frac{a_{2,1}}{a_{3,1}}&0\\\
0&0&\frac{1}{a_{3,1}}&0\\\ 0&0&0&\frac{1}{a_{3,1}}\end{pmatrix},\quad
T.\mathrm{D}.T^{-1}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 1&0&0&0\\\
0&-2&0&0\end{pmatrix}\quad\mbox{and}\quad T_{*}\omega_{h}=e^{14}+2e^{23}.$

Which establishes the cosymplectic JJ-algebra
$({\mathfrak{J}}_{5,1},e^{14}+2e^{23},e^{5})$.

  2. 2.

If $a_{3,1}=0$ and $a_{2,1}\not=0$ , one can take $t=a_{2,1}$ the following
automorphism $T$ of ${\mathfrak{h}}$, satisfies

$T=\begin{pmatrix}\frac{1}{a_{2,1}}&0&0&0\\\ 0&\frac{1}{a_{2,1}^{2}}&0&0\\\
0&0&1&0\\\ 0&0&0&\frac{1}{a_{2,1}}\end{pmatrix},\quad
T.\mathrm{D}.T^{-1}=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\\\ 0&0&0&0\\\
0&0&2&0\end{pmatrix}\quad\mbox{and}\quad T_{*}\omega_{h}=e^{14}+2e^{23}.$

Which gives the cosymplectic JJ-algebra
$({\mathfrak{J}}_{5,2},e^{14}+2e^{23},e^{5})$.

  3. 3.

If $a_{3,1}=0$ and $a_{2,1}=0$. We have directly the cosymplectic JJ-algebra
$({\mathfrak{J}}_{5,3},e^{14}+2e^{23},e^{5})$.

On the other hand, let us consider $(\mathbb{K}^{4},\omega,\mathrm{D})$ to be
a four-dimensional symplectic trivial JJ-algebra, where $\mathrm{D}$ is an
endomorphism of ${\mathfrak{h}}_{0}$ such that $\mathrm{D}^{2}=0$. Then there
exists a basis of $\mathbb{K}^{4}$ where $\mathrm{D}$ has one of the following
canonical forms:

$\mathrm{D}_{0}=(0),\quad\mathrm{D}_{1}=\begin{pmatrix}0&1&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\
0&0&0&0\end{pmatrix}\quad\mbox{and}\quad\mathrm{D}_{2}=\begin{pmatrix}0&1&0&0\\\
0&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\end{pmatrix}.$

Let $\omega=\sum_{1\leqslant i,j\leqslant 4}a_{ij}e^{ij}$ be an arbitrary tow
form. A direct calculation shows that if

$\omega(\mathrm{D}_{2}(e_{i}),e_{j})=\omega(e_{i},\mathrm{D}_{2}(e_{j}))$

holds for $1\leqslant i,j\leqslant 4$. Then, we obtain
$a_{12}=a_{13}=a_{34}=0$ and $a_{14}=a_{23}$. Therefore,
$\omega=a_{23}e^{14}+a_{23}e^{23}+a_{24}e^{24}$. In addition, $\omega$ is non-
degenerate, then $a_{23}\not=0$.

The following automorphism $T$ of $\mathbb{K}^{4}$, satisfies

$T=\begin{pmatrix}-\frac{1}{a_{2,3}}&0&0&0\\\ 0&-\frac{1}{a_{2,3}}&0&0\\\
0&0&1&-\frac{a_{2,4}}{a_{2,3}}\\\ 0&0&0&1\end{pmatrix},\quad
T.\mathrm{D}_{2}.T^{-1}=\mathrm{D}_{2}\quad\mbox{and}\quad
T_{*}\omega=e^{14}+e^{23}.$

Which establishes the cosymplectic JJ-algebra
$({\mathfrak{J}}_{5,0},\omega_{0}=e^{14}+e^{23},\alpha_{0}=e^{5})$.

In the same way,
$\omega(\mathrm{D}_{1}(e_{i}),e_{j})=\omega(e_{i},\mathrm{D}_{1}(e_{j}))$
holds for $1\leqslant i,j\leqslant 4$ we find that $a_{12}=a_{13}=a_{14}=0$.
In this case, $\omega$ is degenerate. It is clear that for $\mathrm{D}_{0}$ we
obtain the trivial five-dimensional cosymplectic JJ-algebra. Note that the JJ-
algebras ${\mathfrak{J}}_{5,1}$, ${\mathfrak{J}}_{5,2}$ and
${\mathfrak{J}}_{5,3}$ are non-isomorphic. In fact it suffices to compare the
dimensions of their centers and derived sub-algebras. Otherwise the
cosymplectic JJ-algebras $({\mathfrak{J}}_{5,0},\omega_{0},\alpha_{0})$ and
$({\mathfrak{J}}_{5,2},\omega,\alpha)$ are isomorphic by the following
isomorphism $\Phi:{\mathfrak{J}}_{5,0}\longrightarrow{\mathfrak{J}}_{5,2}$,
which is given by

$\Phi(e_{1})=2e_{2}-\frac{1}{2}e_{5},\;\Phi(e_{2})=2e_{1},\;\Phi(e_{3})=e_{4},\;\Phi(e_{4})=\frac{1}{2}e_{3},\;\Phi(e_{5})=e_{5}.$

∎

######  Remark 7.

We finish by comparing our JJ-algebras with the ones given in [11], by the
following table.

${\mathfrak{J}}_{5,1}$ | ${\mathfrak{J}}_{5,2}$ | ${\mathfrak{J}}_{5,3}$
---|---|---
${\mathfrak{J}}_{8,5}$ | ${\mathfrak{J}}_{6,5}$ | ${\mathfrak{J}}_{1,4}\oplus\mathbb{K}$

## References

  * [1] Assiry, A.  _Symplectic Forms and the Yang-Baxter Equation in Jacobi-Jordan algebras_ ,  Int. J. Math. Comput. Sci. 19, (2024), 23-36.
  * [2] Baeza, V. R.  _On nuclear Bernstein algebras_ ,  Linear Algebra Appl. 142, (1990), 19-22.
  * [3] Baklouti, A. and Benayadi, S.  _Symplectic Jacobi-Jordan algebras_ ,  Linear Multilinear Algebra 69 (2021), 1557-1578.
  * [4] Burde, D. and Fialowski, A.  _Jacobi-Jordan algebras_ ,  Linear Algebra Appl. 459, (2014), 586-594.
  * [5] de León, M. and Saralegi, M.  _Cosymplectic reduction for singular momentum maps_ ,  J. Phys. A, Math. Gen. 26, No. 19,(1993), 5033-5043 .
  * [6] El Bourkadi, S. and Mansouri, M. W.  _Left-Symmetric Products on Cosymplectic Lie Algebras_ ,  J. of Lie Theory, 34 (2024), No. 2, 249-265
  * [7] Fino, A. and Vezzoni, L.  _Some results on cosymplectic manifolds_ ,  Geom. Dedicata 151 (2011), 41-58.
  * [8] Kamiya, N. and Okubo, S.  _Jordan-Lie super algebra and Jordan-Lie triple system_.  J. Algebra 198, (1997), 388-411.
  * [9] Libermann, P.  _Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact_ ,  Colloque Géom. Diff. Globale (Bruxelles, 1958), Centre Belge Rech. Math., Louvain, 1959, pp. 37-59.
  * [10] Walcher, S.  _On algebras of rank three_ ,  Comm. Algebra 27, (1999), 3401-3438.
  * [11] Yang, Y.  _On cohomology and deformations of Jacobi-Jordan algebras_ ,  Comm. Algebra, 51, (2023), 4702-4721.
  * [12] Zusmanovich, P.  _Special and exceptional mock-Lie algebras_ ,  Linear Algebra Appl. 518 (2017), 79-96.


#  $(C_{\alpha},C_{\beta})$-admissible functions in quasi-pseudometric type
spaces.

Yaé Ulrich Gaba,
<EMAIL_ADDRESS>
Department of Mathematics and Applied Mathematics,
University of Cape Town, Rondebosch 7701, South Africa

###### Abstract

In this article, we give some fixed point results in left $K$-complete quasi-
pseudometric type spaces for self-mappings that are
$(C_{\alpha},C_{\beta})$-admissible.

Keywords: quasi-pseudometric; left(right)$K$-completeness;
$(C_{\alpha},C_{\beta})$-admissible.

##  1 Introduction and Preliminaries

In the last few years the theory of quasi-metric spaces and other related
structures such as quasi-normed cones and asymmetric normed linear spaces (see
for instance [3]) has been at the center of rigorous research activity,
because such a theory provides a convenient framework in the study of several
problems in theoretical computer science, approximation theory and convex
analysis. The existence of fixed points and common fixed points of mappings
satisfying certain contractive conditions in this setting, has also been
discussed.

Recently, in [1], Eniola et al. discussed the newly introduced notion of
quasi-pseudometric type spaces as a logical equivalent to metric type spaces-
introduced by M. A. Khamsi[4]-when the initial distance-like function is not
symmetric.

The aim of this paper is to analyze the existence of fixed points for mapping
defined on a left $K$-complete quasi-pseudometric type space $(X,D,K)$. The
present results generalize another one already obtained by Gaba in [2], where
the author considered $(\alpha,\gamma)$-contractions.

###### Definition 1.1

Let $X$ be a nonempty set, and let the function $D:X\times X\to[0,\infty)$
satisfy the following properties:

  * (D1)

$D(x,x)=0$ for any $x\in X$;

  * (D2)

$D(x,y)\leq K\big{(}D(x,z_{1})+D(z_{1},z_{2})+\cdots+D(x_{n},y)\big{)}$ for
any points $x,y,z_{i}\in X,\ i=1,2,\ldots,n$ and some constant $K>0$.

Then $(X,D,K)$ is called a quasi-pseudometric type space. Moreover, if
$D(x,y)=0=D(y,x)\Longrightarrow x=y$, then $D$ is said to be a $T_{0}$-quasi-
pseudometric type space. The latter condition is referred to as the
$T_{0}$-condition.

###### Remark 1.2

  * •

Let $D$ be a quasi-pseudometric type on $X$, then the map $D^{-1}$ defined by
$D^{-1}(x,y)=D(y,x)$ whenever $x,y\in X$ is also a quasi-pseudometric type on
$X$, called the conjugate of $D$. We shall also denote $D^{-1}$ by $D^{t}$ or
$\bar{D}$.

  * •

It is easy to verify that the function $D^{s}$ defined by $D^{s}:=D\vee
D^{-1}$ , i.e. $D^{s}(x,y)=\max\\{D(x,y),D(y,x)\\}$ defines a metric-type (see
[4]) on $X$ whenever $D$ is a $T_{0}$-quasi-pseudometric type.

  * •

If we substitute the property $(D1)$ by the following property

$(D3)$ : $D(x,y)=0\Longleftrightarrow x=y$,

we obtain a $T_{0}$-quasi-pseudometric type space directly.

  * •

Moreover, for $K=1$, we recover the classical pseudometric, hence quasi-
pseudmetric type spaces generalize quasi-pseudometrics. It is worth mentioning
that if $(X,D,L)$ is a pseudometric type space, then for any $L\geq K$,
$(X,D,L)$ is also a pseudometric type space. We give the following example to
illustrate the above comment.

###### Example 1.3

Let $X=\\{a,b,c\\}$ and the mapping $D:X\times X\to[0,\infty)$ defined by
$D(a,b)=D(c,b)=1/5,\ D(b,c)=D(b,a)=D(c,a)=1/4,\ D(a,c)=1/2$, $D(x,x)=0$ for
any $x\in X$. Since

$\frac{1}{2}=D(a,c)>D(a,b)+D(b,c)=\frac{9}{20},$

then we conclude that $X$ is not a quasi-pseudometric space. Nevertheless,
with $K=2$, it is very easy to check that $(X,D,2)$ is a quasi-pseudometric
type space.

###### Definition 1.4

Let $(X,D,K)$ be a quasi-pseudometric type space. The convergence of a
sequence $(x_{n})$ to $x$ with respect to $D$, called $D$-convergence or left-
convergence and denoted by $x_{n}\overset{D}{\longrightarrow}x$, is defined in
the following way

$x_{n}\overset{D}{\longrightarrow}x\Longleftrightarrow
D(x,x_{n})\longrightarrow 0.$ (1)

Similarly, the convergence of a sequence $(x_{n})$ to $x$ with respect to
$D^{-1}$, called $D^{-1}$-convergence or right-convergence and denoted by
$x_{n}\overset{D^{-1}}{\longrightarrow}x$,

is defined in the following way

$x_{n}\overset{D^{-1}}{\longrightarrow}x\Longleftrightarrow
D(x_{n},x)\longrightarrow 0.$ (2)

Finally, in a quasi-pseudometric type space $(X,D,K)$, we shall say that a
sequence $(x_{n})$ $D^{s}$-converges to $x$ if it is both left and right
convergent to $x$, and we denote it as
$x_{n}\overset{D^{s}}{\longrightarrow}x$ or $x_{n}\longrightarrow x$ when
there is no confusion. Hence

$x_{n}\overset{D^{s}}{\longrightarrow}x\ \Longleftrightarrow\
x_{n}\overset{D}{\longrightarrow}x\ \text{ and }\
x_{n}\overset{D^{-1}}{\longrightarrow}x.$

###### Definition 1.5

A sequence $(x_{n})$ in a quasi-pseudometric type space $(X,D,K)$ is called

  * (a)

left $K$-Cauchy  with respect to $D$ if for every $\epsilon>0$, there exists
$n_{0}\in\mathbb{N}$ such that

$\forall\ n,k:n_{0}\leq k\leq n\quad D(x_{k},x_{n})<\epsilon;$

  * (b)

right $K$-Cauchy with respect to $D$ if for every $\epsilon>0$, there exists
$n_{0}\in\mathbb{N}$ such that

$\forall\ n,k:n_{0}\leq k\leq n\quad D(x_{n},x_{k})<\epsilon;$

  * (c)

$D^{s}$-Cauchy if for every $\epsilon>0$, there exists $n_{0}\in\mathbb{N}$
such that

$\forall n,k\geq n_{0}\quad D(x_{n},x_{k})<\epsilon.$

###### Remark 1.6

  * •

A sequence is left $K$-Cauchy with respect to $d$ if and only if it is right
$K$-Cauchy with respect to $D^{-1}$.

  * •

A sequence is $D^{s}$-Cauchy if and only if it is both left and right
$K$-Cauchy.

###### Definition 1.7

A quasi-pseudometric type space $(X,D,K)$ is called left-complete provided
that any left $K$-Cauchy sequence is $D$-convergent.

###### Definition 1.8

A quasi-pseudometric type space $(X,D,K)$ is called right-complete provided
that any right $K$-Cauchy sequence is $D$-convergent.

###### Definition 1.9

A $T_{0}$-quasi-pseudometric type space $(X,D,K)$ is called bicomplete
provided that the metric type $D^{s}$ on $X$ is complete.

##  2 Main Results

We begin by recalling the following.

###### Definition 2.1

(Compare [2]) Let $(X,D,K)$ be a quasi-pseudometric type space. A function
$T:X\to X$ is called $D$-sequentially continuous or left-sequentially
continuous  if for any $D$-convergent sequence $(x_{n})$ with
$x_{n}\overset{D}{\longrightarrow}x$, the sequence $(Tx_{n})$ $D$-converges to
$Tx$, i.e. $Tx_{n}\overset{D}{\longrightarrow}Tx$.

We then introduce the following definition.

###### Definition 2.2

Let $(X,D,K)$ be a quasi-pseudometric type space, $f:X\to X$ and
$\alpha,\beta:X\times X\to[0,\infty)$ be mappings and
$C_{\alpha}>0,C_{\beta}\geq 0$. We say that $T$ is
$(C_{\alpha},C_{\beta})$-admissible with respect to $K$ if the following
conditions hold:

  1. (C1)

$\alpha(x,y)\geq C_{\alpha}\Longrightarrow\alpha(fx,fy)\geq C_{\alpha},$
whenever $x,y\in X$;

  2. (C2)

$\beta(x,y)\leq C_{\beta}\Longrightarrow\beta(fx,fy)\leq C_{\beta},$ whenever
$x,y\in X$;

  3. (C3)

$0\leq C_{\beta}/C_{\alpha}<1/K$.

We begin by the following lemma.

###### Lemma 2.3

Let $(X,D,K)$ be a quasi-pseudometric type space and let $(x_{n})$ be a
sequence in $X$. Then

$D(x_{0},x_{n})\leq
KD(x_{0},x_{1})+\cdots+K^{n-1}D(x_{n-2},x_{n-1})+K^{n-1}D(x_{n-1},x_{n}).$

From Lemma 2.3, we deduce the following lemma.

###### Lemma 2.4

(Compare [1, Lemma 38]) Let $(X,D,K)$ be a quasi-pseudometric type space and
let $(x_{n})$ be a sequence in $X$ such that

$D(x_{n},x_{n+1})\leq\lambda D(x_{n-1},x_{n})\qquad\text{ for all }\ n\geq 0,$

for some $0<\lambda<1/K$. Then $(x_{n})$ is a left $K$-Cauchy sequence.

Similarly,

###### Lemma 2.5

Let $(X,D,\alpha)$ be a quasi-pseudometric type space and let $(x_{n})$ be a
sequence in $X$ such that

$D(x_{n+1},x_{n})\leq\lambda D(x_{n},x_{n-1})\qquad\text{ for all }\ n\geq 0,$

for some $0<\lambda<1/K$. Then $(x_{n})$ is a right $K$-Cauchy sequence.

We now state our main fixed point theorem.

###### Theorem 2.6

Let $(X,D,K)$ be a Hausdorff left $K$-complete $T_{0}$-quasi-pseudometric type
space. Suppose that $f:X\to X$ is $(C_{\alpha},C_{\beta})$-admissible with
respect to $K$. Assume that

$\alpha(x,y)D(fx,fy)\leq\beta(x,y)D(x,y)\qquad\text{for all }\ \ x,y\in X.$
(3)

If the following conditions hold:

  1. (i)

$f$ is $D$-sequentially continuous;

  2. (ii)

there exists $x_{0}\in X$ such that $\alpha(x_{0},fx_{0})\geq C_{\alpha}$ and
$\beta(x_{0},fx_{0})\leq C_{\beta}$.

Then $f$ has a fixed point.

###### Proof. 2.7

Let $x_{0}\in X$ such that $\alpha(x_{0},fx_{0})\geq C_{\alpha}$ and
$\beta(x_{0},fx_{0})\leq C_{\beta}$. Define the sequence $(x_{n})$ by
$x_{n}=f^{n}x_{0}=fx_{n-1}$. Without loss of generality, we can always assume
that $x_{n}\neq x_{n+1}$ for all $n\in\mathbb{N}$, since if
$x_{n_{0}}=x_{n_{0}+1}$ for some $n_{0}\in\mathbb{N},$ the proof is complete.
Since $f$ is $(C_{\alpha},C_{\beta})$-admissible with respect to $K$ and
$\alpha(x_{0},fx_{0})=(x_{0},x_{1})\geq C_{\alpha},$ we deduce that
$\alpha(x_{1},x_{2})=(fx_{0},fx_{1})\geq C_{\alpha}.$ By continuing this
process, we get that $\alpha(x_{n},x_{n+1})\geq C_{\alpha},$ for all $n\geq
0$. Similarly, we establish that $\beta(x_{n},x_{n+1})\leq C_{\beta},$ for all
$n\geq 0$. Using (3), we get

$\displaystyle C_{\alpha}D(x_{n},x_{n+1})$
$\displaystyle\leq\alpha(x_{n-1},x_{n})D(x_{n},x_{n+1})$
$\displaystyle\leq\beta(x_{n-1},x_{n})D(x_{n-1},x_{n})$ $\displaystyle\leq
C_{\beta}D(x_{n-1},x_{n}),$

and hence

$D(x_{n},x_{n+1})\leq\frac{C_{\beta}}{C_{\alpha}}D(x_{n-1},x_{n})\qquad\text{
for all }\ n\geq 1.$

By Lemma 2.4, since $0\leq\frac{C_{\beta}}{C_{\alpha}}<1/K,$ we derive that
$(x_{n})$ is a left $K$-Cauchy sequence. Since $(X,d)$ is left $K$-complete
and $T$ $D$-sequentially continuous, there exists $x^{*}$ such that
$x_{n}\overset{D}{\longrightarrow}x^{*}$ and
$x_{n+1}\overset{D}{\longrightarrow}fx^{*}$. Since $X$ is Hausdorff,
$x^{*}=fx^{*}$.

###### Corollary 2.8

Let $(X,D,K)$ be a Hausdorff left $K$-complete $T_{0}$-quasi-pseudometric type
space. Suppose that $f:X\to X$ is $(C_{\alpha},C_{\beta})$-admissible with
respect to $K$. Assume that

$\alpha(x,y)D(fx,fy)\leq\beta(x,y)D(x,y)\qquad\text{for all }\ \ x,y\in X.$
(4)

If the following conditions hold:

  1. (i)

$f$ is $D^{-1}$-sequentially continuous;

  2. (ii)

there exists $x_{0}\in X$ such that $\alpha(fx_{0},x_{0})\geq C_{\alpha}$ and
$\beta(fx_{0},x_{0})\leq C_{\beta}$.

Then $f$ has a fixed point.

###### Corollary 2.9

Let $(X,D,K)$ be a bicomplete $T_{0}$-quasi-pseudometric type space. Suppose
that $f:X\to X$ is $(C_{\alpha},C_{\beta})$-admissible with respect to $K$.
Assume that

$\alpha(x,y)D(fx,fy)\leq\beta(x,y)D(x,y)\qquad\text{for all }\ \ x,y\in X.$
(5)

If the following conditions hold:

  1. (i)

$f$ is $D^{s}$-sequentially continuous;

  2. (ii)

there exists $x_{0}\in X$ such that $\alpha(x_{0},fx_{0})\geq C_{\alpha}$ and
$\beta(x_{0},fx_{0})\leq C_{\beta}$;

  3. (iii)

the functions $\alpha$ and $\beta$ are symmetric, i.e.
$\alpha(a,b)=\alpha(b,a)$ for any $a,b\in X$ and $\beta(a,b)=\beta(b,a)$ for
any $a,b\in X$.

Then $f$ has a fixed point.

###### Proof. 2.10

Following the proof of Theorem 2.6, it is clear that the sequence $(x_{n})$ in
$X$ defined by $x_{n+1}=fx_{n}$ for all $n=0,1,2,\cdots$ is $D^{s}$-Cauchy.
Since $(X,D^{s})$ is complete and $f$ $D^{s}$-sequentially continuous, there
exists $x^{*}$ such that $x_{n}\overset{D^{s}}{\longrightarrow}x^{*}$ and
$x_{n+1}\overset{D^{s}}{\longrightarrow}fx^{*}$. Since $(X,D^{s})$ is
Hausdorff, $x^{*}=fx^{*}$.

###### Remark 2.11

In fact, we do not need $\alpha$ and $\beta$ to be symmetric. It is enough,
for the result to be true, to have a point $x_{0}\in X$ for which
$\alpha(x_{0},fx_{0})\geq C_{\alpha}$, $\alpha(fx_{0},x_{0})\geq C_{\alpha}$,
$\beta(x_{0},fx_{0})\leq C_{\beta}$ and $\beta(fx_{0},x_{0})\leq C_{\beta}$.
Otherwise stated, it is enough to have a point $x_{0}\in X$ for which
$\min\\{\alpha(x_{0},fx_{0}),\alpha(fx_{0},x_{0})\\}\geq C_{\alpha}$ and
$\min\\{\beta(x_{0},fx_{0}),\beta(fx_{0},x_{0}\\}\leq C_{\beta}$.

We give the following results which are in fact consequences of the Theorem
2.6.

###### Theorem 2.12

Let $(X,D,K)$ be a Hausdorff left $K$-complete $T_{0}$-quasi-pseudometric type
space. Suppose that $f:X\to X$ is $(C_{\alpha},C_{\beta})$-admissible with
respect to $K$. Assume that

$\alpha(x,y)D(fx,fy)\leq\beta(x,y)D(x,y)\qquad\text{for all }\ \ x,y\in X.$
(6)

If the following conditions hold:

  1. (i)

there exists $x_{0}\in X$ such that $\alpha(x_{0},fx_{0})\geq C_{\alpha}$ and
$\beta(x_{0},fx_{0})\leq C_{\beta}$;

  2. (ii)

if $(x_{n})$ is a sequence in $X$ such that $\alpha(x_{n},x_{n+1})\geq
C_{\alpha}$ and $\beta(x_{n},x_{n+1})\leq C_{\beta}$ for all $n=1,2,\cdots$
and $x_{n}\overset{D}{\longrightarrow}x$, then there exists a subsequence
$(x_{n(k)})$ of $(x_{n})$ such that $\alpha(x,x_{n(k)})\geq C_{\alpha}$ and
$\beta(x,x_{n(k)})\leq C_{\beta}$ for all $k$.

Then $f$ has a fixed point.

###### Proof. 2.13

Following the proof of Theorem 2.6, we know that the sequence $(x_{n})$
defined by $x_{n+1}=fx_{n}$ for all $n=0,1,2,\cdots$ $D$-converges to some
$x^{*}$ and satisfies $\alpha(x_{n},x_{n+1})\geq C_{\alpha}$ and
$\beta(x_{n},x_{n+1})\leq C_{\beta}$ for $n\geq 1$. From the condition $(ii)$,
we know there exists a subsequence $(x_{n(k)})$ of $(x_{n})$ such that
$\alpha(x^{*},x_{n(k)})\geq C_{\alpha}$ and $\beta(x^{*}x_{n(k)})\leq
C_{\beta}$ for all $k$. Since $f$ is satisfies (6), we get

$\displaystyle C_{\alpha}D(fx^{*},x_{n(k)+1})=C_{\alpha}D(fx^{*},fx_{n(k)})$
$\displaystyle\leq\alpha(x^{*},x_{n(k)})D(fx^{*},fx_{n(k)})$
$\displaystyle\leq\beta(x^{*},x_{n(k)})D(x^{*},x_{n(k)})$ $\displaystyle\leq
C_{\beta}D(x^{*},x_{n(k)})$

Letting $k\to\infty$, we obtain $D(fx^{*},x_{n(k)+1})\to 0$. Since $X$ is
Hausdorff, we have that $fx^{*}=x^{*}$. This completes the proof.

###### Corollary 2.14

Let $(X,D,K)$ be a Hausdorff right $K$-complete $T_{0}$-quasi-pseudometric
type space. Suppose that $f:X\to X$ is $(C_{\alpha},C_{\beta})$-admissible
with respect to $K$. Assume that

$\alpha(x,y)D(fx,fy)\leq\beta(x,y)D(x,y)\qquad\text{for all }\ \ x,y\in X.$
(7)

If the following conditions hold:

  1. (i)

there exists $x_{0}\in X$ such that $\alpha(fx_{0},x_{0})\geq C_{\alpha}$ and
$\beta(fx_{0},x_{0})\leq C_{\beta}$;

  2. (ii)

if $(x_{n})$ is a sequence in $X$ such that $\alpha(x_{n+1},x_{n})\geq
C_{\alpha}$ and $\beta(x_{n+1},x_{n})\leq C_{\beta}$ for all $n=1,2,\cdots$
and $x_{n}\overset{D^{-1}}{\longrightarrow}x$, then there exists a subsequence
$(x_{n(k)})$ of $(x_{n})$ such that $\alpha(x_{n(k)},x)\geq C_{\alpha}$ and
$\beta(x_{n(k)},x)\leq C_{\beta}$ for all $k$.

Then $f$ has a fixed point.

###### Corollary 2.15

Let $(X,D,K)$ be a bicomplete $T_{0}$-quasi-pseudometric type space. Suppose
that $f:X\to X$ is $(C_{\alpha},C_{\beta})$-admissible with respect to $K$.
Assume that

$\alpha(x,y)D(fx,fy)\leq\beta(x,y)D(x,y)\qquad\text{for all }\ \ x,y\in X.$
(8)

If the following conditions hold:

  1. (i)

there exists $x_{0}\in X$ such that $\alpha(x_{0},fx_{0})\geq C_{\alpha}$ and
$\beta(x_{0},fx_{0})\leq C_{\beta}$;

  2. (ii)

if $(x_{n})$ is a sequence in $X$ such that $\alpha(x_{m},x_{n})\geq
C_{\alpha}$ and $\beta(x_{m},x_{n})\leq C_{\beta}$ for all $n,m\in\mathbb{N}$
and $x_{n}\overset{D^{s}}{\longrightarrow}x$, then there exists a subsequence
$(x_{n(k)})$ of $(x_{n})$ such that $\alpha(x,x_{n(k)})\geq C_{\alpha}$ and
$\beta(x,x_{n(k)})\leq C_{\beta}$ for all $k$;

  3. (iii)

the functions $\alpha,\beta$ are symmetric.

Then $f$ has a fixed point.

In regard of Remark (2.11), another variant of the above corollary can be
stated as follows:

###### Corollary 2.16

Let $(X,D,K)$ be a bicomplete $T_{0}$-quasi-pseudometric type space. Suppose
that $f:X\to X$ is $(C_{\alpha},C_{\beta})$-admissible with respect to $K$.
Assume that

$\alpha(x,y)D(fx,fy)\leq\beta(x,y)D(x,y)\qquad\text{for all }\ \ x,y\in X.$
(9)

If the following conditions hold:

  1. (i)

there exists $x_{0}\in X$ such that
$\min\\{\alpha(fx_{0},x_{0}),\alpha(x_{0},fx_{0})\\}\geq C_{\alpha}$ and
$\min\\{\beta(fx_{0},x_{0}),\beta(x_{0},fx_{0})\\}\leq C_{\beta}$;

  2. (ii)

if $(x_{n})$ is a sequence in $X$ such that
$\min\\{\alpha(x_{m},x_{n}),\alpha(x_{n},x_{m})\\}\geq C_{\alpha}$ and
$\min\\{\beta(x_{m},x_{n}),\beta(x_{n},x_{m})\\}\leq C_{\beta}$ for all
$n,m\in\mathbb{N}$ and $x_{n}\overset{D^{s}}{\longrightarrow}x$, then there
exists a subsequence $(x_{n(k)})$ of $(x_{n})$ such that
$\min\\{\alpha(x,x_{n(k)}),\alpha(x_{n(k)},x)\\}\geq C_{\alpha}$ and
$\min\\{\beta(x,x_{n(k)}),\beta(x_{n(k)},x)\\}\leq C_{\beta}$ for all $k$.

Then $f$ has a fixed point.

We conclude this paper by the following theorem.

###### Theorem 2.17

Let $(X,D,K)$ be a bicomplete $T_{0}$-quasi-pseudometric space. Suppose that
$f:X\to X$ is $(C_{\alpha},C_{\beta})$-admissible with respect to $K$. Assume
that

$\alpha(x,y)D^{s}(fx,fy)\leq\beta(x,y)D^{s}(x,y)\qquad\text{for all }\ \
x,y\in X.$ (10)

If the following conditions hold:

  1. (i)

there exists $x_{0}\in X$ such that $\alpha(x_{0},fx_{0})\geq C_{\alpha}$ and
$\beta(x_{0},fx_{0})\leq C_{\beta}$;

  2. (ii)

if $(x_{n})$ is a sequence in $X$ such that $\alpha(x_{n+1},x_{n})\geq
C_{\alpha}$ and $\beta(x_{n+1},x_{n})\leq C_{\beta}$ for all $n=0,1,2,\cdots$
and $x_{n}\overset{D^{s}}{\longrightarrow}x$, then $\alpha(x_{n},x)\geq
C_{\alpha}$ and $\beta(x_{n},x)\leq C_{\beta}$ for all $n\in\mathbb{N}$.

Then $f$ has a fixed point.

###### Proof. 2.18

Let $x_{0}\in X$ such that $\alpha(x_{0},fx_{0})\geq C_{\alpha}$ and ;
$\beta(x_{0},fx_{0})\leq C_{\beta}$. Define a sequence $(x_{n})$ by
$x_{n}=f^{n}x_{0}=fx_{n-1}$. Without loss of generality, we can always assume
that $x_{n}\neq x_{n+1}$ for all $n\in\mathbb{N}$, since if
$x_{n_{0}}=x_{n_{0}+1}$ for some $n_{0}\in\mathbb{N},$ the proof is complete.
Following the proof of Theorem 2.6, we see that $(x_{n})$ is a $D^{s}$-Cauchy
sequence $\alpha(x_{n+1},x_{n})\geq C_{\alpha}$ and $\beta(x_{n+1},x_{n})\leq
C_{\beta}$ for all $n=0,1,2,\cdots$. Since $(X,D^{s})$ is complete, there
exists $x^{*}$ such that $x_{n}\overset{D^{s}}{\longrightarrow}x^{*}$ and.
Now, using the contractive condition (10) and condition $(ii)$, we deduce that

$\displaystyle D^{s}(x^{*},fx^{*})$ $\displaystyle\leq
K\left[D^{s}(x^{*},x_{n+1})+\frac{C_{\alpha}}{C_{\alpha}}D^{s}(x_{n+1},fx^{*}))\right]$
$\displaystyle\leq
KD^{s}(x^{*},x_{n+1})+\frac{K}{C_{\alpha}}\alpha(x_{n},x^{*})D^{s}(fx_{n},fx^{*})$
$\displaystyle\leq
KD^{s}(x^{*},x_{n+1})+\frac{K}{C_{\alpha}}\beta(x_{n},x^{*})D^{s}(x_{n},x^{*}))$
$\displaystyle\leq
KD^{s}(x^{*},x_{n+1})+\frac{K\beta}{C_{\alpha}}D^{s}(x_{n},x^{*}).$

Letting $n\to\infty$, we obtain that $D^{s}(x^{*},fx^{*})\leq 0,$ that is
$x^{*}=fx^{*}.$ This complete the proof.

## References

  * [1] E. F. Kazeem, C. A. Agyingi and Y. U. Gaba; _On Quasi-Pseudometric Type Spaces_ , Chinese Journal of Mathematics, vol. 2014, Article ID 198685, 7 pages, 2014. doi:10.1155/2014/198685.
  * [2] Y. U. Gaba _Startpoints and $(\alpha,\gamma)$-contractions in quasi-pseudometric spaces_, Journal of Mathematics, Volume 2014 (2014), Article ID 709253, 8 pages, http://dx.doi.org/10.1155/2014/709253.
  * [3] K. Włodarczyk, R. Plebaniak; _Asymmetric structures, discontinuous contractions and iterative approximation of fixed and periodic points_ , Fixed Point Theory and Applications 2013 (2013) 128, 1-18. http://dx.doi.org/10.1186/1687-1812-2013-128.
  * [4] M. A. Khamsi; _Remarks on Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings_ , Fixed Point Theory and Application, volume 2010(2010), 7 pages, doi:10.1155/2010/315398.


# Perfect Entanglement Transport in Quantum Spin Chain Systems

Sujit Sarkar 1\. PoornaPrajna Institute of Scientific Research, 4
Sadashivanagar, Bangalore 5600 80, India

###### Abstract

We propose a mechanism for perfect entanglement transport in anti-
ferromagnetic (AFM) quantum spin chain systems with modulated exchange
coupling along the xy plane and in the z direction. We use the principle of
adiabatic quantum pumping process for entanglement transfer in the spin chain
systems. In our proposed mechanism, perfect entanglement transfer can be
achieved over an arbitraly long distance. We explain analytically and
physically why the entanglement hops in alternate sites. We solve this problem
by using the Berry phase analysis and Abelian bosonization methods. We find
the condition for blocking of entanglement transport even in the perfect
pumping condition. We also explain physically why entanglement transfer in AFM
chain out performs the ferromagnetic chain. Our analytical solution
interconnects quantum many body physics and quantum information science.

1\. Introduction: Quantum communication between distant co-ordinates in a
quantum network is an important requirement for quantum computation and
information. One can construct the quantum network in different ways. Optical
systems typically employed in quantum communication and cryptography
application to transfer the state between two distinct co-ordinates directly
via photons kie ; ski . Quantum computing applications work with trapped atoms
to transfer information between distant sites , photons in cavity QED zeili ;
raus ; sack ; bayer ; plas . However we would like to study the entanglement
transfer through the quantum spin chain systems bose ; chris ; osbo ; bayat ;
venuti ; eckert1 ; eckert2 ; srini ; hartmann ; amico . The equivalence of
state transferring and teleporation of information transmission has already
been studied in the literature horo ; abol . The potentiality of the spin
chain system, antiferromagnetic(AFM) and ferromagnetic(FM), as a network of
quantum state and entanglement transport has already been studied by many
groups as referred in the literature. The experimental evidence of nanoscale
spin chain and their properties have discussed in Ref. hein . Our approach in
this study is different from the existing studies in the literature bose ;
chris ; osbo ; bayat ; venuti ; eckert1 ; eckert2 ; srini ; hartmann ; amico .
The literature of quantum entanglement study is quite vast in quantum
computation science. Here we mention very briefly the important works that
have already existed in state and entanglement transport in the literature:
The authors of Ref. eckert1  have shown explicitly that the quality of state
and entanglement transfer through all phases of spin-$1$ chain have been
possible. Some AFM phases are more efficient than the FM phase. The authors of
Ref. eckert2  have shown explicitly that dimerized AFM states of spin-$1$
chains are also able to transfer through an adiabatic modulation of exchange
couplings. The authors of Ref. venuti , have shown explicitly that the quantum
information can be efficiently transferred between weakly coupled end spins of
an AFM chain because of an effective coupling between the end spins. The
authors of Ref. srini ; hartmann  have studied the quantum state and
entanglement transfer and the authors of Ref. amico  have studied the
entanglement dynamics, considering initial states deviating from the final
states. The authors of Ref. abol ; bose  have studied the entanglement
transfer in a uniformly coupled spin-$1/2$ AFM/FM spin chain. They have
claimed a curious result that for the AFM spin chain, the entanglement hops to
skip alternate sites. They have also found that the entanglement transfer in
the AFM chain outperforms the FM chain. We explain in our work that these
theoretical predictions are natural. Here we mention very briefly the basic
mechanism of entanglement transfer through the spin chain system based on the
conventional wisdom in the literature bose ; chris ; osbo ; bayat ; venuti ;
eckert1 ; eckert2 ; srini ; hartmann ; amico  and at the same time illustrate
the difference with our approach.
It is well known that entanglement is the manifestation of quantum
correlations between two systems when they are inseparable state. We consider
the spin singlet state as an example of an entangled state.

${|{\psi}^{-}>}_{0,0^{\prime}}=\frac{1}{\sqrt{2}}[{|0>}_{0^{\prime}}{|1>}_{0}~{}-~{}{|1>}_{0^{\prime}}{|0>}_{0}]$
(1)

Typically, the sender holds one member of the state of the pair of qubits
while puting the other member at the near end of the AFM spin chain of length
N. The spin chain is in the ground state. When the spin $0$ starts to interact
with the first spin of the chain then the Hamiltonian includes this additional
interaction term ( ${I}_{0^{\prime}}{\otimes}J{{\sigma}_{0}}.{{\sigma}_{1}}$
), where ${{\sigma}_{0}}$ and ${{\sigma}_{1}}$ are the Pauli spin operators
for the $0$ and $1$ sites respectively and $J$ is the exchange coupling). The
initial state being

$|{\psi}{(0)}>={|{\psi}^{-}>}_{0,0^{\prime}}\otimes|{{\psi}_{g}}>$ (2)

Where $|{\psi}_{g}>$ is the ground state wave function of the AFM Hamiltonian
and $|{\psi}(0)>$ is the ground state wave function of the total Hamiltonian.
This initial state starts to evolve and from that one computes the density
matrix and concurrence to measure the entanglement and purity of states. But
our approach is different. Our main motivation is to interconnect the quantum
many body physics and quantum information science. It is common practice in
quantum many body physics to create a particle at any point in the system and
study the dynamics of that particle to understand the physical behaviour of
the system. Therefore, we consider one of the spin ($\uparrow$ or
$\downarrow$) of the singlet interacts with the spin chain and this spin
itself transports through the chain medium due to the adiabatic variation of
exchange couplings of the Hamiltonian, and reaches the other end of the chain.
Our spin chains are the AFM spin chain with the modulated exchange couplings.
But we consider the monogamous nature of the shared entanglement between the
two spins $0$ and $0^{\prime}$. Before we proceed further we would like to
state the basic aspects of adiabatic pumping process: an adiabatic parametric
quantum pump is a device that generates a dc current by a cyclic variation of
system parameters, the variation being slow enough that the system remains
close to the ground state throughout the pumping cycle thou1 ; thou2 . It is
well known that when a quantum mechanical system evolves, it acquires a time
dependent dynamical phase and time independent geometrical phase berry . The
geometrical phase depends on the geometry of the path in the parameter space.
In the adiabatic entanglement pumping process, the locking potential well
carries a spin of the singlet pairs. As the locking potential well slides
through the adiabatic variation of system parameters, it induces a current
($I$) in the system. In this study we calculate the current of this spin
transport, which transports a spin from one end of the chain to the other and
as a result of which entanglement is transported (because the spin
$0^{\prime}$ and $0$ are singlet and monogonus in nature) from one side to the
other. In our study this entanglement transport is the perfect because the the
adiabatic pumping physics based on Berry phase analysis is topologically
protected against the external perturbations thou1 ; thou2 ; shin .
Here we consider two different Hamiltonian, $H_{1}$ and $H_{2}$ with modulated
exchange coupling in $xy$ and $z$ directions respectively, Hamiltonians of the
systems are the following

$\displaystyle{H_{1}}$ $\displaystyle=$
$\displaystyle-\sum_{n}J(1-(-1)^{n}{{\delta}_{1}}(t))({{S}_{+}}^{n}{{S}_{-}}^{n+1}+{{S}_{+}}^{n+1}{{S}_{-}}^{n})$
(3) $\displaystyle+\sum_{n}{\Delta}{{S}_{z}}^{n}{{S}_{z}}^{n+1}$

This model Hamiltonian has some experimental relevance shin . The other model
Hamiltonian is

$\displaystyle{H_{2}}$ $\displaystyle=$
$\displaystyle-\sum_{n}J({{S}_{x}}^{n}{{S}_{x}}^{n+1}+{{S}_{y}}^{n}{{S}_{y}}^{n+1})$
(4)
$\displaystyle+\sum_{n}{\Delta}{{S}_{z}}^{n}{{S}_{z}}^{n+1}-\frac{1}{2}\sum_{n}{B_{0}}(1-(-1)^{n}{{\delta}_{2}}(t)){{S}_{z}}^{n}$

Here we consider that the fluctuations is periodic over two lattice sites. We
see that this model have essential ingredients to capture the adiabatic
entanglement pumping. One can express spin chain systems to a spinless fermion
systems through the application of Jordan-Wigner transformation. In Jordan-
Wigner transformation the relation between the spin and the electron creation
and annihilation operators are $S_{n}^{z}=\psi_{n}^{\dagger}\psi_{n}-1/2~{}$,
$S_{n}^{-}=\psi_{n}~{}\exp[i\pi\sum_{j=-\infty}^{n-1}n_{j}]~{}$,
$S_{n}^{+}=\psi_{n}^{\dagger}~{}\exp[-i\pi\sum_{j=-\infty}^{n-1}n_{j}]~{}$,
gia2 , where $n_{j}=\psi_{j}^{\dagger}\psi_{j}$ is the fermion number at site
$j$. Spin operators in terms of bosonic field are the following.

$\displaystyle S_{n}^{x}~{}$ $\displaystyle=$
$\displaystyle~{}[~{}c_{2}\cos(2{\sqrt{\pi
K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\cos({\sqrt{\frac{\pi}{K}}}\theta),$
$\displaystyle S_{n}^{y}~{}$ $\displaystyle=$
$\displaystyle~{}-[~{}c_{2}\cos(2{\sqrt{\pi
K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\sin({\sqrt{\frac{\pi}{K}}}\theta),$
$\displaystyle S_{n}^{z}~{}$ $\displaystyle=$
$\displaystyle~{}{\sqrt{\frac{\pi}{K}}}~{}\partial_{x}\phi~{}+~{}(-1)^{n}c_{1}\cos(2{\sqrt{\pi
K}}\phi)~{},$ (5)
${{\psi}_{r}}(x)~{}=~{}~{}\frac{U_{r}}{\sqrt{2\pi\alpha}}~{}~{}e^{-i~{}(r\phi(x)~{}-~{}\theta(x))}$
(6)

$r$ denotes the chirality of the fermionic fields, right (1) or left movers
(-1). The operators $U_{r}$ are operators that commute with the bosonic field.
$U_{r}$ of different species commute and $U_{r}$ of the same species
anticommute. $\phi$ field corresponds to the quantum fluctuations (bosonic) of
spin and $\theta$ is the dual field of $\phi$. They are related by this
relation ${\phi}_{R}~{}=~{}~{}\theta~{}-~{}\phi$ and
${\phi}_{L}~{}=~{}~{}\theta~{}+~{}\phi$.

Using the standard machinery of continuum field theory gia2 , we finally get
the bosonized Hamiltonians as $H_{0}$ is the gapless Tomonoga-Luttinger liquid
part of the Hamiltonian.
After the application of continuum field-theory the Hamiltonian become, in
terms of bosonic fields.

$\displaystyle{H_{1}}$ $\displaystyle=$
$\displaystyle{H_{0}}+\frac{{E_{J_{0}}}{{\delta}_{1}}(t)}{2{{\pi}^{2}}{\alpha}^{2}}\int
dx:cos[2\sqrt{K}{\phi}(x)]:$ (7)
$\displaystyle+\frac{\Delta}{2{{\pi}^{2}}{\alpha}^{2}}\int
dx:cos[4\sqrt{K}{\phi}(x)]:$ $\displaystyle{H_{2}}$ $\displaystyle=$
$\displaystyle{H_{0}}+\frac{{B_{0}}{{\delta}_{2}}(t)}{2{\pi}{\alpha}}\int
dx:cos[2\sqrt{K}{\phi}(x)]:$ (8)
$\displaystyle+\frac{\Delta}{2{{\pi}^{2}}{\alpha}^{2}}\int
dx:cos[4\sqrt{K}{\phi}(x)]:-\frac{B_{0}}{2}\int dx{{\partial}_{x}}{\phi}(x)$

Here, we would like to explain the basic aspects of quantum entanglement
pumping in terms of spin pumping physics of our model Hamiltonians: An
adiabatic sliding motion of one dimensional potential, in gapped Fermi surface
(insulating state), pumps an integer numbers of particle per cycle. In our
case the transport of Jordan-Wigner fermions (spinless fermions) is nothing
but the transport of spin from one end of the chain to the other end because
the number operator of spinless fermions is related to the z-component of spin
density cal . We see that non-zero ${{\delta}_{1}}(t)$ and ${{\delta}_{2}}(t)$
introduce the gap at around the Fermi point and the system is in the
insulating state (Peierls insulator). In this phase spinless fermions form the
bonding orbital between the neighboring sites, which yields a valance band in
the momentum space. It is well known that the physical behavior of the system
is identical at these two Fermi points. We would like to analyse these double
degeneracy point, following the seminal paper of Berry berry : in our model
Hamiltonian there are two adiabatic parameters ${{\delta}_{1}}(t)$ and
${{\delta}_{2}}(t)$. The Hamiltonian starts to evolve under the variation of
these two adiabatic parameters, when the Hamiltonian returns to its original
form after a time $T$, the total geometric phase acquired by the system is
${{\gamma}_{n}}(T)~{}=~{}\frac{i}{2\pi}\int_{C}<{{\psi}_{n}}|{{\nabla}_{R}}|{\psi}_{n}>~{}dR$,
a line integral around a closed loop in two dimensional parameter space. Using
Stokes theorem, one can write
${{\gamma}_{n}}(T)~{}=~{}\frac{i}{2\pi}\int{{\nabla}_{R}}\times<{{\psi}_{n}}|{{\nabla}_{R}}|{\psi}_{n}>~{}dS$.
The flux $\Phi$ through a closed surface C is, $\Phi=\int B.dS$. Therefore one
can think of the Berry phase as flux of a magnetic field. Now we express,
${B_{n}}(K1)={{\nabla}_{K1}}\times{A_{n}}(K1)$, and
${A_{n}}(K1)=\frac{i}{2\pi}<n(K1)|{{\nabla}_{K1}}|n(K1)>$, where
$K1=(k,{\delta}_{1}(t),{\delta}_{2}(t))$. Here $B_{n}$ and $A_{n}$ are the
fictitious magnetic field (flux) and vector potential of the nth Bloch band
respectively. The degenerate points behave as a magnetic monopole in the
generalized momentum space ($K_{1}$) berry , whose magnetic unit can be shown
to be $1$, analytically shin ; berry

$\int_{S1}~{}dS\cdot B_{\pm}~{}=~{}\pm 1$ (9)

positive and negative signs of the above equations are respectively for the
conduction and valance band meet at the degeneracy points. $S_{1}$ represent
an arbitrary closed surface which enclose the degeneracy point. In the
adiabatic process the parameter ${{\delta}_{1}}(t)$ or ${{\delta}_{2}}(t)$ are
changed along a loop ($\Gamma$) enclosing the origin (minima of the system).
We define the expression for spin current ($I$) from the analysis of Berry
phase. It is well known in the literature of adiabatic quantum pumping physics
that two independent parameters are needed to achieve the adiabatic quantum
pumping in a system sharma . Here one may consider these two parameters as the
real and imaginary part of the fourier transform of a modulated coupling
induce potential. When the shape of the potential will change in time, then it
amounts to changing the phase and amplitude in time. The role of adiabatic
parameters are not explicit in our study. Our formalism is different from
others. We define the expression for spin current ($I$) from the analysis of
Berry phase. Then according to the original idea of quantum adiabatic particle
transport thou1 ; thou2 ; shin ; avron , the total number of spinless fermions
($I$) which are transported from one side of this system to the other is equal
to the total flux of the valance band, which penetrates the 2D closed sphere
($S_{2}$) spanned by the $\Gamma$ and Brillioun zone shin .

$I=\int_{S_{2}}dS\cdot B_{+1}~{}=1$ (10)

$B_{+1}$ is directly related with the Berry phase (${{\gamma}_{n}}(T)$) which
is acquired by the system during the adiabatic variation of the exchange
couplings the time period of the adiabatic process. This quantization is
topologically protected against the other perturbation as long as the gap
along the loop remains finite shin ; avron . Therefore the adiabatic
entanglement pumping is constant over the arbitrarily long distance of the
system. This result is in contrast with the existed results in the literature
[8,19]. They have found that the entanglement decay exponentially after a
certain distance.
Now we explain the quantum entanglement transfer for $H_{1}$. The second term
of the Hamiltonian for NN exchange interaction has originated from the $x$ and
$y$ component of exchange interaction. This term implies that infinitesimal
variation of coupling in lattice sites, is sufficient to produce a gap around
the Fermi points. So when ${1/2}<K<1$, only these time dependent exchange
couplings are relevant and lock the phase operator at
${\phi}=0+\frac{n\pi}{\sqrt{K}}$. Now the locking potential slides
adiabatically. The speed of the sliding potential is low enough such that the
system stays in the same valley, i.e., there is no scope to jump onto the
other valley. The system will acquire $2\pi$ phase during one complete cycle
of adiabatic process. This expection is easily verified when we notice the
physical meaning of the phase operator ($\phi$ (x)). Since the spatial
derivative of the phase operator corresponds to the z-component of spin
density, this phase operator is nothing but the minus of the spatial
polarization of the z-component of spin, i.e.,
$P_{s^{z}}~{}=-\frac{1}{N}\sum_{j=1}^{N}j{S_{j}}^{z}$. Shindou has shown
explicitly the equivalence between these two considerations shin . During the
adiabatic process $<{\phi}_{t}>$ changes monotonically and acquires \- $2\pi$
phase. In this process ${P_{s}}^{z}$ increases by 1 per cycle. We define it
analytically as

${\delta}{P_{s}}^{z}=\int_{\Gamma}d{P_{s}}^{z}=-\frac{1}{2\pi}\int
dx{{\partial}_{x}}<{\phi}(x)>=1$ (11)

This physics always hold as far as the system is locked by the sliding
potential and ${\Delta}<1$ shin . The change of the spatial polarization by
unity during a complete evaluation of adiabatic cycle implies that the
transport of entanglement across the system. This is because the spatial
derivative of the phase operator is the Cooper pair density in our system. The
entanglement transport of this scenario can be generalized up to the value of
$\Delta$ for which $K$ is greater than 1/2 . In this limit, z-component of the
exchange interaction has no effect on the entanglement pumping of our system.
But when $K<1/2$ , then the interaction due to $\Delta$ becomes relevant and
creates a gap in the excitation spectrum. This potential profile is static.
Therefore there is no scope to slide the potential and to get a adiabatic
pumping across the system. The authors of Ref. abol ; bose  have also found
that when ${\Delta}>1$ for $XXZ$ AFM spin chain, the fidelity of AFM spin
chain also decreases ,i.e., the entanglement transport decreases in this
limit.
Similarly for the Hamiltonian $H_{2}$, the second term of the Hamiltonian
produce the gap and the pumping process is the same as that of $H_{1}$.
Therefore we conclude that the modulations in the in plane exchange coupling
and also for the modulations in the z-directions yield the same adiabatic
entanglement pumping.
In this pumping process the most favourable states of the system are the
antiferromagnetic configuration $|010101....>$ and $|101010,,,,>$ ($0$ stands
for up spin and $1$ stands for down spin). One may start from any
antiferromagnetic states and transfer the spin of every site to the right by
two sites to achieve the pumping. Therefore our test spin which we introduce
at the one end of the spin, it hops to the right by two sites in every step.
Thus when we study the entanglement transport between the spin $0^{\prime}$
and $0$, then it is natural that entanglement also is transported through
every alternate sites. The authors of Ref. abol ; bose  have observed a very
peculiar behaviour of entanglement transfer for AFM: the nonanalytical
behaviour as a function of time. It is zero for most of the time and it
suddenly grows up and forms a peak at a regular interval of time. But in our
study the entanglement current is constant and it is almost perfect
entanglement pumping. In their case the spin chain has the spin rotational
symmetry. When one member of an entangled pair of qubits is transmitted
through such a channel , then the two qubits states evolve to a Werner state
benn . But our spin chain systems there is no spin rotational invariant
symmetry and the transport mechanism is also different. The physical scenario
of our study is completely different from the existing physical picture. The
quantized entanglement transport of this scenario can be generalized up to the
value of $\Delta$ for which $K$ is greater than 1/2. In this limit, the
z-component of the exchange interaction has no effect on the entanglement
pumping physics of Hamiltonian. . In this limit, z-component of the exchange
interaction has no effect on the entanglement pumping of our system.
Here, we would like to explain the difference of entanglement transport
between the FM and AFM spin chain, it has mentioned in the literature but the
complete physical explanation is not upto the mark bose ; chris ; osbo ; bayat
; venuti ; eckert1 ; eckert2 ; srini ; hartmann ; amico ; abol . As we know
that entanglement is a quantum mechanical property, Schrodinger singled out
many decades ago as ”the characteristic of quantum mechanics sch  and that has
been studied extensively in connection with Bell’s inequality bell . FM ground
state state there is no difference between the classical and quantum
mechanical ground state and the low lying excitations are spin-1 magnons. The
AFM ground state has a complex structure specified by the Bethe-ansatz
solution. There are no similarities between classical and quantum mechanical
ground state and first excited state of the AFM chain and as a result of the
quantum mechanical property of the system the entanglement manifests
prominently in the AFM spin chain. This is the only clear reason why AFM
outperforms the FM spin chain.
Here we discuss possible sources of imperfections in the entanglement pumping
process. The non-adiabatic contributions leave the system in an unknown
superposition of states after the full cycle. Also the appearance of Landau-
Zener transition in the pumping system should be negligible so that the system
is in the ground state. This condition limits the pumping rate of entanglement
by the mathematical relation $\frac{h}{\tau}<<J$. However even then the
entanglement pumping is not perfect due to the non vanishing
$\frac{J}{\Delta}$. Our effort also should take the elimination of
entanglement pumping in the wrong directions. The residual exchange coupling
may lead to a different spin state. An entangled spin transported through a
correct exchange coupling modulation with probability $P$ and through the
residual exchange coupling with the probability $Q=1-P$. Therefore the pumping
error in each site is $\frac{P}{Q}$. Our system consists of $N$ sites.
Therefore the probability of correct entanglement transport is $\sim{P^{N/2}}$
and wrong entanglement transport is $\sim{Q^{N/2}}$. The total pumping error,
$({\frac{Q}{P})}^{N/2}$, decreases with the number of sites in nanoscale spin
chain. Therefore for the spin chain system entanglement transport is better
for larger length compare to the smaller length with same exchange couplings.
Conclusions: we have presented the theoretical explanation of adiabatic
entanglement pumping for our model Hamiltonians. We have found the perfect
entanglement transport condition which cure the existed results in the
literature. We have explained few physical findings of entanglement transport
which were curious before this study.
The author would like to thank, The Center for Condensed Matter Theory of IISc
for extended facility. Finally the author would like to thank Prof. R.
Srikanth, Dr. T. Tulsi and Prof. Indrani Bose.

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