#### Abstract

In this paper, the equivalence of central extensions and is proven in the study in Hom--Jordan Lie triple systems. The concepts of Nijenhuis operators of Hom--Jordan Lie triple systems are given. Moreover, a trivial deformation is got.

#### 1. Introduction

It is well known that Lie triple systems are closely related to geometry. In a symmetric space, its tangent algebra is a Lie triple system. The definitions of the semisimplicity, radicality, and solvability for Lie triple systems are discussed, and the simple Lie triple system is determined by Lister [1]. Cohomologies of Lie triple systems were obtained [2]. Kubo and Taniguchi showed that in Lie triple systems, this kind of cohomology plays an important role in the study of deformations and extensions in 2004 [3]. A generalization of Lie triple systems and -Jordan Lie triple systems was defined in [4] by Okubo and Kamiya, where The case of yields the Lie triple system, and they call the other case of a Jordan Lie triple system. Then, they obtained a method to construct simple Jordan superalgebras by certain triple systems and studied the -type Jordan superalgebra of a Jordan Lie triple system [5]. Recently, the cohomologies, Nijenhuis operators, representations, abelian extensions, and -extensions of -Jordan Lie triple systems were developed by Ma and Chen [6].

The theory of Hom-type algebras has been studied (see [7–16]). In 2012, Yau showed the concept of Hom-Lie triple systems [17]. Later, generalized derivations of Hom-Lie triple systems were determined [18]. The cohomologies, -parameter formal deformations, and central extensions of Hom-Lie triple systems were discussed [19]. In 2019, the generalization of -Jordan Lie triple systems and Hom-Lie triple systems, -parameter formal deformations, and cohomologies of Hom--Jordan Lie triple systems were studied [20]. We pay our main attention to consider central extensions and Nijenhuis operators of Hom--Jordan Lie triple systems.

The paper is organized as follows. In Section 2, we summarize basic concepts and construct a structure of multiplicative Hom--Jordan Lie triple systems. In Section 3, the equivalence of the third cohomology group and the central extensions of a Hom--Jordan Lie triple system is proven. We discuss Nijenhuis operators of Hom--Jordan Lie triple systems and obtain a trivial deformation using a Nijenhuis operator in Section 4.

In this paper, the capital letter denotes an arbitrary field.

#### 2. Preliminaries

We start by recalling the definition of Hom-Lie triple systems.

*Definition 1 [17]. *A Hom-Lie triple system consists of an -vector space , a trilinear map , and linear maps for , called twisted maps, such that for all ,

*Definition 2 [20]. *A Hom--Jordan Lie triple system consists of an -vector space , a trilinear map , and linear maps for , called twisted maps, such that for all ,

*Remark 3. *When , a Hom--Jordan Lie triple system is a Hom-Lie triple system. A Hom--Jordan Lie triple system is a -Jordan Lie triple system if the twisted maps are both equal to the identity map. So Hom-Lie triple systems and -Jordan Lie triple systems are special examples of Hom--Jordan Lie triple systems.

A Hom--Jordan Lie triple system is said to be multiplicative if and and denoted by .

A morphism of the Hom--Jordan Lie triple system is a linear map satisfying and for . An isomorphism is a bijective morphism.

*Definition 4 [20]. *Let be multiplicative Hom--Jordan Lie triple systems, be an -vector space, and . is said to be a -module with respect to if there exists a bilinear map , such that for all ,
where .

Then, is said to be the representation of on with respect to . In the case , is said to be the trivial -module with respect to .

In the case that , i.e., Hom--Jordan Lie triple systems are Hom-Lie triple systems, we can get (8) from (7) by a direct calculation. But it is not true in the other case

Particularly, and (5), (6), and (7) hold, if , , and . In this case, is called the adjoint -module and is called the adjoint representation of on itself with respect to .

In the following, the semidirect product of multiplicative Hom--Jordan Lie triple systems and its module for general algebras were introduced.

Proposition 5. *Assume that is a multiplicative Hom--Jordan Lie triple system on with respect to and is a representation of Then, has a structure of a multiplicative Hom--Jordan Lie triple system.*

*Proof. *We define the operation by and define the twisted map by
By , we get
By (6), (7), and (8), we have

The calculation above shows that (2), (3), and (4) hold.

By (5) and the linearity of ,

Hence, is a multiplicative Hom--Jordan Lie triple system.

Suppose that is an -linear map, which satisfies where is said to be an -Hom-cochain on . The set of all -Hom-cochains is denoted by , for all . (i)If , then (ii)If , then (iii)If , then (iv)If , then

*Definition 6 [20]. *For , the coboundary operator is defined as follows.

The mapping is said to be an -Hom-cocycle if . Denote by the subspace spanned by -Hom-cocycles. For , .

Since , . Define a cohomology space:

#### 3. Central Extensions of Hom--Jordan Lie Triple Systems

Let be a multiplicative Hom--Jordan Lie triple system and be a trivial -module with respect to . Then, is an abelian multiplicative Hom--Jordan Lie triple system with the trivial product. A multiplicative Hom--Jordan Lie triple system is said to be a central extension of by if the following commutative diagram holds with the exact rows of Hom--Jordan Lie triple systems.

where , is a linear map satisfying and , and . Two central extensions and are equivalent, if the following commutative diagram holds.

where is an isomorphism.

Theorem 7. *There is bijective mapping between and equivalent classes of central extensions of by .*

*Proof. *First, we show that there is a bijective mapping between and central extensions of by

Suppose that is a central extension of by . Then, the following commutative diagram holds:

with , , and a linear map satisfying and .

For , since , it follows that . Define a trilinear map by

Since is injective, is well defined, and it follows from that

Note that satisfies and

Hence, . Moreover, since

On the other hand, let and with

Thus, is linear, and

Since that

We have which is a multiplicative Hom--Jordan Lie triple system.

Define three mappings , , and by , , and , respectively. Then,

It is clear that is a subspace of . Hence, is a central extension of by .

Assume that and are equivalent central extensions of by . Then, the following commutative diagram holds:

such that and with an isomorphism and . For their corresponding 3-Hom-cocycles and as above, we have

We have . In fact, since there exists a linear mapping by , for all . Then, that is, . By ,

Then, so .

Suppose and ; i.e., there is satisfying . Then, . Let and , which are defined as above with respect to and , be two central extensions of by . Then, and . There is a linear map: such that and