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 The following commutative diagram holds:
The sufficiency of is an isomorphism that is proven.
If , then ; that is, and ; then, ; hence, is injective; is obviously surjective. Note that
The equivalence of and is proven.
4. Nijenhuis Operators of Hom--Jordan Lie Triple Systems
In this section, the deformation of Hom--Jordan Lie triple systems is studied. The notion of Nijenhuis operators of Hom--Jordan Lie triple systems is introduced, and the trivial deformations of this kind of operators are shown.
Let be a Hom--Jordan Lie triple system and be a trilinear mapping. Consider a -parametrized family of linear operations: where is a formal variable.
We call that generates a -parameter infinitesimal deformation of the Hom--Jordan Lie triple system, if endow with the Hom--Jordan Lie triple system structure which is denoted by . (i) itself defines a Hom--Jordan Lie triple system structure on (ii) is a -cocycle of
Theorem 8. generates a -parameter infinitesimal deformation of the Hom--Jordan Lie triple system ; then, the following two conclusions hold:
Proof. We have From the equality it follows that For the equality the left hand side is equal to and the right hand side is equal to Thus, we have Therefore, defines a Hom--Jordan Lie triple system structure on by (36), (38), and (43). Furthermore, by (42), is a -cocycle.
A deformation is called trivial if there exists a linear map such that for ,
It is clear that
Thus, we have
By the cohomologies discussed in Section 2, equation (46) can be represented in terms of -coboundary as . Furthermore, the following condition holds for by (46) and (47):
In the following, we denote by then, (47) is equivalent to 51
Definition 9. A linear operator is said to be a Nijenhuis operator if and only if (48) and (49) hold.
Theorem 10. Let be a Nijenhuis operator for . Then, a deformation of can be obtained if Moreover, is a trivial deformation.
Proof. Clearly, and . Then, is a -cocycle of . In the following, we show that (4) holds for . By (46), (50), and (51), we have Similarly, a direct computation shows that
Note that ; by (4), (51), and Theorem 8, it follows that
The conclusion of Theorem 10 is proven.
Remark 11. Let be a Nijenhuis operator. If , then
Remark 12. Let be a Nijenhuis operator; by mathematical induction and (48), for any , is also a Nijenhuis operator.
Data Availability
The data used to support the finding of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the NNSF of China (No. 11801211), the Science Foundation of Heilongjiang Province (No. QC2016008), and the Fundamental Research Funds in Heilongjiang Provincial Universities (No. 145109128).