Abstract

Let be a -Jordan Lie supertriple system. We first introduce the notions of generalized derivations and representations of and present some properties. Also, we study the low-dimensional cohomology and the coboundary operator of , and then we investigate the deformations and Nijenhuis operators of by choosing some suitable cohomologies.

1. Introduction

Lie triple systems arose initially in Cartan’s study of Riemannian geometry. Jacobson [1] first introduced Lie triple systems and Jordan triple systems in connection with problems from Jordan theory and quantum mechanics, viewing Lie triple systems as subspaces of Lie algebras that are closed relative to the ternary product. Lister [2] investigated notions of the radical, semisimplicity and solvability as defined for Lie triple systems and determined all simple Lie triple systems over an algebraically closed field. Later, the representation theory, the central extension, the deformation theory, bilinear forms, and the generalized derivation of Lie triple systems and Jordan triple systems have been developed; see [3–11].

In [12], Okubo and Kamiya introduced the notion of -Jordan Lie triple system, where , which is a generalization of both Lie triple systems () and Jordan Lie triple systems (). Later, Kamiya and Okubo [13] studied a construction of simple Jordan superalgebras from certain triple systems. Recently, Ma and Chen [14] discussed the cohomology theory, the deformations, Nijenhuis operators, abelian extensions, and -extensions of -Jordan Lie triple system.

As a natural generalization of Lie triple systems, Okubo [15] introduced the notion of Lie supertriple systems in the study of Yang-Baxter equations. Lie supertriple systems have many applications in high energy physics, and many important results on Lie supertriple systems have been obtained; see [13, 15–17]. In [12], Okubo and Kamiya introduced the notion of -Jordan Lie supertriple system (they still call it Jordan Lie triple system); they presented some nontrivial examples and discussed their quasiclassical property. In the present paper, we hope to study generalized derivations, cohomology theories, and deformations of -Jordan Lie supertriple systems.

This paper is organized as follows. In Section 2, we recall the definition of -Jordan Lie supertriple systems and construct a kind of -Jordan Lie supertriple systems. Also, we study generalized derivation algebra of a -Jordan Lie supertriple system. In Section 3, we introduce notions of the representation and low-dimensional cohomology of a -Jordan Lie supertriple system. In Section 4, we consider the theory of deformations of a -Jordan Lie supertriple system by choosing a suitable cohomology. In Section 5, we study Nijenhuis operators for a -Jordan Lie supertriple system to describe trivial deformations.

2. Generalized Derivations of -Jordan Lie Supertriple Systems

In this section, we start by recalling the definition of -Jordan Lie supertriple systems and then we study its generalized derivations.

Definition 1 ([12]). A -Jordan Lie supertriple system is a -graded vector space together with a triple linear product satisfyingfor all , where and denotes the degree of the element .

Remark 2. Clearly, is an ordinary -Jordan Lie triple system in [14]. The case of defines, especially, a Lie supertriple system while the other case of may be termed an anti-Lie supertriple system as in [18].

Example 3 ([12]). Let be a -Jordan Lie superalgebra. Then becomes a -Jordan Lie supertriple system, where , for all .

Example 4. Let be a -Jordan Lie supertriple system and an indeterminate. Set ; then is a -Jordan Lie supertriple system with a triple linear product defined by for all , where

Definition 5. Let be a -Jordn Lie supertriple system and a nonnegative integer. A homogeneous linear map is said to be a homogeneous -derivation of if it satisfiesfor all , where denotes the degree of .
We denote by , where is the set of all homogeneous -derivations of . Obviously, is a subalgebra of and has a normal Lie superalgebra structure via the bracket product

Definition 6. Let be a -Jordan Lie supertriple system and a nonnegative integer. is said to be a homogeneous generalized -derivation of , if there exist three endomorphisms such thatfor all .

Definition 7. Let be a -Jordan Lie supertriple system and a nonnegative integer. is said to be a homogeneous -quasiderivation of , if there exists an endomorphism such thatfor all .
Let and be the sets of homogeneous generalized -derivations and of homogeneous -quasiderivations, respectively. That is,

Definition 8. Let be a -Jordan Lie supertriple system and a nonnegative integer. The -centroid of is the space of linear transformations on given byWe denote and call it the centroid of .

Definition 9. Let be a -Jordan Lie supertriple system. The quasicentroid of is the space of linear transformations on given by

Remark 10. Let be a -Jordan Lie supertriple system. Then
For any and , we haveIn fact, by the definition of the -Jordan Lie supertriple system, we have Similarly, we have It follows that

Definition 11. Let be a -Jordan Lie supertriple system. is said to be a central derivation of iffor all . Denote the set of all central derivations by .

Remark 12. Let be a -Jordan Lie supertriple system. Then

Definition 13. Let be a -Jordan Lie supertriple system. If , then is called the center of .

Proposition 14. Let be a -Jordan Lie supertriple system; then the following statements hold:(1), and are subalgebras of .(2) is an ideal of .

Proof. We only prove that is a subalgebra of , and similarly for cases of and . For any and , we have Similarly, we have It follows that Obviously, and are contained in ; thus ; that is, is a subalgebra of .
For any , and , we have Also, we have It follows that . That is, is an ideal of .

Proposition 15. Let be a -Jordan Lie supertriple system; then the following statements hold:(1).(2).(3).(4).

Proof. For any , and , we have Similarly, one can check that It follows that ; thus .
Similar to the proof of (1).
For any and , we have Since , we have Similarly, one may check that It follows that Therefore, and .
For any and , we have Thus that is, and .

Theorem 16. Let be a -Jordan Lie supertriple system; then . Moreover, if , then .

Proof. For any , and , we have So and therefore . Moreover, if , then it is easy to see that .

Theorem 17. Let be a -Jordan Lie supertriple system and ; then .

Proof. For any and , we have It follows that . Thus since ; that is, . So .
On the other hand, for any and , we have . Clearly, (6) and (11) hold; that is, , and, therefore, . And this completes the proof.

3. The Cohomology of -Jordan Lie Supertriple Systems

In this section, we introduce the notion of the representation of -Jordan Lie supertriple systems and present its low-dimensional cohomologies.

Definition 18. Let be a -Jordan Lie supertriple system and a -graded vector space. Suppose that there exists a bilinear mapping satisfying the following axioms:for , where ; then is called the representation of ; is called a -module.

Example 19. Let be a -Jordan Lie supertriple system. Define by . It is not hard to check that and itself is a -module. In this case, is said to be the adjoint representation of .

Proposition 20. Let be a -Jordan Lie supertriple system and the representation. Then has a structure of a -Jordan Lie supertriple system.

Proof. Define a triple linear product by for all , where .
Now we check that the operation defined above satisfies axioms in Definition 1. It is easy to see that (1) holds since is a -Jordan Lie supertriple system.
For (2), we take any and compute The last equality holds since
For (3), we have where The second equality holds since . Then we have as desired.
For (4), we take any . First, we calculate the following expression: where Second, we compute the expression : where Third, we compute the expression : where Fourth, we compute the expression : where Finally, by (33), (34), and (35), we have as desired, and this finishes the proof.

Corollary 21. Any -Jordan Lie supertriple system can be considered as a subspace of a -Jordan Lie superalgebra.

Proof. It is straightforward from Example 19 and Proposition 20.

Definition 22. Let be a -Jordan Lie supertriple system and a -module by a bilinear map . If an -linear map satisfies the following axioms(1);(2), then is called an -cochain on . Denote by the set of all -cochains, for .

Definition 23. Let be a -Jordan Lie supertriple system and a -module by a bilinear map . For , the coboundary operator is defined as follows:(i)If , then (ii)If , then (iii)If , then (iv)If , then

Theorem 24. Let be a -Jordan Lie supertriple system and a -module by a bilinear map . The coboundary operator defined above satisfies , .

Proof. From the definition of the coboundary operator, it follows immediately that implies . Then, we only need to prove :By Definition 23, we haveBy (62)–(68), we have where It follows that , as desired. And this finishes the proof.

For , the map is called an -cocycle if . We denote by the subspace spanned by -cocycles and . By Theorem 24, is a subspace of . Therefore, we can define a cohomology space of the -Jordn Lie supertriple system as the factor space

4. -Parameter Formal Deformations of -Jordan Lie Supertriple Systems

Let be a -Jordan Lie supertriple system and the power series ring in one variable with coefficients in . Assume that is the set of formal series whose coefficients are elements of the vector space .

Definition 25. Let be a -Jordan Lie supertriple system. A 1-parameter formal deformation of is a formal power series given by where each is a -trilinear map (extended to be -trilinear) and , satisfying the following axioms:

Remark 26. Equations (72)–(75) are equivalent to ()Furthermore, we can rewrite the deformation Equation (79) by the equality , where When , (79) is equivalent to . When , (79) is equivalent to

By Example 19, is the adjoint representation of itself, where and . It is easy to see that and therefore . Since , we have Similarly, we have It follows that Therefore, we deduce since . Also we can obtain And is called the infinitesimal deformation of .

Definition 27. Let be a -Jordan Lie supertriple system. Two 1-parameter formal deformations and of are said to be equivalent, denoted by , if there exists a formal isomorphism of -modules where , is an linear map (extended to be linear) such that

In particular, if , then is called the null deformation. If , then is called the trivial deformation. If every 1-parameter formal deformation is trivial, then is called an analytically rigid -Jordan Lie supertriple system.

Theorem 28. Let and be two equivalent 1-parameter formal deformations of . Then the infinitesimal deformations and belong to the same cohomology class in

Proof. By the assumption that and are equivalent, there exists a formal isomorphism of -modules satisfying for any Compared with the coefficients of for two sides of the above equation, we have It follows that So , as desired. The proof is completed.

Theorem 29. Let be a -Jordan Lie supertriple system with ; then is analytically rigid.

Proof. Let be a 1-parameter formal deformation of . Then By the assumption , we have ; that is, there exits such that .
Set ; then Similarly, one may check that So is a linear isomorphism. Thus we can define another 1-parameter formal deformation by in the form of Obviously, . Set ; then we have By the above equation, it follows that Therefore, we deduce It follows that . By induction, we have ; that is, is analytically rigid. The proof is finished.

5. Nijenhuis Operators of -Jordan Lie Supertriple Systems

In this section, we introduce the notion of Nijenhuis operators for -Jordan Lie supertriple systems. Also, we give trivial deformations of this kind of operators.

Let be a -Jordan Lie supertriple system and let be an even trilinear map. Consider a -parametrized family of linear operations: for any , where is a formal variable.

If endow with a -Jordan Lie supertriple system structure which is denoted by , then we call that generates a -parameter infinitesimal deformation of the -Jordan Lie supertriple system .

Theorem 30. Let be a -Jordan Lie supertriple system. Then is a -Jordan Lie supertriple system if and only if(i) itself defines a -Jordan Lie supertriple system structure on ;(ii) is a 3-cocycle of .

Proof. Assume that is a -Jordan Lie supertriple system. For any , we have It follows thatFor (3), we have as desired. The last equality holds since and are both -Jordan Lie supertriple systems.
For (3), we take and calculate By similar calculation, we have It follows that Therefore, we haveBy (102), satisfies (3). So defines a -Jordan Lie supertriple system structure on .
Since and , we can rewrite (97) as follows:The last equality holds since is an even trilinear map. So , as required.
Conversely, if satisfies conditions (i) and (ii), it is easy to see that is a -Jordan Lie supertriple system from (96), (97), (101), and (102).

Definition 31. A deformation is said to be trivial if there exists a linear map such that, for all , satisfiesfor any .

The left-hand side of (104) is equal to

The right-hand side of (104) is equal to

Therefore, by (104), we haveBy (108) and (109), we can deduce that

Definition 32. A linear operator is called a Nijenhuis operator if and only if (107) and (110) hold.

Theorem 33. Let be a Nijenhuis operator for . Then, a deformation of can be obtained by putting Moreover, this deformation is trivial.

Proof. The proof is similar to one in the setting of -Jordan Lie triple system in [14].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The paper is partially supported by the Anhui Provincial Natural Science Foundation (nos. 1908085MA03 and 1808085MA14), the outstanding top-notch talent cultivation project of Anhui Province (no. gxfx2017123), the NSF of China (nos. 11761017 and 11801304), the Youth Project for Natural Science Foundation of Guizhou Provincial Department of Education (no. KY155), and the Project funded by China Postdoctoral Science Foundation (no. 2018M630768).