Advances in Mathematical Physics

Advances in Mathematical Physics / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 3295462 | 15 pages | https://doi.org/10.1155/2019/3295462

Derivations and Deformations of -Jordan Lie Supertriple Systems

Academic Editor: Francesco Toppan
Received02 Apr 2019
Accepted16 Jun 2019
Published11 Jul 2019

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 [311].

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, 1517]. 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