On Generalized Jordan Prederivations and Generalized Prederivations of Lie Superalgebras

Abstract

The concepts of (generalized) -prederivations and (generalized) Jordan -prederivations on a Lie superalgebra are introduced. It is proved that Jordan -prederivations (resp., generalized Jordan -prederivations) are -prederivations (resp., generalized -prederivations) on a Lie superalgebra under some conditions. In particular, Jordan -prederivations are -prederivations on a Lie superalgebra.

1. Introduction

Derivations and generalized derivations are interesting subjects both in mathematics and physics, and there has been a great deal of work concerning them. Leger and Luks investigated the structure of the generalized derivations of Lie algebras systematically (cf. [1]). Generalized derivations on rings were studied in [2, 3]. Generalized derivations also play a key role in Benoist's study of Levi factors in derivation algebras of nilpotent Lie algebras (cf. [4]). In 1969, Herstein showed that a Jordan derivation of a prime ring of characteristic not 2 must be a derivation (cf. [5]). Brešar together with Vukman generalized Herstein's result to Jordan -derivations (cf. [6]). In [7], the authors proved that, in a 2-torsion free noncommutative prime ring , a generalized Jordan -derivation is a generalized -derivation when is an automorphism of . We gave some results on generalized derivations of Lie color algebras in [8]. Moreover, Jordan -derivations and generalized Jordan derivations of Lie triple systems were studied in [3, 9, 10].

Prederivations (or Lie triple derivations) of Lie algebras were first introduced by Müller to study bi-invariant semi-Riemannian metrics on Lie groups. Let be a Lie group with a bi-invariant semi-Riemannian metric and its Lie algebra. Then the Lie algebra of the group of isometries of fixing the identity element is a subalgebra of the prederivation algebra of (cf. [11]). Bajo proved that a real or complex Lie algebra admitting a nonsingular prederivation is necessarily nilpotent (cf. [12]). Moens generated this result to Lie algebras over any field of characteristic zero in [13]. Burde showed that the existence of nonsingular prederivations is useful for the construction of the affine structure on Lie algebras (cf. [14]). In recent years, there has been an increasing interest in investigating prederivations (cf. [2, 15–20]).

Lie superalgebras are the natural generalization of Lie algebras and have important applications both in mathematics and in physics. Lie superalgebras are also interesting from a purely mathematical point of view. So it is reasonable to extend the notion of prederivations to Lie superalgebras, which may do the same work in the structure of Lie superalgebras as the prederivations of Lie algebras did. In this paper, we introduce the concepts of (generalized) -prederivations and (generalized) Jordan -prederivations for a Lie superalgebra and obtain some results concerning Jordan -prederivations (resp., generalized Jordan -prederivations) and -prederivations (resp., generalized -prederivations) on a Lie superalgebra.

Throughout this paper, the base field is assumed to be of characteristic not equal to . We now recall some elementary definitions.

Definition 1 (see [12, 19]). A prederivation (Lie triple derivation) of a Lie algebra is a linear mapping such that

Definition 2 (see [21]). A Jordan prederivation (Jordan triple derivation) of a Lie algebra is a linear mapping such that

2. Main Results

Definition 3. Let be a Lie superalgebra and let be homogeneous linear mappings, where . One denotes by the -graded degree of a homogeneous linear mapping of .(1)is called a -prederivation if (2) is called a -prederivation if (3) is called a -prederivation if
In particular, , a -prederivation is called a -prederivation if . It is clear that a -prederivation is a prederivation when .

Definition 4. Let be a Lie superalgebra and let be homogeneous linear mappings, where . (1) is called a Jordan -prederivation if (2) is called a Jordan -prederivation if (3) is called a Jordan -prederivation if
In particular, , a Jordan -prederivation is called a Jordan -prederivation if . It is clear that a Jordan -prederivation is a Jordan prederivation when .

It is clear that if is a -prederivation of , then is a Jordan -prederivation of , where .

In this section, is a Lie superalgebra and are defined to be homogeneous linear mappings of satisfying .

Theorem 5. is a -prederivation of if and only if is a Jordan -prederivation of such that(i),(ii),
where and .

Proof. Assume that is a -prederivation of . Clearly, is a Jordan -prederivation of and ; note that then (i) follows. Since is a -prederivation of , we have ; hence
Conversely, let be a Jordan -prederivation of for which (i) and (ii) hold. Then . It follows that Thus we obtain
By (i), This implies that A similar argument proves By   , we have then that is, where the last equality uses (ii). Since ch, we have , and so ; that is, is a -prederivation of .

Corollary 6. is a -prederivation of if and only if is a Jordan -prederivation of .

Proof. If is a Jordan -prederivation of , then (i) follows immediately. (ii) holds because Therefore, is a -prederivation of by Theorem 5.

Theorem 7. is a -prederivation of if and only if is a Jordan -prederivation of such that(i),(ii),
where and .

Proof. Let be a -prederivation of . Use the fact that as well as the fact that then we have that is, . It is routine to prove (ii).
Suppose, conversely, that is a Jordan -prederivation of satisfying (i) and (ii). Note that In the same way, we can get equalities and . The rest of the proof is the same as the corresponding proof of Theorem 5.

A similar argument proves the following result.

Theorem 8. is a -prederivation of if and only if is a Jordan -prederivation of such that (i),(ii),
where and .

Remark 9. Corollary 6 can also be concluded from Theorem 7 or Theorem 8 since, for any , when is a Jordan -prederivation.

Definition 10. Let be a Lie superalgebra.(1)A generalized -prederivation with respect to a -prederivation is a homogeneous linear mapping such that and (2)A generalized -prederivation with respect to a -prederivation is a homogeneous linear mapping such that and (3)A generalized -prederivation with respect to a -prederivation is a homogeneous linear mapping such that and
In particular, , a generalized -prederivation is called a generalized -prederivation with respect to a -prederivation if . It is clear that is a generalized prederivation when and is a prederivation.

Definition 11. Let be a Lie superalgebra.(1)A generalized Jordan -prederivation with respect to a Jordan -prederivation is a homogeneous linear mapping such that and (2)A generalized Jordan -prederivation with respect to a Jordan -prederivation is a homogeneous linear mapping such that and (3)A generalized Jordan -prederivation with respect to a Jordan -prederivation is a homogeneous linear mapping such that and
In particular, , a generalized Jordan -prederivation is called a generalized Jordan -prederivation with respect to a Jordan -prederivation if . It is clear that is a generalized Jordan prederivation when and is a Jordan prederivation.

Theorem 12. is a generalized -prederivation of with respect to a -prederivation if and only if is a generalized Jordan -prederivation of with respect to a -prederivation such that (i),(ii),
where and .

Proof. Suppose that is a generalized -prederivation of with respect to a -prederivation . Clearly, is a generalized Jordan -prederivation of and . (i) follows from the fact that Since is a generalized -prederivation of with respect to , we have ; hence
Conversely, if is a generalized Jordan -prederivation of with respect to a -prederivation satisfying (i) and (ii), then refer to the proof of Theorem 5; it suffices to prove . In fact, This completes the proof.

Corollary 13. is a generalized -prederivation of with respect to a -prederivation if and only if is a generalized Jordan -prederivation of with respect to a Jordan -prederivation such that

Proof. If is a generalized Jordan -prederivation of , then (i) follows immediately. (ii) holds because Note that is a -prederivation by Corollary 6. Therefore, is a generalized -prederivation of with respect to a -prederivation by Theorem 12.

As before, one can prove the following theorems.

Theorem 14. is a generalized -prederivation of with respect to a -prederivation if and only if is a generalized Jordan -prederivation of with respect to a -prederivation such that (i),(ii),
where and .

Theorem 15. is a generalized -prederivation of with respect to a -prederivation if and only if is a generalized Jordan -prederivation of with respect to a -prederivation such that(i),(ii),
where and .

Remark 16. Corollary 13 can also be concluded from Theorem 14 or Theorem 15 since, for any , when is a generalized Jordan -prederivation.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The paper is supported by NNSF of China (no. 11471090 and no. 11171055), Natural Science Foundation of Jilin Province (no. 201115006), Scientific Research Foundation for Returned Scholars, Ministry of Education of China, and the Fundamental Research Funds for the Central Universities (no. 12SSXT139).