#### Abstract

This paper introduces the category of -triple systems and studies some of their algebraic properties. Also provided is a functor from this category to the category of Leibniz algebras.

#### 1. Introduction

A Triple system is a vector space over a field together with a -trilinear map . Among the many examples known in the literature, one may mention -Lie algebras [1] and Lie triple systems [2] which are the generalizations of Lie algebras to ternary algebras, Jordan triple systems [2] which are the generalizations of Jordan algebras, and Leibniz 3-algebras [3] and Leibniz triple systems [4] which are generalizations of Leibniz algebras [5]. In this paper we enrich the family of triple systems by introducing the concept of -triple systems, presented as another generalization of Leibniz algebras with the particularity that, for all , the map , defined by , is a derivation of , a property of great importance in Nambu Mechanics. We investigate some of their algebraic properties and provide a functorial connection with Leibniz algebras and Lie algebras.

For the remaining of this paper, we assume that is a field of characteristic different to 2 and all tensor products are taken over .

*Definition 1. *A -triple system is a -vector space equipped with a trilinear operation

*Example 2. *Let be an -dimensional vector space with basis . Define on the bracket by
for fixed . It is easy to check that the identity (2) is satisfied. So is a -triple system when endowed with the operation .

Because of the resemblance between the identity (2) and the generalized Leibniz identity [3], it is worth mentioning that, in general, Leibniz 3-algebras do not coincide with -triple systems. The following example provides a Leibniz 3-algebra that is not a -triple system.

*Example 3. *The two-dimensional complex Leibniz 3-algebra (see [6, Theorem 2.14]) with basis , , and brackets
with , is not a -triple system. It is easy to check that its bracket does not satisfy the identity (2).

*Definition 4. *Let be -triple systems. A function is said to be a homomorphism of -triple systems if

We may thus form the category **-TS** of -triple systems and -triple system homomorphisms.

Recall that if is a vector space endowed with a trilinear operation , then a map is called a derivation with respect to if

Lemma 5. *Let be a -triple system and . Then the map defined on by
**
is a derivation with respect to the bracket of .*

*Proof. *By setting and using the identity (2), we have

*Definition 6. *A subspace of a -triple system is a subalgebra of if is a -triple system when endowed with the trilinear operation of .

*Definition 7. *A subalgebra of a -triple system is called ideal (resp., left ideal, resp., right ideal) of if it satisfies the condition (resp., , resp., ). If satisfies the three conditions, then is called a 3-sided ideal.

Note that none of these three conditions implies the others as in the case of Lie triple systems.

*Example 8. *In Example 2, the subspace with basis is an ideal of . However the subspace with basis is not an ideal of , since, for , we have .

*Definition 9. *Given a -triple system , one defines the center of and the derived algebra of , respectively, by

Lemma 10. *For a -triple system , and are ideals of .*

*Proof. *Clearly, . So is an ideal of . That is an ideal follows from the fact that is closed under the operation .

The following theorem classifies a subfamily of two-dimensional complex -triple systems. This result was obtained by Camacho et al. in [6] for Leibniz 3-algebras.

Theorem 11. *Up to isomorphism, there are seven two-dimensional complex -triple systems with one-dimensional derived algebra.*

*Proof. *The proof is similar to [6, Theorem 2.14]. Let be a -triple system with basis , and assume that . Then write , . Then, using the identity (2), the only possible nonzero coefficients yield to the system of equations
for which the solution provides the following -triple systems with bracket operations:
with .

*Definition 12. *Given a -triple system , one defines the left center and the right center of , respectively, by

Lemma 13. *The left center and the right center are 3-sided ideals of .*

*Proof. *To show that is an ideal of , let and let . Then, for every , we have, by the identity (2),
So . The proof that is both left ideal and right ideal is similar, so is the case for .

*Definition 14. *Given a -triple system , we define left and right centralizers of a subalgebra in by
respectively.

Lemma 15. *Let be an ideal of a -triple system . Then and are also ideals of .*

*Proof. *To show that is an ideal of , let , , and . Then, by the identity (2),
So . The proof for is similar.

*Definition 16. *For a -triple system and a subalgebra of , we define the left normalizer of in by
and the right normalizer of in by

Lemma 17. *Let be a subalgebra of a -triple system . Then and are also subalgebras of .*

*Proof. *To show that is a subalgebra of , let , , and . Then, by the identity (2), we have
So . The proof for is similar.

*Remark 18. *If is an ideal, then .

#### 2. From -Triple Systems to Leibniz Algebras

Recall that a Leibniz algebra (sometimes called Loday algebra, named after Jean-Louis Loday) is a vector space with a bilinear product satisfying the Leibniz identity

Proposition 19. *Let be a -triple system. Define on the bracket operation by
**
Then satisfies the Leibniz identity.*

*Proof. *On one hand, we have
Also,
On the other hand,
One checks using the identity (2) that the equality
holds.

Corollary 20. *Let be a -triple system; then endowed with the bilinear map has a Leibniz algebra structure.*

*Proof. *This is a consequence of Proposition 19.

Similarly, we have the following.

Corollary 21. *Let be a -triple system; then has a Leibniz algebra structure, when endowed with the bilinear map defined by
*

These determine two functors from the category **-TS** of -triple systems to the category of Leibniz algebras.

*Definition 22. *Let be a -triple system and a Leibniz algebra. The action of on is a map satisfying
for all and .

Proposition 23. *Let be a -triple system; then the Leibniz algebra acts on via the map defined by .*

*Proof. *The first condition of Definition 22 follows by (2). To show (28), we have

Now let and consider the map defined by , . Clearly, this map is a derivation of as it is induced by the action (Proposition 23) defined above.

Proposition 24. *For a -triple system , the subspace is a Lie algebra with respect to the product
**
More precisely, it is an ideal of the Lie algebra of derivations of .*

*Proof. *To show that is a Lie subalgebra of , let . Then, for all ,
So is closed under the bracket of . Also, for any derivation , we have, for all ,
Hence .

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.