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 .