Introduction to -Triple Systems

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