LIE CENTRAL TRIPLE RACKS

This paper introduces Lie c-triple racks. These triple systems generalize both left and right racks to ternary algebras, and locally differentiates to gb-triple systems [3]. Mathematics Subject Classification (2010): 05B07, 18E10, 17A99


Introduction
The generalization of Lie algebras to algebras such as Lie triple systems, Jordan triple systems [8] and 3-Lie algebras [7] suggests a natural generalization of Leibniz algebras (non commutative Lie algebras) [11] to ternary algebras.One generalization is provided by Leibniz 3-algebras [5] for which the characteristic identity expresses the adjoint action as a derivation of the algebra.A second generalization of Leibniz algebras to ternary algebras is provided by Leibniz triple systems [4].They are defined in such a way that Lie triple systems are a particular case.Recently, the author introduced gb-triple systems [3], another generalization of Leibniz algebras in which the ternary operation T : g ⊗3 → g expresses the map T a,b (x) = T (a, x, b) as a derivation of g for all a, b ∈ g.
The local integration problem of these algebras generated from Lie's third theorem, which states that every finite-dimensional Lie algebra over the real numbers is associated with a Lie group.Partial solutions to this problem for Leibniz algebras (dubbed by Loday as the Coquecigrue problem) have been provided by several authors (see M. Kinyon [10], S. Covez [6]).The author extended Kinyon's results to Leibniz 3-algebras using Lie 3-racks [2].In this paper we open the problem of integration of gb-triple systems.We follow Kinyon's approach [10] to open a path to a solution by defining an algebraic structure that locally differentiates to a gb-triple systems.We refer to these algebras as Lie c-triple racks.They appear to generalize both left and right Lie racks to ternary algebras; a particularity not supported by Lie 3-racks.
For the remainder of this paper, we assume that K is a field of characteristic different to 2.

c-Triple racks
In this section we define c-triple racks and provide some examples.We also provide funtorial connections with the category of groups and the category of racks.
Recall that a gb-triple system [3] is a K -vector space g equipped with a trilinear operation [−, −, −] g : g ⊗3 −→ g satisfying the identity Note that these generalize the notions of racks and quandles [9] to ternary operations.It is also clear that c-triple quandles are weak c-triple quandles but the converse is not true.See Example 2.4.
This provides a category c pRACK of pointed c-triple racks and pointed c-triple Therefore the c-distributive property is satisfied.For the second axiom, given a, c, d ∈ Example 2.5.Let G be a group with identity 1, and define on G the operation Then (G, [−, −, −], 1) is a pointed weak c-triple quandle.Indeed, we have on one hand On the other hand, Therefore the c-distributive property is satisfied.For the second axiom, given As a consequence, we have the following: Proposition 2.6.There is a faithful functor F from the category of groups to the category of pointed c-triple racks.
Now by the universal property of quotient groups, there is a unique morphism of groups α * : F (R) −→ G such that the following diagram commutes.It is easy to show that (R, [−, −, −], 1) is a pointed c-triple rack.
As a consequence we have the following: Proposition 2.8.There is a faithful functor H from the category of pointed racks to the category of c-triple racks.Let us observe that in the proof of Proposition 2.8 the set R ×(3) is a quandle if R is a c-triple quandle.

From Lie c-triple racks to gb-triple systems
In this section we define the notion of Lie c-triple racks.We show that the tangent functor T 1 locally (at a specific point) maps Lie c-triple racks to gb-triple system.Proof.
Remark 3.5.Note that for all a, c ∈ R, R acts on itself (considered as a differentiable manifold) via the maps φ (a,c) by Proposition 3.4.Also, and denote by X 1 := φ (a,c) * (X) the vector field extension of X.Then X 1 is generated by a one-parameter family of diffeomorphisms γ X : R → R with initial point γ X (0) = 1 and initial tangent vector dγ X (0) = X.The corresponding exponential map (see [12,Chapter 9]) denoted exp 1 : T 1 (R) → R is then defined by exp 1 (X) = γ X (1).
Hence D (A,C) * is a derivation of g and the map T 1 (Φ) is exactly D (A,C) * .
From the calculations performed in the proofs of Theorem 3.6 and Theorem 3.7, we deduce that the ternary operation [−, −, −] g satisfies the identity (1).We then have the following result: The equality holds by the Jacoby identity.
Proof.Define F by F(G) = (G, [−, −, −], 1) as in Example 2.5.Its left adjoint F is defined as follows: Given a pointed c-triple rack R, consider the quotient group G R =< R > /I where < R > is the free group on R and I is the normal subgroup of < R > generated by the set {(a −1 c −1 b −1 ca)([a, b, c] R ) : with a, b, c ∈ R}.Indeed, given a morphism of c-triple racks α : R −→ F(G), there is a unique morphism of groups β :< R >−→ G such that α = β| R by the universal property of free groups.So