Systems Simplicity

A simple system is a system who has no proper ideals. We prove that every simple system J have one of the following assertion: (1) J is h−irreducible. (2) J = J1 ⊕ J̃1 is the direct summation of two h−invariant and h−irreducible subsystems.


Introduction
All the vector spaces considered in this work are of finite dimensions on a field K = R or C. An n-ary algebra E is a vector space with an n−linear map from E × E × ... × E on E. A Jordan triple system is defined by the following conditions on the trilinear map: {x, y, {u, v, w}} − {u, v, {x, y, w}} = {{x, y, u}, v, w} − {u, {y, x, v}, w} (1.1) {u, v, w} = {w, v, u}. (1.2) It is well known that from a triple Jordan system, we can build a system Lie triple on the same underlying vector space. We go to study other connections between all of these structures already mentioned. More precisely, using the construction of Koecher-Kantor -Tits applied to an algebra of Jordan J, we can construct a Lie algebra (J). On the other hand, using similar constructions, also called the KKT construction, one can construct a Lie algebra from a triple Lie or Jordan system. These constructions are due to Kurt Meyberg. The theory of groups and Lie algebras begins at the end of the 19th century with the work of the Norwegian mathematician Sophus Lie. She has known many ramifications (non-Euclidean geometries, homogeneous spaces, harmonic analysis, representation theory, algebraic groups, quantum groups ...) and still remains very active. In addition, these objects also intervene in branches that are a priori more distant from mathematics: in number theory, by means of automorphs and the Langlands program , and in theoretical physics, in particular in particle physics or general relativity. The triple Lie system represents the linear infinitesimal analog of symmetrical space (Loos, 1969). In other words, the local study of symmetric spaces is equivalent to that of triple Lie systems. The topicality of the system classification problem Lie triples follows from the importance of symmetrical spaces that play a role considerable in several fields of modern science such as physics cosmological (Weinberg, 1982), theoretical mechanics (Doubrovine et al., 1982), differential geometry (Cartan, 1926-27;Helgason, 1962), the theory of continuous loops and that of continuous groups etc.
We are interested in these types of n-ary algebras given their importance on the algebraic level and geometric. In particular, if J is a Euclidean Jordan algebra, then the interior topological of the set C = {x 2 , x ∈ J} is a symmetric cone.
The main upshot of this paper is to prove that every simple triple system J have one of the following assertion: (1) J is h−irreducible.
(2) J = J 1 J 1 is the direct summation of two h−invariant and h−irreducible subsystems.

Systems Symplicity
Definition 1. A Jordan triple system is a triple system (J, { , , }) which satisfies ∀u, v, w, x, y ∈ J: Example 1. 1. Let V be a linear pace and B be a bilinear form of V. Then, V endowed with the product defined by: is a Jordan triple system.

Now we can easly obtain,
Theorem 2. If J is a Jordan triple system. We consider L(h, J) is a Lie algebra called the KKT-algebra of J. The Lie bracket on L(h, J) is defined as follow: Remarks 1.
and L is the −1 propre subspace of J.
7. If E = (id, −id) ∈ h, then (ad E ) 3 = ad E and h, J, J are respectively the propre spaces of ad E with respective eigenvalues 0, +1, −1. If we suppose that the application α is equal to the identity map, we find the old notion of Lie triple system [?, ?]. where U 1 et U 2 are the projections of U on J and J respectively.

Corollary 2.
U ⊆ J is an ideal of J if and only if U U is an ideal of L.
1. Let U be a linear subspace of J. Then, U is said to be h-invariant if {J, J, U} ⊆ U.
2. J is said h-irreducible if it contains no h-invariant non trivial ideal.

Lemma 2.
If U a h−invariant linear subspace of J. Then, U is a h−invariant linear subspace of J and U ⊆ U.
Corollary 3. If L is simple, then J is h−irreducible.