Associative Triple Systems with Nondegenerate Bilinear Forms

Associative algebras with n-ary compositions in general and associative triple system in particular play important roles in Lie and Jordan theories, geometry, analysis. For instance, associative triple systems give rise to Jordan triple systems [1-3] and Jordan triple systems give rise to 3-graded Lie algebras through the Tists Kantor Koecher construction [4,5], from which most simple Lie algebras can be obtained. Jordan triple systems also give rise to Lie triple systems through the K. Meyberg construction [6,7]. On the other hand, Lie triple systems give rise to graded Lie algebras, which are exactly the kind of Lie algebras associated to symmetric spaces. In geometry and analysis, various types of Jordan triple systems are used in the classifications of different classes of symmetric spaces [2,8-10].


D({a,b,c})={D(a),b,c}+{a,D(b),c}+{a,b,D(c)},∀a,b,c∈V
The set of all derivation of V is denoted by Der(A). The set Der(A) of derivations of A is a Lie algebra of linear transformations, we call it the derivation algebra of A. Further, if a,b∈A, the linear maps L(a,b) (resp.R(a,b)) defined on A by L(a,b)c=(a,b,c) (resp. R(a,b)c=(c,a,b)),∀c∈A, is a derivation of A. The linear maps L(a,b) (resp.R(a,b)) are called the left (resp.right) multiplications of A. It was proved [11] that if B is a symmetric and right invariant bilinear form on A, then B is left invariant. Therefore, a symmetric bilinear form B is invariant if and only if it is right invariant.
(2) We say that (A,B) is a symmetric associative triple system if B is a non-degenerate symmetric invariant bilinear form on A. Here B is called a symmetric structure on A. A symmetric associative triple system (A,B) is said to be reducible (or B-reducible) if it admits an ideal J such that the restriction of B to J×J is non-degenerate. Otherwise, we will say (T,B) is irreducible.
(3)We say that (A,ω) is a symplectic associative triple system if ω is a non-degenerate skew-symmetric bilinear form on T such that the identity.
Then D  is an invertible derivation which is skew-symmetric with respect to B. Hence, the symmetric associative triple system (T n ,B) admits a symplectic structure.
Proposition 1: Let V be an associative triple system and Z(V)={aV;{a,b,c}=0,∀b,c∈V} be the center of V. Then,

Definition 6
Let V be an associative triple system and B be an invariant scalar product on V.
2. An ideal U of V is said to be: (b) Solvable (resp. nipotent) if it is solvable (resp. nilpotent) as a asoociative triple system.
3. The largest solvable ideal of V is called the radical of V and denoted Rad(V).
(a) Semi-simple if it has no non trivial solvable ideal. That is The following lemma is straightforward. Lemma 1: Let (V,B) be an associative triple system and U be an ideal of V. Then, If U is nondegenerate, then A=U⊕U ⊥ and U ⊥ is also nondegenerate. Lemma 2: Let (V,B) be an associative triple system. Then,  where r∈N and such that for i∈{1,…,r}, V i is B-irreducible as an associative triple system.
For i≠j and (x,y)∈V i ×V j , we have B(x,y)=0.
We precede by induction on n=dim(V). If n=1, then the assertion is true. Suppose that every asoociative triple system of dimension less than n satisfies the proposition. Let (V,B) be an associative triple system of dimension n+1. If V does not contain any non trivial nondegenerate exists δ∈Hom (A,A), such that ω(a,b)=B((a),b),a,b∈A. Further, since ω is symplectic, then;  c d B  a b c d B a b c d B a b c d  a b c The fact that B is nondegenerate implies that is a derivation of A. Conversely, if δ is a B-antisymmetric invertible derivation of A, then it is clear that the bilinear form ω:A×AK defined by: ω(a,b)=B((a),b),a,b∈A, is a symplectic form of A.

b c t r a b c t r a b c t r a b c t r a b c t r a b c t r
∀a,b,c,t,r∈A. Furthermore, the bilinear form B defined on ( ) The constructed symmetric asoocidtive triple system ( ( ), ) [12,13]. .

D a f D x f D a A and f A
. We get the result by Theorem 0.1. If I is a simple ideal of V, then there exists i 0 ∈{1,…,r} such that
(i) Let I be a non-trivial simple ideal of V. Assume that for all i∈{1,…,r} we have I∩V i ={0}. Since The previous Proposition shows that, in the case of semisimples triples systems, the decompsition into the direct sum of orthogonal nondegenerate ideals coincides with the decomposition into a direct sum of simple ideals.
The following theorem presents a process of construction of a symmetric asoociative triple systems. is not nilpotent too. Consequently, the family of a symmetric asoociative triple systems contains strictly the of semi-simple asoociative triple systems and the symmetric nilpotent asoociative triple systems. Let I be an isotropic ideal of J of dimension n/2. Since B is nondegenerate, the ideal I is abelian. Let us consider V an isotropic complementary to I. Then, J=I⊕V and V ⊥ =V. Let x,y,w∈V. Put {x,y,z}=(x,y,z)+β(x,y,z) where α(x,y,z)∈I and β(x,y,z)∈V. It is easy to check that (V,β) is a asoociative triple system. Now, since B is nondegenerate, the linear map v: I→V * ;i→B(I,.) is invertible. Furthermore, dim(I)=n/2=dim(V * ). Thus, v is an isomorphism of vector spaces. We consider the T * -extension