Examples of Freudenthal-Kantor triple systems

It seems that the concept of a triple system (or called a ternary algebra) in nonassociative algebras started from the metasymplectic geometry due to Freudenthal. After a generalization of the concept has been studied by Tits, Koecher, Kantor, Yamaguti, Allison and authors ([1,2] for many earlier references on the subject). Also it is well known the object of investigation of Jordan and Lie algebras with application to symmetric spaces or domains [3] and to physics [4,5].


Introduction
It seems that the concept of a triple system (or called a ternary algebra) in nonassociative algebras started from the metasymplectic geometry due to Freudenthal. After a generalization of the concept has been studied by Tits, Koecher, Kantor, Yamaguti, Allison and authors ( [1,2] for many earlier references on the subject). Also it is well known the object of investigation of Jordan and Lie algebras with application to symmetric spaces or domains [3] and to physics [4,5].
Nonassociative algebras are rich in of mathematics, not only for pure algebra differential geometry, but also for representation theory and algebraic geometry. Specialy, the Lie algebras and Jordan algebras plays an important role in many mathematical and physical objects. As a construction of Lie algebras as well as Jordan algebras, we are interested in characterizing the Lie algebras from view point of triple systems [6][7][8][9]). These imply that we are considering to structure of the subspase L 1 of the five graded Lie (super)algebra L (ϵ,δ) = L -2 ⊕ L -1 ⊕ L 0 ⊕ L 1 ⊕ L 2 satisfying [L i , L j ] ⊆ L i+j ,, associated with an (ϵ,δ) Freudenthal-Kantor triple system which contains a class of Jordan triple systems related 3 graded Lie algebra L -1 ⊕ L 0 ⊕ L 1 . For these consideretions without untilizing properties of root systems or Cartan matrices, we would like to refer to the articles of the present author and earlier references quoted therein [1,2,10,11].
In particular, for an characterizing of Lie algebras, an applying to geometry and physics, we will introduce couple topics about a symmetry of Lie algebras (section 1) and a definition of hermitian triple systems (section 2) in this note, which is a survey and an announcement of new results.
More precisely speaking, first, the symmetry group of Lie algebras and superalgebras constructed from (ϵ,δ) Freudenthal-Kantor triple systems has been studied. Especially, for a special (ϵ, ϵ) Freudenthal-Kantor triple, it is SL(2) group. Secondly, we will give a definition of hermitian * generalized Jordan triple systems and the examples of their tripotents defined by elements c of triple systems satisfying CCC=C.

Symmetry of Lie algebras associated with triple sytems
A triple system V is a vector space over a field F of characteristic ≠ 2 or 3 with a trilinear map V V V ⊗ ⊗ V → . We denote the trilinear product ( or ternary product) by juxtaposition (xyz) ϵ V(or<xyz>, [xyz],etc).
Since (ϵ,δ) Freudenthal-Kantor triple systems offer a simple method of constructing Lie algebras (for the case of (δ=+1)) and Lie superalgebras (for the case of δ = -1) with 5-graded structure, it may be of some interest to study its symmetry group in this note. In order to facilitate the discussion, let us briefly sketch its definition.
We introduce two linear mappings L and K:V⊕V →End V by for any u,v,x,y ϵ V and ϵ =+1we call the triple system to be (ϵ,δ) FKTS.
We note that (-1,1)FKTS is said to be a generalized Jordan triple system of second order [12] in section 2, becaue that is a generalization of concept of Jordan triple systems.
We can then construct a Lie algebra for δ =+1 and a Lie superalgebra for δ =-1 as follows: Let W be a space of 2 × 1matrices over V and define a tri-linear product: Then, it defines a Lie triple system for δ =+1 and an anti-Lie triple system for δ =-1 We then note 0 ( , ) ( , ) = s | , , , , , is a Lie subalgebra of Mat 2 (End(V))where Bfor an associative algebra B implies a Lie algebra with bracket; [x,y] = xy-yx We note also then ( , ), is a derivation of the triple system. Setting then L defined by Note that the endomorphism L (X,Y) is then an inner derivation of the triple system.
we will mainly discuss a subsystem L of L, given by rather than the larger L . Then, L is 5 -graded Here, we utilized the following Proposition for some of its proof.
Let (V, (xyz))) be a (ϵ,δ) -Freudenthal-Kantor triple system with an endomorphism P such that P 2 = -ϵδId and P(xyz) = (PxPyPz) Then, (V,[xyz)] is a Lie triple system (for δ=1) and anti-Lie triple system (for δ=-1) with respect to the product In passing, we note that the standard for X,Y,Z ϵ W We then extend their actions to the whole of L in a natural way to show that they will define automorphism of ˆ.
L They moreover satisfy where Id is the identity mapping (ii) We call the group generated by σ(λ) and θ satisfying these conditions simply as D (ϵ,δ) due to a lack of better terminology. If the field F contains ω ϵ F satisfying ω 3 = 1 but ω ≠ 1, then a finite sub-group of D (ϵ,δ) generated by θ and σ (ω ) defines DiC 3 group for ϵ = δ but S 3 for ϵ = -δ, as they have noted already [13].
We note that the corresponding local symmetry of D (ϵ,δ) yields a derivation of ˆ, L given by We can find a larger automorphism group of ˆ, L if we impose some additional conditions. First suppose that K(x,y) is now expresed as for any x,y ϵ V. We call then the triple system to be a special (ϵ, δ) FKTS [13]. Moreover for the case of special (ϵ, ϵ) FKTS (i.e. ϵ = δ), the automorphism group of L turns out to be a larger SL(2,F)(=Sp(2,F)) group which contains D(ϵ,ϵ) as its subgroup. In this case, the triple system [W,W,W] becomes invariant under Also, the associated Lie algebras or superalgebras are BC 1 -graded algebra of type C 1.
Finally, we consider a ternary system ( , , ) V xy xyz (V,xy,xyz) where xy and xyz are binary and ternary products, respectively, in the vector space V. Suppose that they satisfy (1) the triple system (V,xyz) is a (-1,1) FKTS. We may call the ternary system (V,xy,xyz) to be Allison-ternary algebra or simply A -ternary algebra, since A= (V,xy) is then the structurable algebra [14][15][16]. This case is of great interest, first because structurable algebras exhbit a triality relation [16,17], and second because we can construct another type of Lie algebras independently of the standard construction of (-1,1) FKTS, which is S 4 -invariant and of BC 1 graded Lie algebra of type B 1 . The relationship between the Lie algebra constructed in the new way and that given as in Eq.(1.17) is by no means transparent. Note that the group D(-1,1) contains S 3 but not S 4 symmetry. We may show that if the field F contains the square root The Lie algebras constructed as in Eqs.(1.13) and (1.14) is also a BC 1 -graded Lie algebra of type B 1 without assuming ). Also, if F is an algebraically closed field of characteristic zero, then any simple Lie algebra is known to be S 4 -invariant and can be constructed by some structurable algebra, so that any such Lie algebra is also a BC 1graded Lie algebra of type B 1 , (as well as of type C 1 ). Of course, the underlying sl(2) symmetry is different for both B 1 and C 1 cases. Roughly speaking, it seems that these concept are a version of Lie algebras theory corresponding with a Galois group of algebraic numbers theory.
The contents of this section is a cowork with Prof. Okubo, and the details will be discussed in other papers.

Hermitian triple systems
For a geometrical object based on triple systems in this section, first we note that the symmetric bounded domains are a one to one, correspondence to hermite Jordan triple systems, such that a certain trace form is positive definite hermitian. Hence as a generalization of these triple systems, we are interesting to investigate for structure theory of hermitian generalized Jordan triple systems, in particular, the case of hermitian generalized Jordan triple systems of second order (i.e. hermitian (-1,1) -Freudenthal-Kantor triple system ).
We shall concerned with algebras and triple systems which are infinite over a complex number field, unless otherwise specified in this section.
Definition: A triple system V is said to be a *-generalized Jordan triple system of second order if relations (0)--(iv) satisfy;

ii) K (<abc>,d)+K(c,<abd>)+K(a,K(c,d)b)= 0, where L<(a,b)c = <abc> and K(a,b)c =<acb>-<bca>,
iii) <xyz>is C-linear operator on x,z and C-anti-linear operator on y, iv) <abc> continuous with respect to a norm ║║that is, there exists x y y x . In particular, if a triple system V satisfies the condition (o),(i),(iii),(iv) and (v), then it is said to be a hermitain *-generalized Jordan triple system. Furthermore, as a generalization of the generalized Jordan triple system of second order, it is said to be a . (ϵ,δ) -Freudenthal-Kantor triple system if the following relations satisfy;

Example 1:
Let V be a J*-triple (for the definition, to Loos [3]. Then V is a hermitian *-generalized Jordan triple system of second order, because they satisfy the condition (i) and a special case of (ii) , that is, K (x,y)z = <xyz> -<yzx>≡0 (identicallly zero).

Example 2:
Let T* n,n be the space of diagonal matrix of n×n with element of the complex number. Then T* n,n is a *-generalized Jordan triple system of second order, with respect to the product and the norm (real number) and e i are matrix unit element of T* n,n and x T is the transpose of x.
Next let V be a *-generalized Jordan triple system. Then we may define the notation of a tripotent and a bitripotent as follows. For *-generalized Jordan triple system V, we can define a norm ║║as follows; where i e are tripotents or bitripotents. We note that ║x║ ≥ 0and ║x+y║≤║x║+║y║. Moreprecisely speaking, we have where the notations are denoted by λij,µij ϵ R (real number field) and also E ij means that (i,j) element is 1 and othere element is zero. E ij and These show that * , n k D is a hermilian *generalized Jordan triple system of second order, that is, a hermitian (-1,1)-FKTS.

Examples 4:
Let S* 2n,k be the set of all 2n ×k matrices with operation Then S* 2n,k is a hermitian *-generalized Jordan triple system of second order.
In fact, the elements For any element X of S 2n,k ,we may represent as follows; ( where x 1 is a k×n matrix and x 2 is a n × l matrix with operation given by foumula  For any element of A * kn ⊕ A * nl , we may represent as follows; (1) (1)  (1) , E jt (2) ) ϵ A * kn ⊕ A * nl is a tripotent. Furthermore, the norm is defined by (1) ( is a skew-symmetric n×n matrix and x 2 is a column,i.e.,is a n ×1 matrix, with operation given by formula, we may represent as follow; and the norm is defined by In the end of this section, we note a Peirce decomposition as follows ( [12] for the case of δ = 1). This section is a cowork with Dr M.Sato with an application to physics and the details will be considered in future paper [18].

Concluding Remark
In this section, we shall briefy describe a correspondence with the triple systems and the Lie algebras or superalgebras of simple classical type associated with their triple systems [1,2,7,10,11].
Let V be the matrix set of Mat(m,n;F) and the triple product is defined by