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**Noriaki Kamiya ^{*}**

Department of Mathematics, University of Aizu, Aizuwakamatsu, 965-8580, Japan

- *Corresponding Author:
- Norikai kamiya

Department of Mathematics University

of Aizu, Aizuwakamatsu

965-8580, Japan

**Tel:**+0242-37-2500

**E-mail:**[email protected]

**Received date:** January 26, 2014; **Accepted date:** June 02, 2014; **Published date:** June 10, 2013

**Citation:** Kamiya N (2014) Examples of Freudenthal-Kantor triple systems. J Generalized Lie Theory Appl 8:210. doi:10.4172/1736-4337.1000210

**Copyright:** © 2014 Kamiya N. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

Symmetry group of Lie algebras and superalgebras constructed from (ϵ,δ) Freudenthal-Kantor triple systems has been studied. Also, the definition and examples of hermitian triple systems is introduced in this note.

Triple systems; Lie algebras; Symmety group

Triple systems; Lie algebras; Symmety group

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 [6-9]). These imply that we are considering to structure of the subspase
L_{1} of the five graded Lie (super)algebra satisfying associated with an (ε,δ) Freudenthal-Kantor
triple system which contains a class of Jordan triple systems related
3 graded Lie algebra . 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 . We denote the trilinear product ( or ternary product) by juxtaposition (xyz) ε V(or<xyz>,[xyz],etc).

A well studied triple system is the (ε,δ) Freudenthal-Kantor triple systems ( abbreviated hereafter as to (ε,δ) FKTS) with ε and δ being either +1 or -1 [1,2,9,12].

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

L (x,y) = xyz, K (x,y)z= - δyzx(1.1)

for δ = +1 or -1 If they satisfy

[L(u,v), L(x,y)] = L(L(u,v)x,y)+ εL(x,L(v,u)y), (1.2)

K(K(u,v)x,y) = L(y,x)K(u,v)- ε K(u,v)L(x,y)(1.3)

for any u,v,x,y ε V and ε =+1we call the triple system to be (ε,δ) FKTS

One consequence of Eqs.(1.2) and (1.3) is the validity of the following important identity (see [Y-O.] Eqs.(2.9) and (2.10))

K(u,v)K(x,y) = ε δL(K(u,v)y,x- εL(K(u,v)x,y)(1.4)

= L(v,K(x,y)u) – δL(u,K(x,y)v) (1.5)

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:

W ⊕ W ⊕ W→ W by

Then, it defines a Lie triple system for δ =+1 and an anti-Lie triple system for δ =-1 We then note

is a Lie subalgebra of Mat2(End(V))- where B- for 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

gives a Lie algebra for δ =+1 and a Lie superalgebra for δ =-1 where

i.e., D satisfies

= D[X_{1},X_{2},X_{3}]

[DX_{1},X_{2},X_{3}]+ [X_{1}, D X_{2},X_{3}]+ [X_{1}, X_{2}, D X_{3}] (1.12a)

and hence induces also

[D,[X,Y]] = [DX,Y]+[X,DY], (1.2b)

if we define the bracket by

[D_{1}⊕X_{1},D_{1}⊕X_{2}] = ([D_{1},D_{2}] + L(X_{1},X_{2})) ⊕ (D_{1}X_{2} - D_{2}X_{1}). (1.13)

where

[D_{1},D_{2}] = D_{1},D_{2} – D_{2},D_{1}

and

Note that the endomorphism L (X,Y) is then an inner derivation of the triple system.

Since we will mainly discuss a subsystem of L, given by

rather than the larger L . Then, is 5 -graded

where

Here, we utilized the following Proposition for some of its proof.

**Theorem 1: **([K-O.], [K-M-O.]) 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

[xyz]=(xPyz)- δ(yPxz)+ δ(xPzy)-(yPzx).

In passing, we note that the standard is a result of Theorem 1 immediately with

Next, we introduce for any λ ε F (λ ≠ 0), being non-zero constant by

in We may easily verify that they are automorphism of [W,W,W], i.e, we have for example

θ ([X,Y,Z]) =[ θ X, θ Y, θ Z]

for X,Y,Z ε W We then extend their actions to the whole of in a natural way to show that they will define automorphism of . They moreover satisfy

(i) σ

σ (1)=Id, θ^{4} = Id(1.19a)

where Id is the identity mapping

(ii)

(iii) σ(μ)σ(v)= σ(μv) for μ,v ε F, μv ≠ 0

(iv) σ(λ) θ σ(λ) = θ for any λ ε F, λ ≠ 0. (1.19d)

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].

Conversly any 5 -graded Lie algebra (or Lie superalgebra) with
such automorphism θ and σ(λ) satisfying Eqs.(1.19) lead essentially to
a (ε δ) FKTS in L_{1} with a triple product defined by {x,y,z} = [[x, θ y,]z]
for x,y, z ε L_{1} [6].

We note that the corresponding local symmetry of D (ε,δ) yields a derivation of , given by

which satisfies

h[X,Y,Z] =[ hX,Y,Z]+[X,hY,Z]+[X,Y,hZ](1.21a)

as well as [h,[X,Y]] = [hX,Y] + [X,hY] (1.21b) for X,Y,Z ε W .

We can find a larger automorphism group of , if we impose some additional conditions. First suppose that K(x,y) is now expresed as

K(x,y) = ε δ L(y,x) - ε L(x,y) (1.22)

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 ˆL 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

for any 2×2 SL(2,F) matrixU,i.e.

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.

(2) The binary algebra (V,xy) is unital and involution with the involutive map

(3) The triple product xyz is expressed in terms of the bi-linear products by

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 [14-16]. 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 of-1, then Eqs.(1.17) can be prolonged to yield the Lie algebra for
the structurable algebra.

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 ([6]).
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_{1}-
graded 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;

0) V is a Banach space,

i) [L(a,b),L(c,d)] = L(<abc>,d) –L(c,<bad>),

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 ex_{i}sts
K>0 such that

║<xxx>║≤K║x║^{3} for all x ε V.

Furthermore a *-generalized Jordan triple system of second order is said to be hermitian if it satisfies the following condition,

v) all operator L(x,y) is a positive hermitian operator with a hermitian metrix

(x,y) = trace L(x,y),(2.1)

that is, (L(x,y)z,w) = (z,L*(x,y)w), and . 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;

i) ' [L(a,b),L(c,d)] = L(<abc>,d) εL(c,<bad>)

ii) 'K(<abc>,d)+K(c,<abd>) + δK(a,K(c,d)b) =0,

where K(a,b)c = <abc> -δ<bca>, ε= ±1, δ = ±1.

We note that these identities (i)' and (ii)' are equivalent to the identities (1.2) and (1.3) in section 1 [2,7].

**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 ║║

for all x,y,z ε T*_{n,n}

where , λ_{i},μ_{i}∈R (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.

**Definition: **It is said to be a tripotent of V if

<ccc> = c,c ε V(2.2).

**Definition: **It is said to be a strongly bitripotent of V if a pair (c_{1},c_{2}) of

<c_{1}c_{1}c_{2}> = -1/2c_{2}, < c_{1}c_{1}c_{2}> = -1/2c_{1}, and other porducts are zero.

**Definition:** It is said to be a bitripotent of V if a pair 1 2 (c ,c ) of
the tripotens satisfy the relations

<c_{1}c_{1}c_{2}> = αc_{2}, <c_{2}c_{2}c_{1}> = αc_{1} ,

< c_{1}c_{2}c_{1}> =βc_{2},< c_{2}c_{1}c_{2}>=βc_{1},

<c_{2}c_{1}c_{1}> = γc_{2}, <c_{1}c_{2}c_{2}> = γc_{1},

and other products are zero, where

α^{2}+ β^{2}+γ^{2} ≠ 0,α,β,γ ε R (real number).

For *-generalized Jordan triple system V, we can define a norm ║║as follows;

║x║= max |λ_{i}|,ifx = Σ λ_{i}e_{i} ε V,

where e_{i} are tripotents or bitripotents. We note that

║x║ ≥ 0and ║x+y║≤║x║+║y║.

**Example 3: **(a generalization of Example 2.2) Let D*_{n,k} be the set of
all n×k matrices with operation

where x^{T} and mean transpose and conjugation of x respectiverly.

Then D*_{nk} is a hermitian *-generalized Jordan triple system of
second order. In fact, the conditions (i),(ii) and (iii) are evident, by
straightfoward calculations.

For (v) considering, we have (x,y): = tr (x,y) linear on x and anti linear on y by the condition (iii). Thus we may get as follows.

From the relations;

it follows that L(x,y)+L(y,x), and L(y,x)-L(y,x) have are a real trace form and an imagenary trace form respectively. Thus we have

To prove positive definite, we consider

Since we get this means that the trace from (x,y) is positive definite. It is enough to show the condition (iv). By means of result of the property of matrix, we can write

x=Σμ_{i}e_{i}, ║x║ = max |μ_{i}|,

e_{i} are tripotents or bitripotens if e_{i} is the unit element of matrix.

Moreprecisely speaking, we have

where the notations are denoted by λ_{ij},μ_{ij} ε R (real number field) and also E_{ij} means that (i,j) element is 1 and othere element is zero. E_{ij} and are tipotents, i.e., <E_{ij}E_{ij}E_{ij}> = E_{ij} , and Furthermore, we note are bitripotents.

Then we have

These show that D* _{n,k} 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

where x_{i} is an n×k- matrix, and

Then S*_{2n,k} is a hermitian *-generalized Jordan triple system of
second order.

In fact, the elements are tripotents.

For any element X of S_{2n,k},we may represent as follows;

and the norm is defined by

where E_{ij} (1) is the matrix unit of and
E_{lm}^{(2)} E_{lm}^{(2)} is the matrix unit of

By straightforward calculations as well as Example 2.3, we can show
that S*_{2n,k} is a hermitian *-generalized Jordan triple system of second
order.

**Example 4: **Let A* _{kn} ⊕ A* _{nl} A* _{kn} be the set fo all pairs where x_{1} is a k×n matrix and x_{2} is a

n × l matrix with operation given by foumula

Then A*_{kn} ⊕ A* _{nl} is a hermitian *-generalized Jordan triple system
of second order.

In fact, for we have

hence it follows that the trace form ( , ) is positive definite

For any element of A* _{kn} ⊕ A* _{nl} , we may represent as follows;

and (E_{ij} ^{(1)}, E_{jt} ^{(2)}) ε A* _{kn} ⊕ A* _{nl} is a tripotent.

Furthermore, the norm is defined by

**Example 6: **Let C*_{nn} ⊕ A*_{nl} be the set of all pairs where x_{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,

Then C*_{nn} ⊕ A*_{nl} is a hermitian *-generalized Jordan triple system
of second order.

In fact, we put c_{ij}:= the matrix of (i,j) -element is 1,(j,i) -element
is -1 and other element is zero (1≤ i ≤ n), further more ek := (0- - - ,1, -
- - 0)T,(k th element is 1). Then are tripotents and also are
tripotents but these are not bitripotents. For any element of C*_{nn} ⊕ A*_{nl},
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).

**Theorem 1: **For δ =±1 and hermitian (-1, δ) Freudenthal-Kantor
triple system V with a tripotent C s.t. <ccc> = c we have

where denoted by , and also defined by R(x,y)z=<zxy>, N_{δ} = L (c,c)
– δR(c,c) and T_{δ} = (1+δ)L(c,c)-R(c,c)+Id.,

This section is a cowork with Dr M.Sato with an application to physics and the details will be considered in future paper [18].

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

<xyz> = xy^{T}z+ δ (zy^{T}x-μzx^{T}y) x,y,z ε V.(*)

Then there ex_{i}sts 4 cases of (-1, δ)-Freudentahl-Kantor triple
systems with μ=0 or 1. and δ =±1.

The standard embeding Lie algebras or superalgebras , where W = V ⊕ V (cf. section 1) associated with the triple product (*) are appeared by 4 types as follows;

(i) 5 graded Lie algebra (δ =1,μ =1)

B_{m+1} = so(2m+2l+1) (n= 2l+1), D_{m+1} = so(2m+2l) (n=2l).

(ii) 3 graded Lie algebra (δ =1,μ =0)

A_{n+m-1} =sl(n+m).

(iii) 5 graded Lie superalgebra (δ =-1,μ =1)

. . .B(l,m) =osp(2l+1|2m) (n =2l+1), D(l,m) = osp(2l|2m) (n=2l).

In particular, if m=1, then the triple product

<xyz> = <x,y> z-<z,y> x+<z,x> y

is a (-1,-1)-Freudentahl-Kantor triple system satisfying
K(x,z)=<xyz>+<zyx>=2<x,z>y, and so dim{(x,z}span =1 (called a
balanced triple system), where <x,y> = x_{1}y_{1}+. . . +x_{n}y_{n}.

(iv) 3 graded Lie superalgebra (δ=-1,μ=0)

A(m-1,n-1) = sl(m|n) (m≠n), A(m-1|m-1) = psl (m|m) (m= n).

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