#### Abstract

We define Hermitian -Freudenthal-Kantor triple systems and prove a structure theorem. We also give some examples of triple systems that are generalizations of the and Hermitian 3-algebras. We apply a -generalized Jordan triple system to a field theory and obtain a Chern-Simons gauge theory. We find that the novel Higgs mechanism works, where the Chern-Simons gauge theory reduces to a Yang-Mills theory in a certain limit.

#### 1. Introduction

The structure theory of -Freudenthal-Kantor triple systems of finite dimensions, which contain the concept of Jordan triple systems, has been studied in [1–11].

On the other hand, it is well known that symmetric bounded domains have one-to-one correspondence to Hermitian Jordan triple systems [12–15], for which a certain trace form is positive definite Hermitian. Moreover, structure theorems of Hermitian generalized Jordan triple systems of the second-order can be found in [16]. Hence, as a generalization of these concepts, it is interesting to investigate the structure theory of Hermitian -Freudenthal-Kantor triple systems (HFKTSs). In this generalization, Hermitian generalized Jordan triple systems of the second-order or the so-called Kantor triple systems are included as Hermitian -Freudenthal-Kantor triple systems.

From the viewpoint of string theory, HFKTSs are generalizations (A different generalization, the so-called differential crossed module, was extensively investigated in [17–19].) of Hermitian 3-algebras [20–62], which have played crucial roles in M-theory. The field theories applied with Hermitian 3-algebras are the Chern-Simons gauge theories that describe the effective actions of membranes in M-theory. In a certain limit, the novel Higgs mechanism works, where the Chern-Simons theories become the Yang-Mills theories that describe the effective actions of D-branes in string theory. Moreover, 3-algebra models of M-theory itself have been proposed and were studied in [63–67]. Therefore, it is interesting to apply HFKTSs to field theories. It is particularly important to study whether the novel Higgs mechanism works in a field theory with an HFKTS.

The Hermitian 3-algebras are special cases, where or, equivalently, , of Hermitian -Freudenthal-Kantor triple systems of the second order. Also, the Hermitian 3-algebras are classified into and Hermitian 3-algebras [34, 36, 37, 43, 67]. Therefore, it is natural to extend these triple systems to more general Hermitian -Freudenthal-Kantor triple systems or Hermitian generalized Jordan triple systems.

Here, we are concerned with algebras and triple systems that are finite or infinite over a complex number field, unless otherwise specified. We refer to [68–72] for nonassociative algebras and to [73–75] for Lie superalgebras, for example.

#### 2. Definitions and Preamble

A triple system is a vector space over a field of characteristic with a trilinear map . In this paper, we are concerned with triple systems over the complex number , and we denote the trilinear product by for , assuming that is -linear on and and -antilinear on . Also we use the notations of two operations End and End with respect to the triple systems [1, 2], where and .

*Definition 1. *A triple system is said to be a *--Freudenthal-Kantor triple system if the following relations (0)–(iv) are satisfied: is a Banach space,(i), where , , and ,(ii),
(iii) is a -linear operator on and and a -antilinear operator on ,(iv) is continuous with respect to a norm, ; that is, there exists such that
Furthermore, a *--Freudenthal-Kantor triple system is said to be Hermitian if it satisfies the following condition.(v)All operators are positive Hermitian operators with a Hermitian metric
that is, and .

This definition is a generalization of the concept of known -Freudenthal-Kantor triple systems (see [1–10]) to that of Hermitian triple systems. Note that there are many simple Lie algebras (the case of ) and simple Lie superalgebras (the case of ) constructed from -Freudenthal-Kantor triple systems [4, 7, 8].

Let be a *--Freudenthal-Kantor triple system. Then we may define the notations of tripotent and bitripotent as follows.

*Definition 2. *It is said to be a tripotent of if

*Definition 3. *It is said to be a strong bitripotent of if a pair of tripotents satisfies the relations,
and other products are zero.

*Definition 4. *It is said to be a bitripotent of if a pair of tripotents satisfies the relations
and other products are zero, where

From now on, we will consider a finite-dimensional triple system equipped with a tripotent, unless otherwise specified.

Following [16], theorems concerning tripotents (Theorems 5 and 6) are proved for Hermitian generalized Jordan triple systems. Actually, the proofs of [16] are valid in Hermitian -Freudenthal-Kantor triple systems because they are independent of . Here, we show the theorems below for completeness.

Theorem 5. *Let be a Hermitian -Freudenthal-Kantor triple system. If is flat (i.e., for all ), then we have the decomposition
**
where are tripotents or bitripotents.*

*Proof. *Let be the** R**-linear span of all , where .

From
it follows that belongs to for all . Choose a scalar product, , for example, , satisfying .

By taking the real part of and restricting , we obtain a Euclidean scalar product on such that consists of self-adjoint transformations. Furthermore, is commutative since .

By a standard calculation, may be simultaneously diagonalized. Therefore, there exists a basis of such that for all . Hence, from , we have
Furthermore, after sign changes, we may assume that .

Indeed, if , we write , then such that is positive.

Replacing by , we may assume that are tripotents.

On the other hand, if , we have

These imply that and are tripotents or bitripotents.

This completes the proof.

We define odd powers of inductively as follows:

By using this theorem, we have the following.

Theorem 6. *Let be a Hermitian -Freudenthal-Kantor triple system. Then every can be written uniquely as
**
where are tripotents or bitripotents and satisfy
*

*Proof. *Applying Theorem 5 to the subspace spanned by the powers of , from definition (i), we have for , and thus . Hence, is flat.

Without loss of generality, we can assume that . This implies that any element can be represented by , with real numbers , after permutations, and we may assume that , where are tripotents or bitripotents.

Furthermore, because are tripotents or bitripotents, the powers of are

That is, we obtain
where are polynomials in with their coefficients in .

These expressions may also be written as
This completes the proof.

For the HFKTS , we can define a norm as follows: where are tripotents or bitripotents. Note that

#### 3. Peirce Decomposition

In this section, we briefly consider the Peirce decomposition of a *--Freudenthal-Kantor triple system equipped with the tripotent .

We recall conditions (i) and (ii) in Section 2, which are equivalent to the following conditions and :

From condition with , we have where we define . From condition with , we have We write (if is a tripotent element, ), and then we obtain

When , we obtain

Summarizing the results, we have the following.

Theorem 7. *Let be a *--Freudenthal-Kantor triple system. Then, we have the following decomposition with respect to a tripotent (i.e., ):
**
where
*

#### 4. Examples

Hermitian 3-algebras are classified into and Hermitian 3-algebras [34, 36, 37, 43, 67]. In this section, we extend these 3-algebras to Hermitian -Freudenthal-Kantor triple systems and *-generalized Jordan triple systems.

*Example 1. *Let be the set of all matrices with the operation
where and denote the transpose and conjugation of , respectively.

Then is a Hermitian -Freudenthal-Kantor triple system. In fact, it satisfies conditions (0), (i), (ii), (iii), (iv), and (v) in Section 2. This is an extension of the Hermitian 3-algebra (the metric of the Hermitian 3-algebra is defined as , which is different from our Definition 1(v)), which is a basis for the effective action of multiple membranes in M-theory.

One of the tripotents is given by
where is the identity matrix (). Because any element can be decomposed as
the Peirce decomposition is given by

As in Theorem 5, we can expand any element as , where denotes that element is 1 and other elements are zero, and and are tripotents, that is, and .

*Example 2. *Let be the set of all matrices with the operation
where , is an × matrix and .

Then is a Hermitian -Freudenthal-Kantor triple system. In fact, it satisfies conditions (0), (i), (ii), (iii), (iv), and (v) in Section 2. This triple system with reduces to the Hermitian 3-algebra.

In this example, we can show that , where are tripotents and any element can be expanded by using part of them.

*Example 3. *Let be the set of all matrices with the operation

Then is a *-generalized Jordan triple system. In fact, it satisfies conditions (0), (i) with , (iii), and (iv) in Section 2 but does not satisfy (ii) and (v). This is an extension of the triple system in Example 1.

#### 5. Application to a Field Theory

In this section, we apply a *-generalized Jordan triple system to a field theory.

We start with where and are matter and gauge fields, respectively. runs from 1 to , whereas runs from 0 to 2. satisfies . This action is invariant under the transformations generated by the operator . This action describes the bosonic parts of the effective actions of supermembranes in M-theory if appropriate potential terms of are added and a Lorentzian Lie 3-algebra or a Hermitian 3-algebra is applied. Here, we apply the *-generalized Jordan triple system in Example 3 to this action.

The covariant derivative is explicitly written as where and are real antisymmetric matrices, which generate the and Lie algebras, respectively. The action can be rewritten in a covariant form with respect to and and we obtain a Chern-Simons gauge theory: In this action, and transform as adjoint representations of and , respectively, whereas transforms as a bifundamental representation of , where gauge parameters and are defined in the same way as and , respectively.

Next, let us examine whether the novel Higgs mechanism works in this theory when . By redefining the gauge fields as we can separate a nondynamical mode as where We divide into two real matrices as and consider fluctuations around a background solution as . If we rescale and as we obtain By using the equation of motion of , the action reduces to as , where and run from 1 to . Therefore, we conclude that the novel Higgs mechanism works in the Chern-Simons gauge theory with the *-generalized Jordan triple system in Example 3 with , and we obtain a Yang-Mills theory in this limit.

#### 6. Concluding Remarks

For our triple systems, we emphasized in this paper that there exists a generalized concept of Hermitian 3-algebras, which have played crucial roles in M-theory. In particular, we find the novel Higgs mechanism also works in the generalized concept.

HFKTSs and *-generalized Jordan triple systems can be regarded as left and right actions on by two Lie algebras, as one can see in the examples. For the novel Higgs mechanism, it is necessary that the two Lie algebras are the same because we need to define summation between the Lie algebras as in Section 5. Thus, the novel Higgs mechanism works only when Example 3 is applied among the examples in this paper.

Our principal physical motivation for generalizing the Hermitian 3-algebras to HKFTSs is to regularize the Nambu 3-algebra, which is defined by a Nambu bracket. The area-preserving diffeomorphism (APD) algebra defined by a Poisson bracket on a torus is regularized by the algebra in the 't Hooft base. On the other hand, the APD algebra defined on a sphere is regularized by the algebra in another base. That is, regularizations of infinite-dimensional algebras depend on the basis of the corresponding finite algebras. Although Hermitian 3-algebras ( and Hermitian 3-algebras) are strong candidates for the regularization of the Nambu 3-algebra because of their large symmetries and their relations to M2-branes, it is not clear how to choose a base for the regularization. To study this systematically, we generalized the Hermitian 3-algebras to HFKTSs ( to Example 1 and to Example 2), and we found a nilpotent basis for them. The next step is to generalize the Nambu 3-algebra to HFKTSs and to find a corresponding nilpotent base. Then by restricting the HFKTSs to the 3-algebras, we may prove that the Nambu 3-algebra is regularized by a Hermitian 3-algebra.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The work of Matsuo Sato is supported in part by Grant-in-Aid for Young Scientists (B) no. 25800122 from JSPS.