^{1}

^{2}

^{1}

^{2}

^{3}.

We define Hermitian

The structure theory of

On the other hand, it is well known that symmetric bounded domains have one-to-one correspondence to Hermitian Jordan triple systems [

From the viewpoint of string theory, HFKTSs are generalizations (A different generalization, the so-called differential crossed module, was extensively investigated in [

The Hermitian 3-algebras are special cases, where

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 [

A triple system

A triple system

All operators

This definition is a generalization of the concept of known

Let

It is said to be a tripotent of

It is said to be a strong bitripotent of

It is said to be a bitripotent of

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

Following [

Let

Let

From

By taking the real part of

By a standard calculation,

Indeed, if

Replacing

On the other hand, if

These imply that

This completes the proof.

We define odd powers of

By using this theorem, we have the following.

Let

Applying Theorem

Without loss of generality, we can assume that

Furthermore, because

That is, we obtain

These expressions may also be written as

For the HFKTS

In this section, we briefly consider the Peirce decomposition of a *-

We recall conditions (i) and (ii) in Section

From condition

When

From conditions

Summarizing the results, we have the following.

Let

Hermitian 3-algebras are classified into

Let

Then

One of the tripotents is given by

As in Theorem

Let

Then

In this example, we can show that

Let

Then

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

We start with

The covariant derivative is explicitly written as

Next, let us examine whether the novel Higgs mechanism works in this theory when

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

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

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

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