Quadratic Jordan Triple Systems

In this work, We show that every Jordan triple system can be viewed as a T∗extension of another one or an ideal of co-dimension one of a Jordan triple system whose represent the T∗extension of another Jordan triple system. Moreover, several result involving the structure of quadratic Jordan triple systems are given.

Representations of simple anti-Jordan triple systems of m × n matrices

We show that the universal associative envelope of the simple anti-Jordan triple system of all [Formula: see text] ([Formula: see text] is even, [Formula: see text]) matrices over an algebraically closed field of characteristic 0 is finite-dimensional. The monomial basis and the center of the universal envelope are determined. The explicit decomposition of the universal envelope into matrix algebras is given. The classification of finite-dimensional irreducible representations of an anti-Jordan triple system is obtained. The semi-simplicity of the universal envelope is shown. We also show that the universal associative envelope of the simple polarized anti-Jordan triple system of [Formula: see text] matrices is infinite-dimensional.

Universal Enveloping Algebras of Simple Symplectic Anti-Jordan Triple Systems

In this work we are concerned with the universal associative envelope of a finite-dimensional simple symplectic anti-Jordan triple system (AJTS). We prove that if 𝕋 is a triple system as above, then there exists an associative algebra U(𝕋) and an injective homomorphism ε : 𝕋 → U(𝕋), where U(𝕋) is an AJTS under the triple product defined by (a,b,c) = abc - cba. Moreover, U(𝕋) is a universal object with respect to such homomorphisms. We explicitly determine the PBW-basis of U(𝕋), the center Z(U(𝕋)) and the Gelfand-Kirillov dimension of U(𝕋).

A class of Hermitian generalized Jordan triple systems and Chern–Simons gauge theory

We find a class of Hermitian generalized Jordan triple systems (HGJTSs) and Hermitian (ϵ, δ)-Freudenthal–Kantor triple systems (HFKTSs). We apply one of the most simple HGJTSs which we find to a field theory and obtain a typical u(N) Chern–Simons gauge theory with a fundamental matter.

Hermitian generalized Jordan triple systems and certain applications to field theory

We define Hermitian generalized Jordan triple systems and prove a structure theorem. We also give some examples of the systems and study mathematical properties. We apply a Hermitian generalized Jordan triple system to a field theory and obtain a Chern–Simons gauge theory.

Hermitian(ϵ,δ)-Freudenthal-Kantor Triple Systems and Certain Applications of*-Generalized Jordan Triple Systems to Field Theory

We define Hermitian(ϵ,δ)-Freudenthal-Kantor triple systems and prove a structure theorem. We also give some examples of triple systems that are generalizations of theu(N)⊕u(M)andsp(2N)⊕u(1)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.