Freudenthal-Kantor Triple Systems Research Articles

On constructions of Lie (super) algebras and (𝜀,δ)-Freudenthal–Kantor triple systems defined by bilinear forms

In this work, we discuss a classification of [Formula: see text]-Freudenthal–Kantor triple systems defined by bilinear forms and give all examples of such triple systems. From these results, we may see a construction of some simple Lie algebras or superalgebras associated with their Freudenthal–Kantor triple systems. We also show that we can associate a complex structure into these ([Formula: see text]-Freudenthal–Kantor triple systems. Further, we introduce the concept of Dynkin diagrams associated to such [Formula: see text]-Freudenthal–Kantor triple systems and the corresponding Lie (super) algebra construction.

Symmetry of Lie algebras associated with (ε, δ)-Freudenthal-Kantor triple system

AbstractSymmetry groups of Lie algebras and superalgebras constructed from (∈, δ)-Freudenthal-Kantor triple systems have been studied. In particular, for a special (ε, ε)-Freudenthal–Kantor triple, it is the SL(2) group. Also, the relationship between two constructions of Lie algebras from structurable algebras has been investigated.

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.

3-qubit entanglement: A Jordan algebraic perspective

It is by now well known that three qubits can be totally entangled in two physically distinct ways. Here we review work classifying the physically distinct forms of 3-qubit entanglement using the elegant framework of Jordan algebras, Freudenthal-Kantor triple systems and groups of type E7. In particular, it is shown that the four Freudenthal-Kantor ranks correspond precisely to the four 3-qubit entanglement classes: (1) Totally separable A-B-C, (2) Biseparable A-BC, B-CA, C-AB, (3) Totally entangled W, (4) Totally entangled GHZ. The rank 4 GHZ class is regarded as maximally entangled in the sense that it has non-vanishing quartic norm, the defining invariant of the Freudenthal-Kantor triple system. While this framework is specific to three qubits, we show here how the essential features may be naturally generalised to an arbitrary number of qubits.

Examples of Freudenthal-Kantor Triple Systems

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

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.

Left unital Kantor triple systems and structurable algebras

Left unital Kantor triple systems will be shown to coincide with the triple systems attached to structurable algebras endowed with an involutive automorphism , with triple product given by the formula . A related result is proved for Freudenthal-Kantor triple systems. Some consequences for the associated five-graded Lie algebras and superalgebras are deduced too. In particular, left unital Freudenthal-Kantor triple systems are shown to be closely related to Lie superalgebras strictly graded over the root system of type .

Special Freudenthal–Kantor triple systems and Lie algebras with dicyclic symmetry

We study Lie algebras endowed with an action by automorphisms of the dicyclic group of degree 3. The close connections of these algebras with Lie algebras graded over the non-reduced root system BC1, with J-ternary algebras and with Freudenthal–Kantor triple systems are explored.

A CHARACTERIZATION OF (−1, −1)-FREUDENTHAL–KANTOR TRIPLE SYSTEMS

AbstractIn this paper, we discuss a connection between (−1, −1)-Freudenthal–Kantor triple systems, anti-structurable algebras, quasi anti-flexible algebras and give examples of such structures. The paper provides the correspondence and characterization of a bilinear product corresponding a triple product.

Representations of (α, β, γ) triple system

We introduce a notion of (α, β, γ) triple system which generalizes the familiar generalized Jordan triple system. We then discuss its realization by some bilinear algebras and vice versa. We also give a characterization of the structurable algebra of Allison in terms of (−1, 1) Freudenthal–Kantor triple system by imposing some additional triple product constraints.

Dialgebras and related triple systems

We consider some algebraical systems that lead to various nearly associative triple systems. We deal with a class of algebras which contains Leibniz-Poisson algebras, dialgebras, conformal algebras, and some triple systems. We describe all homogeneous structures of ternary Leibniz algebras on a dialgebra. For this purpose, in particular, we use the Leibniz-Poisson structure on a dialgebra. We then find a corollary describing the structure of a Lie triple system on an arbitrary dialgebra, a conformal associative algebra and a classical associative triple system. We also describe all homogeneous structures of an (e, δ)-Freudenthal-Kantor triple system on a dialgebra.

The Wedderburn principal theorem, nilpotency and solvability for Freudenthal-Kantor triple systems

The Wedderburn principal theorem, nilpotency and solvability for Freudenthal-Kantor triple systems

On classical Yang-Baxter equations associated with Freudenthal-Kantor triple systems

A modified classical Yang-Baxter equation has been studied ??? triple product form and its relevance to well-known Freudenthal-Kantor triple systems is discussed with some examples.

On the peirce decompositions for freudenthal—kantor triple systems

It is the object of this paper to investigate the Peirce decomposition for Freudenthal-Kantor triple systems, by making use of the concept of anti-derivation of the triple systems.

A structure theory of Freudenthal-Kantor triple systems

The notion of the radical of a Freudenthal-Kantor triple system is introduced. Using this, it is shown that a semisimple Freudenthal-Kantor triple system is decomposed into a direct sum of simple ideals. Also this paper defines and characterizes the structure groups for Freudenthal-Kantor triple systems.

A construction of Lie algebras and Lie superalgebras by Freudenthal-Kantor triple systems, I

A construction of Lie algebras and Lie superalgebras by Freudenthal-Kantor triple systems, I