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.

Classification of simple linearly compact Kantor triple systems over the complex numbers

Simple finite-dimensional Kantor triple systems over the complex numbers are classified in terms of Satake diagrams. We prove that every simple and linearly compact Kantor triple system has finite dimension and give an explicit presentation of all the classical and exceptional systems.

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.

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.

Compact Exceptional Simple Kantor Triple Systems Defined on Tensor Products of Composition Algebras*

In this article we give the classification of compact exceptional simple Kantor triple systems defined on tensor products of composition algebras A = 𝔸 1⊗ 𝔸 2 such that their Kantor algebras ℒ(φ, A) are real forms of exceptional simple Lie algebras.

On compact realifications of exceptional simple Kantor triple systems

Let A be the realification of the matrix algebra determined by Jordan algebra of hermitian matrices of order three over a complex composition algebra. We define an involutive automorphism on A with a certain action on the triple system obtained from A which give models of simple compact Kantor triple systems. In addition, we give an explicit formula for the canonical trace form and the classification for these triples and their corresponding exceptional real simple Lie algebras. Moreover, we present all realifications of complex exceptional simple Lie algebras as Kantor algebras for a compact simple Kantor triple system defined on a structurable algebra of skew-dimension one

On compact realifications of exceptional simple Kantor triple systems

On compact realifications of exceptional simple Kantor triple systems

Compact realifications of exceptional simple Kantor triple systems defined on tensor products of composition algebras

In this paper we give by a unified formula the classification of exceptional compact simple Kantor triple systems defined on tensor products of composition algebras corresponding to realifications of exceptional simple Lie algebras.

Models of Compact Simple Kantor Triple Systems Defined on a Class of Structurable Algebras of Skew-Dimension One

Let (A,−): = ℳ(J) be the 2 × 2-matrix algebra determined by Jordan algebra J: = H 3(𝔸) of hermitian 3 × 3-matrices over a real composition algebra 𝔸, where (−) is the standard involution on A. We show that the triple systems B A (x, ∼,z), x,y,z ∈ A, are models of simple compact Kantor triple systems satisfying the condition (A), where B A (x,y,z) is the triple system obtained from the algebra (A,−) and (∼) denotes a certain involution on A. In addition, we obtain an explicit formula for the canonical trace form for the triple systems B A (x, ∼,z).

A realization of the Lie algebra associated to a Kantor triple system

We present a nonlinear realization of the 5-graded Lie algebra associated to a Kantor triple system. Any simple Lie algebra can be realized in this way, starting from an arbitrary 5-grading. In particular, we get a unified realization of the exceptional Lie algebras f4,e6,e7,e8, in which they are respectively related to the division algebras R,C,H,O.

STRUCTURABLE ALGEBRAS AND MODELS OF COMPACT SIMPLE KANTOR TRIPLE SYSTEMS DEFINED ON TENSOR PRODUCTS OF COMPOSITION ALGEBRAS#

ABSTRACT Let (A, −) be a structurable algebra. Then the opposite algebra (A op , −) is structurable, and we show that the triple system B op A(x, y, z):=Vopx,y(z)=x(y¯z)+z(y¯x)−y(x¯z), x, y, z ∈ A, is a Kantor triple system (or generalized Jordan triple system of the second order) satisfying the condition (A). Furthermore, if A=𝔸1⊗𝔸2 denotes tensor products of composition algebras, (−) is the standard conjugation, and (∧) denotes a certain pseudoconjugation on A, we show that the triple systems B op 𝔸1⊗𝔸2 ( x , y¯∧, z) are models of compact Kantor triple systems. Moreover these triple systems are simple if (dim𝔸1, dim𝔸2) ≠ (2, 2). In addition, we obtain an explicit formula for the canonical trace form for compact Kantor triple systems defined on tensor products of composition algebras.

Ternary algebraic approach to extended superconformal algebras

The construction of extended ( N = 2 and N = 4) superconformal algebras (SCA) over very general classes of ternary algebras (triple systems) is given. For N = 2 this construction leads to superconformal algebras corresponding to certain Kählerian coset spaces of Lie groups with non-vanishing torsion. In general, a given Lie group admits more than one coset space of this type. The construction and a complete classification of N = 2 SCAs over Kantor triple system is given. In particular, the division algebras and their tensor products lead to N = 2 superconformal algebras associated with the coset spaces of the groups of the Magic Square. For a very special class of ternary algebras, namely the Freudenthal triple (FT) systems, the N = 2 superconformal algebras can be extended to N = 4 superconformal algebras with the gauge group SU(2)×SU(2)×U(1). The realization and a complete classification of N = 2 and N = 4