Ternary Algebras Research Articles

Multi- C ∗ $C^{*}$ -ternary algebras and applications

In this paper, we look at the concept of multi- $C^{*}$ -ternary algebras and consider some properties. As an application we approximate multi- $C^{*}$ -ternary algebra homomorphisms and derivations in these spaces.

Ternary generalization of Heisenberg's algebra

A concise study of ternary and cubic algebras with Z3 grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, S3, and its abelian subgroup Z3 may play an important role in quantum physics. We show then how most of important algebras with Z2 grading can be generalized with ternary composition laws combined with a Z3 grading. We investigate in particular a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, , one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra.An analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator, and some properties of its eigenfunctions are briefly discussed.

LIE CENTRAL TRIPLE RACKS

This paper introduces Lie c-triple racks. These triple systems generalize both left and right racks to ternary algebras, and locally differentiates to gb-triple systems.

Noncommutative Galois Extensions and Ternary Clifford Analysis

In this paper we introduce the idea of Galois extension for a class of associative algebras and discuss binary and ternary Clifford algebras by such an algebraic construction. Nonion algebra is characterized by Galois extensions and a ternary structure is proposed for \({\mathfrak{su}(3)}\) leading to a duality for certain binary and ternary differential operators.

Ternary Z3 -graded generalization of Heisenberg's algebra

We investigate a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, j = e 2πi/3, one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra.A cubic analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator. A set of eigenstates in coordinate representation is constructed in terms of functions satisfying linear differential equation of third order.

Random C ∗ -ternary algebras and application

In this paper, we introduce the concept of random \(C^{*}\)-ternary algebras and consider some properties of them. As an application we approximate a random \(C^{*}\)-ternary algebra homomorphism in these spaces.

Non-Commutative Ternary Nambu-Poisson Algebras and Ternary Hom-Nambu-Poisson Algebras

The main purpose of this paper is to study non-commutative ternary Nambu-Poisson algebras and their Homtype version. We provide construction results dealing with tensor product and direct sums of two (non- commutative) ternary (Hom-) Nambu-Poisson algebras. Moreover, we explore twisting principle of (non-commutative) ternary Nambu-Poisson algebras along with algebra morphism that lead to construct (non-commutative) ternary Hom- Nambu-Poisson algebras. Furthermore, we provide examples and a 3-dimensional classification of non-commutative ternary Nambu-Poisson algebras.

A fixed point approach to almost ternary homomorphisms and ternary derivations associated with the additive functional equation of n-Apollonius type in fuzzy ternary Banach algebras

A fixed point approach to almost ternary homomorphisms and ternary derivations associated with the additive functional equation of n-Apollonius type in fuzzy ternary Banach algebras

Group theory aspects of chaotic strings

Chaotic strings are a special type of non-hyperbolic coupled map lattices, exhibiting a rich structure of complex dynamical phenomena with a surprising correspondence to physical contents. Chaotic strings are generated by the Chebyshev maps T2(ϕ) and T3(ϕ). In this paper we connect the Chebyshev maps via the Galois theory to the cyclic groups Z2 and Z3 and give some ideas how this fundamental connection might lead to the emergence of the familiar Lie group structure of particle physics and, finally, even to the emergence of space-time. The Z3-graded cubic and ternary algebras presented here have been introduced by R. Kerner in 1991 and then developed and elaborated in collaboration with many researches. We present here the most important results associated with these papers.

Nearly higher ternary derivations in Banach ternary algebras :An alternative fixed point approach

We say a functional equation () is stable if any function g satisfying the equation () approximatelyis near to true solution of (). Using xed point methods, we investigate approximately higherternary derivations in Banach ternary algebras via the Cauchy functional equationf(1x + 2y + 3z) = 1f(x) + 2f(y) + 3f(z) :

Homomorphisms and derivations on C∗-ternary algebras associated with a generalized Cauchy-Jensen type additive functional equation

In this paper, we prove the generalized Hyers-Ulam stability of C∗-ternary homomorphisms and C∗-ternary derivations on C∗-ternary algebras associated with the generalized Cauchy-Jensen type additive functional equationfor all xi; xj ∈ X where n ∈ Z+ is a fixed integer with n ≥ 3.ABSTRACTIn this paper, we prove the generalized Hyers-Ulam stability of ∁*-ternary homomorphisms and ∁*-ternary derivations on∁*-ternary algebras associated with the generalized Cauchy-Jensen type additive functional equation for all where  is a fixed integer with

On $(n+1)$-ary derivations of simple $n$-ary Mal′tsev algebras

We defined (n+1)-ary derivations of n-ary algebras. We described 3-derivations of simple Malcev algebras and 4-ary derivations of simple ternary Malcev algebra M_8.

English

English

C *-Ternary 3-Homomorphisms on C *-Ternary Algebras

In this paper, we investigate the Ulam-Hyers stability of C*-ternary algebra 3-homomorphisms for the functional equation $$f(x_1 + x_2 + x_3, y_1 + y_2 + y_3, z_1 + z_2 + z_3) = \sum_{1\leq i,j,k\leq 3} f(x_i, y_j, z_k)$$ in C*-ternary algebras.

Notes on the Variety of Ternary Algebras

In this work we review the class T of ternary algebras introduced by J. A. Brzozowski and C. J. Serger in [1]. We determine properties of the congruence lattice of a ternary algebra A. The most important result refers to the construction of the free ternary algebra on a poset. In particular, we describe the poset of the join irreducible elements of the free ternary algebra with two free generators.

Ternary (sigma,tau,xi)-derivations on Banach ternary algebras

Let A be a Banach ternary algebra over a scalar eld R or C and X be a Banach ternary A-module.Let ; and  be linear mappings on A, a linear mapping D : (A; [ ]A) ! (X; [ ]X) is called a ternary(; ; )-derivation, ifD([xyz]A) = [D(x) (y)(z)]X + [(x)D(y)(z)]X + [(x) (y)D(z)]Xfor all x; y; z 2 A.In this paper, we investigate ternary (; ; )-derivation on Banach ternary algebras, associatedwith the following functional equationf(x + y + z4) + f(3x 􀀀 y 􀀀 4z4) + f(4x + 3z4) = 2f(x) :Moreover, we prove the generalized Ulam{Hyers stability of ternary (; ; )-derivations on Banachternary algebras.

Basics of Ternary Algebras and their underlying Nambu Brackets

Ternary algebras amount to closing systems of antisymmetrized trinomials of operators. The Filippov conditions (FI, which are not identities) for ternary algebras are contrasted to Bremner's identities dictated by associativity of operator products, and thus analogous to Jacobi identities.Maps of the known FI-compliant ternary algebras to underlying classical Nambu brackets are constructed, which then explain this compliance: FI-compliant ternary algebras are essentially classical Nambu brackets in disguise.In some cases involving infinite algebras, we show the classical limit may be obtained by a contraction of the quantal ternary algebra, and then explicitly realized through classical Nambu brackets. We illustrate this classical-contraction method on our Virasoro-Witt ternary algebra paradigm.The content of the talk is in the two references.

AUTOMATIC CONTINUITY OF 3-HOMOMORPHISMS ON TERNARY BANACH ALGEBRAS

In this paper we consider automatic continuity of 3-homomorphism and anti-3-homomorphism between non-unital ternary Banach algebras. We show that every surjective 3-homomorphism (anti-3-homomorphism) from ternary Banach algebra A into semisimple ternary Banach algebra B is continuous.

APPROXIMATE BI-HOMOMORPHISMS AND BI-DERIVATIONS IN C*-TERNARY ALGEBRAS: REVISITED

Bae and W. Park [3] proved the Hyers-Ulam stability of bi-homomorphisms and bi-derivations in <TEX>$C^*$</TEX>-ternary algebras. It is easy to show that the definitions of bi-homomorphisms and bi-derivations, given in [3], are meaningless. So we correct the definitions of bi-homomorphisms and bi-derivations. Under the conditions in the main theorems, we can show that the related mappings must be zero. In this paper, we correct the statements and the proofs of the results, and prove the corrected theorems.

Ulam Stability Generalizations of 4th- Order Ternary Derivations Associated to a Jmrassias Quartic Functional Equation on Fréchet Algebras

Let be a Banach ternary algebra over a scalar field R or C and be a ternary Banach -module. A quartic mapping is called a - order ternary derivation if for all . In this paper, we prove Ulam stability generalizations of - order ternary derivations associated to the following JMRassias quartic functional equation on frche algebras: .