Academic literature on the topic 'Ternary Jordan algebra'

We determine the identities of degree ≤ 6 satisfied by the symmetric (Jordan) product a○b = ab + ba and the associator [a,b,c] = (ab)c - a(bc) in every nonassociative algebra. In addition to the commutative identity a○b = b○a we obtain one new identity in degree 4 and another new identity in degree 5. We demonstrate the existence of further new identities in degree 6. These identities define a variety of binary-ternary algebras which generalizes the variety of Jordan algebras in the same way that Akivis algebras generalize Lie algebras.

Novel nonassociative octonionic ternary gauge field theories are proposed based on a ternary bracket. This paves the way to the many physical applications of exceptional Jordan Strings/Membranes and Octonionic Gravity. The old octonionic gravity constructions based on the split octonion algebra Os (which strictly speaking is not a division algebra) is extended to the full fledged octonion division algebra O. A real-valued analog of the Einstein–Hilbert Lagrangian [Formula: see text] involving sums of all the possible contractions of the Ricci tensors plus their octonionic-complex conjugates is presented. A discussion follows of how to extract the Standard Model group (the gauge fields) from the internal part of the octonionic gravitational connection. The role of exceptional Jordan algebras, their automorphism and reduced structure groups which play the roles of the rotation and Lorentz groups is also re-examined. Finally, we construct (to our knowledge) generalized novel octonionic string and p-brane actions and raise the possibility that our generalized 3-brane action (based on a quartic product) in octonionic flat backgrounds of 7,8 octonionic dimensions may display an underlying E7, E8 symmetry, respectively. We conclude with some final remarks pertaining to the developments related to Jordan exceptional algebras, octonions, black-holes in string theory and quantum information theory.

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

When Loday and Ronco studied ternary planar trees, they introduced types of algebras,called trioids and trialgebras. A trioid is a nonempty set equipped with three binary associativeoperations satisfying additional eight axioms relating these operations, while a trialgebra is justa linear analog of a trioid. If all operations of a trioid (trialgebra) coincide, we obtain the notionof a semigroup (associative algebra), and if two concrete operations of a trioid (trialgebra)coincide, we obtain the notion of a dimonoid (dialgebra) and so, trioids (trialgebras) are ageneralization of semigroups (associative algebras) and dimonoids (dialgebras). Trioids andtrialgebras have close relationships with the Hopf algebras, the Leibniz 3-algebras, the Rota-Baxter operators, and the post-Jordan algebras. Originally, these structures arose in algebraictopology. One of the most useful concepts in algebra is the free object. Every variety containsfree algebras and free objects in any variety of algebras are important in the study of thatvariety. Loday and Ronco constructed the free trioid of rank 1 and the free trialgebra. Recently,the free trioid of an arbitrary rank, the free commutative trioid, the free n-nilpotent trioid, thefree rectangular triband, the free left n-trinilpotent trioid and the free abelian trioid wereconstructed and the least dimonoid congruences as well as the least semigroup congruence onthe first four free algebras were characterized. However, just mentioned congruences on freeleft (right) n-trinilpotent trioids and free abelian trioids were not considered. In this paper, wecharacterize the least dimonoid congruences and the least semigroup congruence on free left(right) n-trinilpotent trioids and free abelian trioids.

Abstract The purpose of the following is to try to make sense of the stereo- graphic projection in a non-commutative setup. To this end, we consider the open unit ball of a ternary ring of operators, which naturally comes equipped with a non-commutative version of a hyperbolic metric and ask for a manifold onto which the open unit ball can be mapped so that one might think of this situation as providing a noncommutative analog to mapping the open disk of complex numbers onto the hyperboloid in three space, equipped with the restriction of the Minkowskian metric. We also obtain a related result on the Jordan algebra of self-adjoint operators

In this work, by using some orthogonally fixed point theorem, we prove the stability and hyperstability of orthogonally C ∗ -ternary Jordan homomorphisms between C ∗ -ternary Banach algebras and orthogonally C ∗ -ternary Jordan derivations of some functional equation on C ∗ -ternary Banach algebras.

C. Park et al. proved the stability of homomorphisms and derivations in Banach algebras, Banach ternary algebras, C*-algebras, Lie C*-algebras and C*-ternary algebras. In this paper, we improve and generalize some results concerning derivations. We first introduce the following generalized Jensen functional equation [Formula: see text] and using fixed point methods, we prove the stability of n-ary derivations and n-ary Jordan derivations in n-ary Banach algebras. Secondly, we study this functional equation with *-n-ary derivations in C*-n-ary algebras.

L'objectif de cette thèse est d'étudier les opérateurs de Rota-Baxter relatifs sur les algèbres ternaires de type Lie et de type Jordan. L'étude porte sur leur structure, leur cohomologie, leurs déformations et leur lien avec les équations de Yang-Baxter. Ce travail est divisé en trois parties. La première partie est consacrée à l'étude de l'algèbre de contrôle des systèmes triples de Lie, et à son application à la théorie existante de la cohomologie. De plus, nous introduisons la notion d'opérateur de Rota-Baxter relatif sur les systèmes triples de Lie et construisons une 3-algèbre de Lie comme cas spécial des L∞-algèbres dont les éléments de Maurer-Cartan sont des opérateurs de Rota-Baxter relatifs. Dans la deuxième partie, nous introduisons la notion d'opérateur de Rota-Baxter relatif twisté sur les algèbres 3-Lie et construisons une L∞-algèbre dont les éléments de Maurer-Cartan sont des opérateurs de Rota-Baxter relatifs twistés. Cela nous permet de définir la cohomologie de Chevalley-Eilenberg d'un opérateur de Rota-Baxter relatif twisté. Dans la dernière partie, nous étudions la représentation des algèbres ternaires de Jordan, ce qui nous permet d'introduire la notion d'algèbres ternaires de Jordan cohérentes. Ensuite, les opérateurs de Rota-Baxter relatifs des algèbres ternaires de Jordan sont introduits et les solutions de l'équation de Yang-Baxter de Jordan ternaire sont discutées en impliquant des opérateurs de Rota-Baxter relatifs
The goal of this thesis is to explore relative Rota-Baxter operators in the context of ternary algebras of both Lie and Jordan types. We mainly consider Lie triple systems, 3-Lie algebras and ternary Jordan algebras. The study covers their structure, cohomology, deformations, and their connection with the Yang-Baxter equations. The work is divided into three main parts. The first part aims first to introduce and study a graded Lie algebra whose Maurer-Cartan elements are Lie triple systems. It turns out to be the controlling algebra of Lie triple systems deformations and fits with the adjoint cohomology theory of Lie triple systems introduced by Yamaguti. In addition, we introduce the notion of relative Rota-Baxter operators on Lie triple systems and construct a Lie 3-algebra as a special case of L∞-algebras, where the Maurer-Cartan elements correspond to relative Rota-Baxter operators. In the second part, we introduce the concept of twisted relative Rota-Baxter operators on 3-Lie algebras and construct an L∞-algebra, where the Maurer-Cartan elements are twisted relative Rota-Baxter operators. This allows us to define the Chevalley-Eilenberg cohomology of a twisted relative Rota-Baxter operator. In the last part, we deal with a representation theory of ternary Jordan algebras. In particular, we introduce and discuss the concept of coherent ternary Jordan algebras. We then define relative Rota-Baxter operators for ternary Jordan algebras and discuss solutions ofthe ternary Jordan Yang-Baxter equation involving relative Rota-Baxter operators. Moreover, we investigate ternary pre-Jordan algebras as the underlying algebraic structure of relative Rota-Baxter operators