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Journal articles on the topic 'Ternary Jordan algebra'

Author: Grafiati

Published: 26 April 2025

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1

BREMNER, MURRAY R., and IRVIN R. HENTZEL. "IDENTITIES RELATING THE JORDAN PRODUCT AND THE ASSOCIATOR IN THE FREE NONASSOCIATIVE ALGEBRA." Journal of Algebra and Its Applications 05, no. 01 (February 2006): 77–88. http://dx.doi.org/10.1142/s0219498806001594.

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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.

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2

CASTRO, CARLOS. "ON OCTONIONIC GRAVITY, EXCEPTIONAL JORDAN STRINGS AND NONASSOCIATIVE TERNARY GAUGE FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 09, no. 03 (May 2012): 1250021. http://dx.doi.org/10.1142/s0219887812500211.

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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.

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3

Šlapal, Josef. "Digital Jordan Curves and Surfaces with Respect to a Closure Operator." Fundamenta Informaticae 179, no. 1 (February 9, 2021): 59–74. http://dx.doi.org/10.3233/fi-2021-2013.

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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.

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4

Zhuchok, A. V. "The least dimonoid congruences on relatively free trioids." Matematychni Studii 57, no. 1 (March 31, 2022): 23–31. http://dx.doi.org/10.30970/ms.57.1.23-31.

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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.

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5

Stachó, Lászlo L., and Wend Werner. "On non-commutative Minkowski spheres." Analele Universitatii "Ovidius" Constanta - Seria Matematica 20, no. 2 (June 1, 2012): 159–70. http://dx.doi.org/10.2478/v10309-012-0047-y.

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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

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6

Keshavarz, Vahid, and Sedigheh Jahedi. "Orthogonally C ∗ -Ternary Jordan Homomorphisms and Jordan Derivations: Solution and Stability." Journal of Mathematics 2022 (December 26, 2022): 1–7. http://dx.doi.org/10.1155/2022/3482254.

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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.

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7

GHARETAPEH, S. KABOLI, MADJID ESHAGHI GORDJI, M. B. GHAEMI, and E. RASHIDI. "TERNARY JORDAN HOMOMORPHISMS IN C∗ -TERNARY ALGEBRAS." Journal of Nonlinear Sciences and Applications 04, no. 01 (February 12, 2011): 1–10. http://dx.doi.org/10.22436/jnsa.004.01.01.

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8

GORDJI, M. ESHAGHI, R. KHODABAKHSH, and H. KHODAEI. "ON APPROXIMATE n-ARY DERIVATIONS." International Journal of Geometric Methods in Modern Physics 08, no. 03 (May 2011): 485–500. http://dx.doi.org/10.1142/s0219887811005245.

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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.

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9

Savadkouhi, M. Bavand, M. Eshaghi Gordji, J. M. Rassias, and N. Ghobadipour. "Approximate ternary Jordan derivations on Banach ternary algebras." Journal of Mathematical Physics 50, no. 4 (April 2009): 042303. http://dx.doi.org/10.1063/1.3093269.

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10

Kaygorodov, Ivan, Alexander Pozhidaev, and Paulo Saraiva. "On a ternary generalization of Jordan algebras." Linear and Multilinear Algebra 67, no. 6 (March 5, 2018): 1074–102. http://dx.doi.org/10.1080/03081087.2018.1443426.

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11

Elduque, Alberto, and Susumu Okubo. "Lie algebras with S3- or S4-action and generalized Malcev algebras." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 2 (March 25, 2009): 321–57. http://dx.doi.org/10.1017/s0308210508000164.

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Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied. In the case of S3-symmetry, the Lie algebras are coordinatized by some non-associative systems, which are termed generalized Malcev algebras, as they extend the classical Malcev algebras. These systems are endowed with a binary and a ternary product, and include both the Malcev algebras and the Jordan triple systems.

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12

Shestakov, A. I. "Ternary derivations of separable associative and jordan algebras." Siberian Mathematical Journal 53, no. 5 (September 2012): 943–56. http://dx.doi.org/10.1134/s0037446612050199.

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13

GÜNAYDIN, MURAT, and SEUNGJOON HYUN. "TERNARY ALGEBRAIC CONSTRUCTION OF EXTENDED SUPERCONFORMAL ALGEBRAS." Modern Physics Letters A 06, no. 19 (June 21, 1991): 1733–43. http://dx.doi.org/10.1142/s0217732391001871.

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We give a construction of extended (N = 2 and N = 4) superconformal algebras over a very general class of ternary algebras (triple systems). For N = 2 this construction leads to superconformal algebras corresponding to certain coset spaces of Lie groups with non-vanishing torsion and generalizes a previous construction over Jordan triple systems which are associated with Hermitian symmetric spaces. In general, a given Lie group admits more than one coset space of this type. We give examples for all simple Lie groups. In particular, the division algebras and their tensor products lead to N = 2 superconformal algebras associated with 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). We give a complete list of the FT systems and the corresponding N = 4 models. They are associated with the unique quaternionic symmetric spaces of Lie groups.

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14

O’Regan, Donal, John Michael Rassias, and Reza Saadati. "Approximations of ternary Jordan homomorphisms and derivations in multi-C ∗ ternary algebras." Acta Mathematica Hungarica 134, no. 1-2 (May 21, 2011): 99–114. http://dx.doi.org/10.1007/s10474-011-0116-0.

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15

Vlasov, Alexander Yurievich. "Clifford Algebras, Spin Groups and Qubit Trees." Quanta 11, no. 1 (December 1, 2022): 97–114. http://dx.doi.org/10.12743/quanta.v11i1.199.

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Representations of Spin groups and Clifford algebras derived from the structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deletion of superfluous branches. The usual Jordan–Wigner construction also may be formally obtained in this approach by bringing the process up to trivial qubit chain (trunk). The methods can also be used for effective simulation of some quantum circuits corresponding to the binary tree structure. The modeling of more general qubit trees, as well as the relationship with the mapping used in the Bravyi–Kitaev transformation, are also briefly discussed.Quanta 2022; 11: 97–114.

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16

Park, Choonkil, and M. Eshaghi Gordji. "Comment on “Approximate ternary Jordan derivations on Banach ternary algebras” [Bavand Savadkouhiet al.J. Math. Phys. 50, 042303 (2009)]." Journal of Mathematical Physics 51, no. 4 (April 2010): 044102. http://dx.doi.org/10.1063/1.3299295.

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17

Ebadian, A., N. Ghobadipour, and M. Eshaghi Gordji. "A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C∗-ternary algebras." Journal of Mathematical Physics 51, no. 10 (October 2010): 103508. http://dx.doi.org/10.1063/1.3496391.

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18

YUN, SUNGSIK, MADJID ESHAGHI GORDJI, and JEONG PIL SEO. "ERRATUM: “A FIXED POINT METHOD FOR PERTURBATION OF BIMULTIPLIERS AND JORDAN BIMULTIPLIERS IN C*-TERNARY ALGEBRAS” [J. MATH. PHYS. 51, 103508 (2010)]." Pure and Applied Mathematics 23, no. 3 (August 31, 2016): 237–46. http://dx.doi.org/10.7468/jksmeb.2016.23.3.237.

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19

El Bachraoui, Mohamed. "Convolution over Lie and Jordan algebras." Contributions to Discrete Mathematics 1, no. 1 (March 24, 2006). http://dx.doi.org/10.55016/ojs/cdm.v1i1.61883.

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Given a ternary relation C on a set U and an algebra A, we present a construction of a convolution algebra A(U, C) of U = (U, C) over A. This generalises bothmatrix algebras and algebras obtained from convolution of monoids. To any class of algebras corresponds a class of convolution structures. Our study cases are the classes of commutative, associative, Lie, and Jordan algebras. In each of these classes we give conditions on (U, C) under which A(U, C) is in the same class as A. It turns out that in some situations these conditions are even necessary.

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20

Arfa, Anja, Abdelkader Ben Hassine, and Sami Mabrouk. "Ternary Hom-Jordan algebras induced by Hom-Jordan algebras." Linear and Multilinear Algebra, December 17, 2020, 1–25. http://dx.doi.org/10.1080/03081087.2020.1859441.

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21

Chtioui, Taoufik, Atef Hajjaji, and Sami Mabrouk. "𝒪-operators of ternary Jordan algebras and ternary Jordan Yang-Baxter equations." Asian-European Journal of Mathematics, June 4, 2022. http://dx.doi.org/10.1142/s1793557123500262.

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22

Kaygorodov, Ivan, Mykola Khrypchenko, and Pilar Páez-Guillán. "The geometric classification of non-associative algebras: a survey." Communications in Mathematics Volume 32 (2024), Issue 2... (December 6, 2024). https://doi.org/10.46298/cm.14458.

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This is a survey on the geometric classification of different varieties of algebras (nilpotent, nil-, associative, commutative associative, cyclic associative, Jordan, Kokoris, standard, noncommutative Jordan, commutative power-associative, weakly associative, terminal, Lie, Malcev, binary Lie, Tortkara, dual mock Lie, $\mathfrak{CD}$-, commutative $\mathfrak{CD}$-, anticommutative $\mathfrak{CD}$-, symmetric Leibniz, Leibniz, Zinbiel, Novikov, bicommutative, assosymmetric, antiassociative, left-symmetric, right alternative, and right commutative), $n$-ary algebras (Fillipov ($n$-Lie), Lie triple systems and anticommutative ternary), superalgebras (Lie and Jordan), and Poisson-type algebras (Poisson, transposed Poisson, Leibniz-Poisson, generic Poisson, generic Poisson-Jordan, transposed Leibniz-Poisson, Novikov-Poisson, pre-Lie Poisson, commutative pre-Lie, anti-pre-Lie Poisson, pre-Poisson, compatible commutative associative, compatible associative, compatible Novikov, compatible pre-Lie). We also discuss the degeneration level classification.

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