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9 pages, 224 KB

Open AccessArticle

On Some Unification Theorems: Yang–Baxter Systems; Johnson–Tzitzeica Theorem

by Florin Felix Nichita

Axioms 2025, 14(3), 156; https://doi.org/10.3390/axioms14030156 - 21 Feb 2025

Viewed by 525

This paper investigates the properties of the Yang–Baxter equation, which was initially formulated in the field of theoretical physics and statistical mechanics. The equation’s framework is extended through Yang–Baxter systems, aiming to unify algebraic and coalgebraic structures. The unification of the algebra structures [...] Read more.

This paper investigates the properties of the Yang–Baxter equation, which was initially formulated in the field of theoretical physics and statistical mechanics. The equation’s framework is extended through Yang–Baxter systems, aiming to unify algebraic and coalgebraic structures. The unification of the algebra structures and the coalgebra structures leads to an extension for the duality between finite dimensional algebras and finite dimensional coalgebras to the category of finite dimensional Yang–Baxter structures. In the same manner, we attempt to unify the Tzitzeica–Johnson theorem and its dual version, obtaining a new theorem about circle configurations. Full article

(This article belongs to the Special Issue New Perspectives in Lie Algebras)

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24 pages, 356 KB

Open AccessArticle

Set-Theoretical Solutions for the Yang–Baxter Equation in GE-Algebras: Applications to Quantum Spin Systems

by Ibrahim Senturk, Tahsin Oner, Abdullah Engin Çalık, Hüseyin Şirin, Metin Bilge and Neelamegarajan Rajesh

Axioms 2024, 13(12), 846; https://doi.org/10.3390/axioms13120846 - 2 Dec 2024

Viewed by 1194

Abstract

This manuscript presents set-theoretical solutions to the Yang–Baxter equation within the framework of GE-algebras by constructing mappings that satisfy the braid condition and exploring the algebraic properties of GE-algebras. Detailed proofs and the use of left and right translation operators are provided to [...] Read more.

This manuscript presents set-theoretical solutions to the Yang–Baxter equation within the framework of GE-algebras by constructing mappings that satisfy the braid condition and exploring the algebraic properties of GE-algebras. Detailed proofs and the use of left and right translation operators are provided to analyze these algebraic interactions, while an algorithm is introduced to automate the verification process, facilitating broader applications in quantum mechanics and mathematical physics. Additionally, the Yang–Baxter equation is applied to spin transformations in quantum mechanical spin-

\frac{1}{2}

systems, with transformations like rotations and reflections modeled using GE-algebras. A Cayley table is used to represent the algebraic structure of these transformations, and the proposed algorithm ensures that these solutions are consistent with the Yang–Baxter equation, offering new insights into the role of GE-algebras in quantum spin systems. Full article

(This article belongs to the Special Issue Yang-Baxter Equations, Nonassociative Structures and Applications—In Memoriam, Stefan Papadima)

6 pages, 239 KB

Open AccessProceeding Paper

Solutions of Yang–Baxter Equation of Mock-Lie Algebras and Related Rota Baxter Algebras

by Amir Baklouti

Comput. Sci. Math. Forum 2023, 7(1), 24; https://doi.org/10.3390/IOCMA2023-14397 - 28 Apr 2023

Viewed by 1107

Abstract

This paper discusses the relationship between Mock-Lie algebras, Lie algebras, and Jordan algebras. It highlights the importance of the Yang–Baxter equation and symplectic forms in the study of integrable systems, quantum groups, and topological quantum field theory. The paper also proposes studying the [...] Read more.

This paper discusses the relationship between Mock-Lie algebras, Lie algebras, and Jordan algebras. It highlights the importance of the Yang–Baxter equation and symplectic forms in the study of integrable systems, quantum groups, and topological quantum field theory. The paper also proposes studying the admissible associative Yang–Baxter equation in algebras and characterizing Q-Admissible Solutions of the Associative Yang–Baxter Equation in Rota–Baxter Algebras. Finally, it discusses the interplay of coalgebra, r-matrix, and pseudo-Euclidean forms in the context of Jordan algebras and its applications in quantum groups and integrable systems. Full article

(This article belongs to the Proceedings of The 1st International Online Conference on Mathematics and Applications)

7 pages, 264 KB

Open AccessCommunication

Unification Theories: Rings, Boolean Algebras and Yang–Baxter Systems

by Florin F. Nichita

Axioms 2023, 12(4), 341; https://doi.org/10.3390/axioms12040341 - 31 Mar 2023

Cited by 2 | Viewed by 1575

Abstract

This paper continues a series of papers on unification constructions. After a short discussion on the Euler’s relation, we introduce a matrix version of the Euler’s relation,

E^{I π} + U = O

. We refer to a related equation, [...] Read more.

This paper continues a series of papers on unification constructions. After a short discussion on the Euler’s relation, we introduce a matrix version of the Euler’s relation,

E^{I π} + U = O

. We refer to a related equation, the Yang–Baxter equation, and to Yang–Baxter systems. The most consistent part of the paper is on the unification of rings and Boolean algebras. These new structures are related to the Yang–Baxter equation and to Yang–Baxter systems. Full article

(This article belongs to the Special Issue Yang-Baxter Equations, Nonassociative Structures and Applications—In Memoriam, Stefan Papadima)

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17 pages, 353 KB

Open AccessArticle

Solutions of the Yang–Baxter Equation and Automaticity Related to Kronecker Modules

by Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa and Adolfo Ballester-Bolinches

Computation 2023, 11(3), 43; https://doi.org/10.3390/computation11030043 - 21 Feb 2023

Cited by 1 | Viewed by 1828

Abstract

The Kronecker algebra

K

is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over

K

are said to be Kronecker [...] Read more.

The Kronecker algebra

K

is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over

K

are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form

A x = B x^{'}

, where A and B are

m \times n

,

F

-matrices with

F

an algebraically closed field. On the other hand, researching the Yang–Baxter equation (YBE) is a topic of great interest in several science fields. It has allowed advances in physics, knot theory, quantum computing, cryptography, quantum groups, non-associative algebras, Hopf algebras, etc. It is worth noting that giving a complete classification of the YBE solutions is still an open problem. This paper proves that some indecomposable modules over

K

called pre-injective Kronecker modules give rise to some algebraic structures called skew braces which allow the solutions of the YBE. Since preprojective Kronecker modules categorize some integer sequences via some appropriated snake graphs, we prove that such modules are automatic and that they induce the automatic sequences of continued fractions. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications in Computing)

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16 pages, 390 KB

Open AccessArticle

Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

by Ashim Kumar, Dijana Mosić, Predrag S. Stanimirović, Gurjinder Singh and Lev A. Kazakovtsev

Mathematics 2022, 10(15), 2738; https://doi.org/10.3390/math10152738 - 2 Aug 2022

Cited by 4 | Viewed by 1937

Abstract

This paper investigates new solution sets for the Yang–Baxter-like (YB-like) matrix equation involving constant entries or rational functional entries over complex numbers. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as [...] Read more.

This paper investigates new solution sets for the Yang–Baxter-like (YB-like) matrix equation involving constant entries or rational functional entries over complex numbers. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as

{2, 5}

-inverses) of a matrix, which commute with it. This class of

{2, 5}

-inverses is defined based on resolving appropriate matrix equations and inner inverses. In general, solutions to such matrix equations represent optimization problems and require the minimization of corresponding matrix norms. We decided to analytically extend the obtained results to the derivation of explicit formulae for solving the YB-like matrix equation. Furthermore, algorithms for computing the solutions are developed corresponding to the suggested methods in some computer algebra systems. The main features of the proposed approach are highlighted and illustrated by numerical experiments. Full article

(This article belongs to the Section E: Applied Mathematics)

13 pages, 906 KB

Open AccessArticle

Zeroing Neural Network Approaches Based on Direct and Indirect Methods for Solving the Yang–Baxter-like Matrix Equation

by Wendong Jiang, Chia-Liang Lin, Vasilios N. Katsikis, Spyridon D. Mourtas, Predrag S. Stanimirović and Theodore E. Simos

Mathematics 2022, 10(11), 1950; https://doi.org/10.3390/math10111950 - 6 Jun 2022

Cited by 15 | Viewed by 2435

Abstract

This research introduces three novel zeroing neural network (ZNN) models for addressing the time-varying Yang–Baxter-like matrix equation (TV-YBLME) with arbitrary (regular or singular) real time-varying (TV) input matrices in continuous time. One ZNN dynamic utilizes error matrices directly arising from the equation involved [...] Read more.

This research introduces three novel zeroing neural network (ZNN) models for addressing the time-varying Yang–Baxter-like matrix equation (TV-YBLME) with arbitrary (regular or singular) real time-varying (TV) input matrices in continuous time. One ZNN dynamic utilizes error matrices directly arising from the equation involved in the TV-YBLME. Moreover, two ZNN models are proposed using basic properties of the YBLME, such as the splitting of the YBLME and sufficient conditions for a matrix to solve the YBLME. The Tikhonov regularization principle enables addressing the TV-YBLME with an arbitrary input real TV matrix. Numerical experiments, including nonsingular and singular TV input matrices, show that the suggested models deal effectively with the TV-YBLME. Full article

(This article belongs to the Special Issue Selected Papers from the International Conference of Numerical Analysis and Applied Mathematics (ICNAAM))

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8 pages, 311 KB

Open AccessArticle

On the Johnson–Tzitzeica Theorem, Graph Theory, and Yang–Baxter Equations

by Florin F. Nichita

Symmetry 2021, 13(11), 2070; https://doi.org/10.3390/sym13112070 - 2 Nov 2021

Cited by 3 | Viewed by 2333

Abstract

This paper presents several types of Johnson–Tzitzeica theorems. Graph diagrams are used in this analysis. A symmetric scheme is derived, and new results are obtained and open problems stated. We also present results relating the graphs and the Yang–Baxter equation. This equation has [...] Read more.

This paper presents several types of Johnson–Tzitzeica theorems. Graph diagrams are used in this analysis. A symmetric scheme is derived, and new results are obtained and open problems stated. We also present results relating the graphs and the Yang–Baxter equation. This equation has certain symmetries, which are used in finding solutions for it. All these constructions are related to integrable systems. Full article

(This article belongs to the Special Issue Graph Algorithms and Graph Theory)

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10 pages, 272 KB

Open AccessArticle

On the Colored and the Set-Theoretical Yang–Baxter Equations

by Laszlo Barna Iantovics and Florin Felix Nichita

Axioms 2021, 10(3), 146; https://doi.org/10.3390/axioms10030146 - 2 Jul 2021

Cited by 8 | Viewed by 2196

Abstract

This paper is related to several articles published in AXIOMS, SCI, etc. The main concepts of the current paper are the colored Yang–Baxter equation and the set-theoretical Yang–Baxter equation. The Euler formula, colagebra structures, and means play an important role in our study. [...] Read more.

This paper is related to several articles published in AXIOMS, SCI, etc. The main concepts of the current paper are the colored Yang–Baxter equation and the set-theoretical Yang–Baxter equation. The Euler formula, colagebra structures, and means play an important role in our study. We show that some new solutions for a certain system of equations lead to colored Yang–Baxter operators, which are related to an Euler formula for matrices, and the set-theoretical solutions to the Yang–Baxter equation are related to means. A new coalgebra is obtained and studied. Full article

(This article belongs to the Special Issue Non-associative Structures, Yang–Baxter Equations and Related Topics)

20 pages, 1064 KB

Open AccessArticle

The Quantum Yang-Baxter Conditions: The Fundamental Relations behind the Nambu-Goldstone Theorem

by Ivan Arraut

Symmetry 2019, 11(6), 803; https://doi.org/10.3390/sym11060803 - 17 Jun 2019

Cited by 18 | Viewed by 4175

Abstract

We demonstrate that when there is spontaneous symmetry breaking in any system, relativistic or non-relativistic, the dynamic of the Nambu-Goldstone bosons is governed by the Quantum Yang-Baxter equations. These equations describe the triangular dynamical relations between pairs of Nambu-Goldstone bosons and the degenerate [...] Read more.

We demonstrate that when there is spontaneous symmetry breaking in any system, relativistic or non-relativistic, the dynamic of the Nambu-Goldstone bosons is governed by the Quantum Yang-Baxter equations. These equations describe the triangular dynamical relations between pairs of Nambu-Goldstone bosons and the degenerate vacuum. We then formulate a theorem and a corollary showing that these relations guarantee the appropriate dispersion relation and the appropriate counting for the Nambu-Goldstone bosons. Full article

(This article belongs to the Special Issue Symmetries in the Universe)

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34 pages, 402 KB

Open AccessArticle

R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation

by Victoria Lebed

Axioms 2013, 2(3), 443-476; https://doi.org/10.3390/axioms2030443 - 5 Sep 2013

Cited by 4 | Viewed by 6546

Abstract

In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel0d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case [...] Read more.

In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel0d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the “braided” aspects of YD module structure, exhibiting a braided system encoding all the axioms from the definition of YD modules. The functoriality and several generalizations of this construction are studied using the original machinery of YD systems. As consequences, we get a conceptual interpretation of the tensor product structures for YD modules, and a generalization of the deformation cohomology of YD modules. This homology theory is thus included into the unifying framework of braided homologies, which contains among others Hochschild, Chevalley-Eilenberg, Gerstenhaber-Schack and quandle homologies. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2013)

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6 pages, 143 KB

Open AccessCommunication

Yang-Baxter Systems, Algebra Factorizations and Braided Categories

by Florin F. Nichita

Axioms 2013, 2(3), 437-442; https://doi.org/10.3390/axioms2030437 - 3 Sep 2013

Cited by 6 | Viewed by 4820

Abstract

The Yang-Baxter equation first appeared in a paper by the Nobel laureate, C.N. Yang, and in R.J. Baxter’s work. Later, Vladimir Drinfeld, Vaughan F. R. Jones and Edward Witten were awarded Fields Medals for their work related to the Yang-Baxter equation. After a [...] Read more.

The Yang-Baxter equation first appeared in a paper by the Nobel laureate, C.N. Yang, and in R.J. Baxter’s work. Later, Vladimir Drinfeld, Vaughan F. R. Jones and Edward Witten were awarded Fields Medals for their work related to the Yang-Baxter equation. After a short review on this equation and the Yang-Baxter systems, we consider the problem of constructing algebra factorizations from Yang-Baxter systems. Our sketch of proof uses braided categories. Other problems are also proposed. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2013)

12 pages, 183 KB

Open AccessArticle

Hopf Algebra Symmetries of an Integrable Hamiltonian for Anyonic Pairing

by Jon Links

Axioms 2012, 1(2), 226-237; https://doi.org/10.3390/axioms1020226 - 20 Sep 2012

Cited by 4 | Viewed by 6120

Abstract

Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of [...] Read more.

Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of its coupling parameters. While the integrable structure of the model relates to the well-known six-vertex solution of the Yang–Baxter equation, the Hopf algebra symmetries are not in terms of the quantum algebra Uq(sl(2)). Rather, they are associated with the Drinfel’d doubles of dihedral group algebras D(Dn). Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

5 pages, 201 KB

Open AccessCommunication

Introduction to the Yang-Baxter Equation with Open Problems

by Florin Nichita

Axioms 2012, 1(1), 33-37; https://doi.org/10.3390/axioms1010033 - 26 Apr 2012

Cited by 22 | Viewed by 9956

Abstract

The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter’s work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, [...] Read more.

The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter’s work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry, etc. Many scientists have found solutions for the Yang-Baxter equation, obtaining qualitative results (using the axioms of various algebraic structures) or quantitative results (usually using computer calculations). However, the full classification of its solutions remains an open problem. In this paper, we present the (set-theoretical) Yang-Baxter equation, we sketch the proof of a new theorem, we state some problems, and discuss about directions for future research. Full article

(This article belongs to the Special Issue Axioms: Feature Papers)

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