Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 09/2017, Volume 86, Issue 307, pp. 2519 - 2534

Braces were introduced by Rump to study non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. We generalize Rump's braces to the...

MATHEMATICS, APPLIED | SET-THEORETIC SOLUTIONS | CONJECTURE

MATHEMATICS, APPLIED | SET-THEORETIC SOLUTIONS | CONJECTURE

Journal Article

Journal of Algebra, ISSN 0021-8693, 08/2017, Volume 483, pp. 163 - 187

In this paper we obtain new solutions of the Yang–Baxter equation that are left non-degenerate through left semi-braces, a generalization of braces introduced...

Quantum Yang–Baxter equation | Set-theoretical solution | Skew brace | Semi-brace | SET-THEORETICAL SOLUTIONS | MATHEMATICS | REGULAR SUBGROUPS | Quantum Yang-Baxter equation | AFFINE GROUP

Quantum Yang–Baxter equation | Set-theoretical solution | Skew brace | Semi-brace | SET-THEORETICAL SOLUTIONS | MATHEMATICS | REGULAR SUBGROUPS | Quantum Yang-Baxter equation | AFFINE GROUP

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 05/2019, Volume 348, pp. 523 - 530

Let A ∈ Cl × l be a diagonalizable matrix with spectrum contained in the set {1, α, 0}. In this paper we derive a general and explicit expression for the...

Eigenvalues | Yang–Baxter-like matrix equation | Diagonalizable matrix | MATHEMATICS, APPLIED | Yang-Baxter-like matrix equation

Eigenvalues | Yang–Baxter-like matrix equation | Diagonalizable matrix | MATHEMATICS, APPLIED | Yang-Baxter-like matrix equation

Journal Article

Advances in Mathematics, ISSN 0001-8708, 11/2018, Volume 338, pp. 649 - 701

We involve simultaneously the theory of braided groups and the theory of braces to study set-theoretic solutions of the Yang–Baxter equation (YBE). We show the...

Yang–Baxter | Braces | Braided groups | QUADRATIC ALGEBRAS | HOPF-ALGEBRAS | EXTENSIONS | BICROSSPRODUCT | Yang-Baxter | MATHEMATICS | SKEW POLYNOMIAL-RINGS | SEMIGROUPS | MAPS | MATCHED PAIRS | HOMOLOGY | BINOMIAL RELATIONS | Algebra

Yang–Baxter | Braces | Braided groups | QUADRATIC ALGEBRAS | HOPF-ALGEBRAS | EXTENSIONS | BICROSSPRODUCT | Yang-Baxter | MATHEMATICS | SKEW POLYNOMIAL-RINGS | SEMIGROUPS | MAPS | MATCHED PAIRS | HOMOLOGY | BINOMIAL RELATIONS | Algebra

Journal Article

Journal of Algebra and its Applications, ISSN 0219-4988, 01/2018, Volume 17, Issue 1

In this work, we compute solutions of the Yang-Baxter associative equation in dimensions one and two. For these solutions, we describe the double constructions...

Yang-Baxter equation | Frobenius algebra | Connes cocycle | dendriform algebra | D -equation | Associative algebra | D-equation | MATHEMATICS, APPLIED | HOPF-ALGEBRAS | MATHEMATICS | TREES | POISSON ALGEBRAS | HOMOLOGY | BIALGEBRAS

Yang-Baxter equation | Frobenius algebra | Connes cocycle | dendriform algebra | D -equation | Associative algebra | D-equation | MATHEMATICS, APPLIED | HOPF-ALGEBRAS | MATHEMATICS | TREES | POISSON ALGEBRAS | HOMOLOGY | BIALGEBRAS

Journal Article

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, ISSN 1751-8113, 08/2019, Volume 52, Issue 35, p. 355201

An elliptic Bailey lemma is formulated on the basis of the univariate rarefied elliptic beta integral. It leads to a generalized operator star-triangle...

BETA-INTEGRALS | star-triangle relation | Yang-Baxter equation | MODELS | PHYSICS, MULTIDISCIPLINARY | Bailey lemma | PHYSICS, MATHEMATICAL | PARAMETER

BETA-INTEGRALS | star-triangle relation | Yang-Baxter equation | MODELS | PHYSICS, MULTIDISCIPLINARY | Bailey lemma | PHYSICS, MATHEMATICAL | PARAMETER

Journal Article

7.
Full Text
Iterative methods for finding commuting solutions of the Yang–Baxter-like matrix equation

Applied Mathematics and Computation, ISSN 0096-3003, 09/2018, Volume 333, pp. 246 - 253

The main goal of this paper is the numerical computation of solutions of the so-called Yang–Baxter-like matrix equation AXA=XAX, where A is a given complex...

Yang–Baxter-like matrix equation | Stability | Idempotent matrix | Iterative methods | Fréchet derivative | Convergence | MATHEMATICS, APPLIED | Yang-Baxter-like matrix equation | Frechet derivative

Yang–Baxter-like matrix equation | Stability | Idempotent matrix | Iterative methods | Fréchet derivative | Convergence | MATHEMATICS, APPLIED | Yang-Baxter-like matrix equation | Frechet derivative

Journal Article

Journal of Algebra, ISSN 0021-8693, 10/2016, Volume 463, pp. 80 - 102

Given a left brace B, a method is given to construct explicitly all the non-degenerate involutive set-theoretic solutions (X,r) of the Yang–Baxter equation...

Yang–Baxter equation | Brace | Set-theoretic solution | MATHEMATICS | ALGEBRAS | Yang Baxter equation | I-TYPE | SET-THEORETIC SOLUTIONS

Yang–Baxter equation | Brace | Set-theoretic solution | MATHEMATICS | ALGEBRAS | Yang Baxter equation | I-TYPE | SET-THEORETIC SOLUTIONS

Journal Article

Journal of Pure and Applied Algebra, ISSN 0022-4049, 10/2019, Volume 223, Issue 10, pp. 4477 - 4493

We describe the indecomposable involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation as dynamical extensions of non-degenerate left...

Yang–Baxter equation | Cycle set | Set-theoretic solution | MATHEMATICS | MATHEMATICS, APPLIED | Yang-Baxter equation

Yang–Baxter equation | Cycle set | Set-theoretic solution | MATHEMATICS | MATHEMATICS, APPLIED | Yang-Baxter equation

Journal Article

Journal of Algebra, ISSN 0021-8693, 01/2019, Volume 517, pp. 1 - 18

We introduce a new family of classical r-matrices for the Lie algebra sln that lies in the Zariski boundary of the Belavin–Drinfeld space M of quasi-triangular...

Frobenius functionals | Cremmer–Gervais r-matrices | Parabolic subalgebras | Classical Yang–Baxter equation | Principal elements | Frobenius Lie algebras | LIE-ALGEBRAS | MATHEMATICS | Classical Yang-Baxter equation | INDEX | Cremmer-Gervais r-matrices | Subprime loans | Algebra

Frobenius functionals | Cremmer–Gervais r-matrices | Parabolic subalgebras | Classical Yang–Baxter equation | Principal elements | Frobenius Lie algebras | LIE-ALGEBRAS | MATHEMATICS | Classical Yang-Baxter equation | INDEX | Cremmer-Gervais r-matrices | Subprime loans | Algebra

Journal Article

Advances in Mathematics, ISSN 0001-8708, 02/2019, Volume 343, pp. 273 - 315

We show that all strongly non-degenerate trigonometric solutions of the associative Yang–Baxter equation (AYBE) can be obtained from triple Massey products in...

Associative Yang–Baxter equation | Fukaya category | Square-tiled surface | MATHEMATICS | Associative Yang-Baxter equation | GENUS | TRIGONOMETRIC SOLUTIONS | CURVES | MIRROR SYMMETRY

Associative Yang–Baxter equation | Fukaya category | Square-tiled surface | MATHEMATICS | Associative Yang-Baxter equation | GENUS | TRIGONOMETRIC SOLUTIONS | CURVES | MIRROR SYMMETRY

Journal Article

Journal of Algebra, ISSN 0021-8693, 04/2016, Volume 451, pp. 494 - 525

In this paper, we initiate the study of the interplay between k-graphs and the Yang–Baxter equation. For this, we provide two very different perspectives. On...

Yang–Baxter equation | k-Graph | Set-theoretic solution | Yang-Baxter equation | K-Graph | SET-THEORETICAL SOLUTIONS | MATHEMATICS | SEMIGROUPS | BRACES | INVARIANTS | C-ASTERISK-ALGEBRAS | PRODUCT SYSTEMS | HIGHER-RANK GRAPHS | HOMOLOGY

Yang–Baxter equation | k-Graph | Set-theoretic solution | Yang-Baxter equation | K-Graph | SET-THEORETICAL SOLUTIONS | MATHEMATICS | SEMIGROUPS | BRACES | INVARIANTS | C-ASTERISK-ALGEBRAS | PRODUCT SYSTEMS | HIGHER-RANK GRAPHS | HOMOLOGY

Journal Article

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, ISSN 1751-8113, 11/2019, Volume 52, Issue 48, p. 485201

We construct birational maps that satisfy the parametric set-theoretical Yang?Baxter equation and its entwining generalisation. For this purpose, we employ...

Darboux transformations | NLS type equations | YANG-BAXTER MAPS | PHYSICS, MULTIDISCIPLINARY | entwining parametric Yang?Baxter maps | PHYSICS, MATHEMATICAL | Liouville integrability

Darboux transformations | NLS type equations | YANG-BAXTER MAPS | PHYSICS, MULTIDISCIPLINARY | entwining parametric Yang?Baxter maps | PHYSICS, MATHEMATICAL | Liouville integrability

Journal Article

Journal of Algebra, ISSN 0021-8693, 05/2020, Volume 549, pp. 268 - 290

We use ring-theoretic methods and methods from the theory of skew braces to produce set-theoretic solutions to the reflection equation. We also use...

Yang-Baxter equation | Radical ring | Brace | Reflection equation | Skew brace

Yang-Baxter equation | Radical ring | Brace | Reflection equation | Skew brace

Journal Article

Theoretical and Mathematical Physics, ISSN 0040-5779, 10/2016, Volume 189, Issue 1, pp. 1472 - 1485

We survey the matrix product solutions of the Yang–Baxter equation recently obtained from the tetrahedron equation. They form a family of quantum R-matrices of...

Yang–Baxter map | tetrahedron equation | generalized quantum group | Yang–Baxter equation | Theoretical, Mathematical and Computational Physics | Applications of Mathematics | Physics | CRYSTAL BASES | Yang-Baxter equation | ANALOG | PHYSICS, MULTIDISCIPLINARY | Yang-Baxter map | VERTEX MODELS | PHYSICS, MATHEMATICAL

Yang–Baxter map | tetrahedron equation | generalized quantum group | Yang–Baxter equation | Theoretical, Mathematical and Computational Physics | Applications of Mathematics | Physics | CRYSTAL BASES | Yang-Baxter equation | ANALOG | PHYSICS, MULTIDISCIPLINARY | Yang-Baxter map | VERTEX MODELS | PHYSICS, MATHEMATICAL

Journal Article

Applied Categorical Structures, ISSN 0927-2852, 2019, Volume 27, Issue 4, pp. 323 - 363

We introduce a new type of categorical object called a hom-tensor category and show that it provides the appropriate setting for modules over an arbitrary...

Hom-bialgebra | Yang–Baxter equation | Tensor category | LIE-ALGEBRAS | MATHEMATICS | Yang-Baxter equation | CYCLIC HOMOLOGY | VIRASORO ALGEBRA | Computer science

Hom-bialgebra | Yang–Baxter equation | Tensor category | LIE-ALGEBRAS | MATHEMATICS | Yang-Baxter equation | CYCLIC HOMOLOGY | VIRASORO ALGEBRA | Computer science

Journal Article

Communications in Mathematical Physics, ISSN 0010-3616, 7/2016, Volume 345, Issue 2, pp. 507 - 543

We consider finite-dimensional reductions of an integral operator with the elliptic hypergeometric kernel describing the most general known solution of the...

Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Physics | VERTEX-FACE CORRESPONDENCE | BETA-INTEGRALS | MATRIX | FUSION | SYMMETRY | INTEGRABLE QUANTUM-SYSTEMS | MODELS | IDENTITIES | 6J-SYMBOLS | PHYSICS, MATHEMATICAL | OPERATORS | Algebra

Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Physics | VERTEX-FACE CORRESPONDENCE | BETA-INTEGRALS | MATRIX | FUSION | SYMMETRY | INTEGRABLE QUANTUM-SYSTEMS | MODELS | IDENTITIES | 6J-SYMBOLS | PHYSICS, MATHEMATICAL | OPERATORS | Algebra

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 09/2018, Volume 76, Issue 5, pp. 1085 - 1098

Let A=I−PQT, where P and Q are two n×2 complex matrices of full column rank and det(QTP)≠0. All the commuting solutions with A of the quadratic matrix equation...

Rank-two updated matrix | Yang–Baxter-like matrix equation | Commuting solutions | SPLITTING ITERATION METHODS | MATHEMATICS, APPLIED | Yang-Baxter-like matrix equation | POSITIVE-DEFINITE | ALGORITHM | ALGEBRAIC RICCATI-EQUATIONS | SOLVING SYLVESTER EQUATIONS | SPECTRAL SOLUTIONS | AX PLUS XB | Questions and answers | Numerical analysis

Rank-two updated matrix | Yang–Baxter-like matrix equation | Commuting solutions | SPLITTING ITERATION METHODS | MATHEMATICS, APPLIED | Yang-Baxter-like matrix equation | POSITIVE-DEFINITE | ALGORITHM | ALGEBRAIC RICCATI-EQUATIONS | SOLVING SYLVESTER EQUATIONS | SPECTRAL SOLUTIONS | AX PLUS XB | Questions and answers | Numerical analysis

Journal Article

Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, 09/2016, Volume 49, Issue 39, p. 395202

In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on the R-matrix description which...

noncommutative integrable systems | elliptic integrable systems | Painlevé VI equation | Euler-Arnold tops | YANG-BAXTER EQUATION | QUADRATIC ALGEBRAS | PHYSICS, MULTIDISCIPLINARY | LIE | CALOGERO-MOSER SYSTEMS | MODEL | PHYSICS, MATHEMATICAL | SYMMETRY | Painleve VI equation | PAIRS

noncommutative integrable systems | elliptic integrable systems | Painlevé VI equation | Euler-Arnold tops | YANG-BAXTER EQUATION | QUADRATIC ALGEBRAS | PHYSICS, MULTIDISCIPLINARY | LIE | CALOGERO-MOSER SYSTEMS | MODEL | PHYSICS, MATHEMATICAL | SYMMETRY | Painleve VI equation | PAIRS

Journal Article

Letters in Mathematical Physics, ISSN 0377-9017, 4/2019, Volume 109, Issue 4, pp. 843 - 856

We introduce the notion of N-reflection equation which provides a generalization of the usual classical reflection equation describing integrable boundary...

Geometry | Classical reflection equation | 17B63 | Theoretical, Mathematical and Computational Physics | Complex Systems | Non-skew-symmetric r -matrices | Group Theory and Generalizations | Classical Yang–Baxter equation | Gaudin models | Physics | 37K10 | Non-skew-symmetric r-matrices | INTEGRABLE MODELS | BOUNDARY-CONDITIONS | Classical Yang-Baxter equation | HAMILTONIAN STRUCTURES | PHYSICS, MATHEMATICAL | R-MATRICES | Analysis | Models | Algebra | Mathematics | Nonlinear Sciences | Exactly Solvable and Integrable Systems | Mathematical Physics

Geometry | Classical reflection equation | 17B63 | Theoretical, Mathematical and Computational Physics | Complex Systems | Non-skew-symmetric r -matrices | Group Theory and Generalizations | Classical Yang–Baxter equation | Gaudin models | Physics | 37K10 | Non-skew-symmetric r-matrices | INTEGRABLE MODELS | BOUNDARY-CONDITIONS | Classical Yang-Baxter equation | HAMILTONIAN STRUCTURES | PHYSICS, MATHEMATICAL | R-MATRICES | Analysis | Models | Algebra | Mathematics | Nonlinear Sciences | Exactly Solvable and Integrable Systems | Mathematical Physics

Journal Article

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