Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups

DOI10.1016/j.aim.2018.09.005zbMath1437.16028arXiv1507.02602OpenAlexW2963154383MaRDI QIDQ1789498

Tatiana Gateva-Ivanova

Publication date: 10 October 2018

Published in: Advances in Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1507.02602

zbMATH Keywords

bracesYang-Baxterbraided groups

Mathematics Subject Classification ID

Quantum groups and related algebraic methods applied to problems in quantum theory (81R50) Actions of groups and semigroups; invariant theory (associative rings and algebras) (16W22) Noncommutative geometry in quantum theory (81R60) Graded rings and modules (associative rings and algebras) (16W50) Subgroups of symmetric groups (20B35) Quadratic and Koszul algebras (16S37) Yang-Baxter equations (16T25)

Related Items

Indecomposable involutive solutions of the Yang-Baxter equation of multipermutational level 2 with abelian permutation group ⋮ Primitive set-theoretic solutions of the Yang–Baxter equation ⋮ Algebraic approach to Rump’s results on relations between braces and pre-lie algebras ⋮ Reflection equation as a tool for studying solutions to the Yang-Baxter equation ⋮ Braces and symmetric groups with special conditions ⋮ Enumeration of set-theoretic solutions to the Yang–Baxter equation ⋮ Medial and semimedial left quasigroups ⋮ On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation ⋮ Classification of non-degenerate involutive set-theoretic solutions to the Yang-Baxter equation with multipermutation level two ⋮ The matched product of the solutions to the Yang-Baxter equation of finite order ⋮ On skew braces (with an appendix by N. Byott and L. Vendramin) ⋮ A family of irretractable square-free solutions of the Yang-Baxter equation ⋮ Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation ⋮ On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang-Baxter equation ⋮ Left non-degenerate set-theoretic solutions of the Yang–Baxter equation and dynamical extensions of q-cycle sets ⋮ Skew-braces and \(q\)-braces ⋮ Finite skew braces of square-free order and supersolubility ⋮ The construction of multipermutation solutions of the Yang-Baxter equation of level 2 ⋮ Nilpotency of skew braces and multipermutation solutions of the Yang–Baxter equation ⋮ Set-theoretic type solutions of the braid equation ⋮ Schur covers of skew braces ⋮ On some Garsideness properties of structure groups of set-theoretic solutions of the Yang-Baxter equation ⋮ The Yang–Baxter equation and Thompson’s group F ⋮ Two-component Yang-Baxter maps and star-triangle relations ⋮ From braces to Hecke algebras and quantum groups ⋮ Set-theoretical solutions to the Hom-Yang-Baxter equation and Hom-cycle sets ⋮ Skew left braces and 2-reductive solutions of the Yang-Baxter equation ⋮ A note on set theoretical solutions of the Yang-Baxter equation with trivial retraction ⋮ Near braces and p$p$‐deformed braided groups ⋮ Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation ⋮ From braces to pre-Lie rings ⋮ Set-theoretical solutions of the Yang-Baxter and pentagon equations on semigroups ⋮ A note on set-theoretic solutions of the Yang-Baxter equation ⋮ Involutive Latin solutions of the Yang-Baxter equation ⋮ Extensions of linear cycle sets and cohomology ⋮ Indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level 2 with non-abelian permutation group ⋮ Skew braces and the Yang–Baxter equation ⋮ Set-theoretic solutions of the pentagon equation ⋮ Set-theoretic solutions of the Yang-Baxter equation, associated quadratic algebras and the minimality condition ⋮ Computing skew left braces of small orders ⋮ Set-theoretic Yang-Baxter \& reflection equations and quantum group symmetries ⋮ Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation ⋮ A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equation ⋮ Constructing finite simple solutions of the Yang-Baxter equation ⋮ Set-theoretic solutions of the Yang-Baxter equation and new classes of \(\mathrm{R}\)-matrices ⋮ L-algebras with duality and the structure group of a set-theoretic solution to the Yang-Baxter equation ⋮ Distributive biracks and solutions of the Yang–Baxter equation ⋮ Combinatorial solutions to the reflection equation ⋮ Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces ⋮ A new formula for Lazard's correspondence for finite braces and pre-Lie algebras ⋮ The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation ⋮ Asymmetric product of left braces and simplicity; new solutions of the Yang–Baxter equation ⋮ Two theorems on balanced braces ⋮ Factorizations of skew braces ⋮ On the passage from finite braces to pre-Lie rings ⋮ Some braces of cardinality \(p^4\) and related Hopf-Galois extensions ⋮ Set-theoretic Yang–Baxter equation, braces and Drinfeld twists ⋮ New simple solutions of the Yang-Baxter equation and solutions associated to simple left braces

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Cohomology and extensions of braces
• Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation
• On the Yang-Baxter equation and left nilpotent left braces
• A family of solutions of the Yang-Baxter equation.
• Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity
• Quantum Yang-Baxter equation, braided semigroups, and dynamical Yang-Baxter maps.
• Quantum spaces associated to multipermutation solutions of level two
• Multipermutation solutions of the Yang-Baxter equation
• Braces, radical rings, and the quatum Yang-Baxter equation.
• Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova.
• Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction
• Retractability of set theoretic solutions of the Yang-Baxter equation
• Skew polynomial rings with binomial relations
• Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations
• Semigroups of \(I\)-type
• Quadratic algebras of skew type and the underlying monoids.
• On the set-theoretical Yang-Baxter equation
• Braces and symmetric groups with special conditions
• A family of irretractable square-free solutions of the Yang-Baxter equation
• A note on set-theoretic solutions of the Yang-Baxter equation
• Dynamical braces and dynamical Yang-Baxter maps
• Set-theoretical solutions to the quantum Yang-Baxter equation
• More examples of bicrossproduct and double cross product Hopf algebras
• Braces and the Yang-Baxter equation
• Set-theoretic solutions of the Yang-Baxter equation, graphs and computations
• Matched pairs approach to set theoretic solutions of the Yang-Baxter equation.
• Solutions of the Yang-Baxter equation associated with a left brace.
• Hopf braces and Yang-Baxter operators
• Matched pairs of groups and bismash products of hopf algebras
• On some unsolved problems in quantum group theory
• Noetherian Properties of Skew Polynomial Rings with Binomial Relations
• On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation
• Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles
• QUADRATIC ALGEBRAS OF SKEW TYPE SATISFYING THE CYCLIC CONDITION
• A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation
• The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition