Set theoretic Yang-Baxter equation, braces and Drinfeld twists

Abstract

We consider involutive, non-degenerate, finite set theoretic solutions of the Yang-Baxter equation. Such solutions can be always obtained using certain algebraic structures that generalize nil potent rings called braces. Our main aim here is to express such solutions in terms of admissible Drinfeld twists substantially extending recent preliminary results. We first identify the generic form of the twists associated to set theoretic solutions and we show that these twists are admissible, i.e. they satisfy a certain co-cycle condition. These findings are also valid for Baxterized solutions of the Yang-Baxter equation constructed from the set theoretical ones.

arXiv comment23 pages, LaTex. A few comments added

References in corpus (11)

• On set-theoretical solutions of the quantum Yang-Baxter equation
• Set-theoretical solutions to the quantum Yang-Baxter equation
• Classification of integrable equations on quad-graphs. The consistency approach
• Skew braces and the Yang-Baxter equation
• Yang-Baxter maps and integrable dynamics
• On skew braces (with an appendix by N. Byott and L. Vendramin)
• Permutation-type solutions to the Yang-Baxter and other n-simplex equations
• On Quadrirational Yang-Baxter Maps
• Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
• From Braces to Hecke algebras & Quantum Groups
• A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equation

Cited by in corpus (9)

• Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
• The Floquet Baxterisation
• Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices
• Near braces and p-deformed braided groups
• Novel non-involutive solutions of the Yang-Baxter equation from (skew) braces
• Drinfeld Twists on Skew Braces
• Generalized digroups, di-skew braces, and solutions of the set-theoretic Yang-Baxter equation
• Algebraic structures in set-theoretic Yang-Baxter & reflection equations
• Self-distributive structures, braces & the Yang-Baxter equation