Self-distributive structures, braces & the Yang-Baxter equation

Abstract

The theory of the set-theoretic Yang-Baxter equation is reviewed from a purely algebraic point of view. We recall certain algebraic structures called shelves, racks and quandles. These objects satisfy a self-distributivity condition and lead to solutions of the Yang-Baxter equation. The quantum algebra as well as the integrability associated to Baxterized involutive set-theoretic solutions is briefly discussed. We then present the theory of the universal algebras associated to rack and general set-theoretic solutions. We show that these are quasi-triangular Hopf algebras and we derive the universal set-theoretic Drinfel'd twist. It is shown that this is an admissible twist allowing the derivation of the universal set-theoretic R-matrix.

arXiv comment31 pages, LaTex. Generalisations added in Section 5 and minor typos corrected

References in corpus (20)

• 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)
• Yang-Baxter maps and symmetries of integrable equations on quad-graphs
• N-soliton collision in the Manakov model
• On structure groups of set-theoretic solutions to the Yang-Baxter equation
• Soliton Cellular Automata Associated With Crystal Bases
• Hopf braces and Yang-Baxter operators
• On Quadrirational Yang-Baxter Maps
• Factorizations of skew braces
• Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
• The retraction relation for biracks
• Set theoretic Yang-Baxter equation, braces and Drinfeld twists
• Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses
• A Survey of Racks and Quandles: Some recent developments
• Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices
• Reflection equation as a tool for studying solutions to the Yang-Baxter equation
• Near braces and p-deformed braided groups
• Novel non-involutive solutions of the Yang-Baxter equation from (skew) braces