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

TL;DR

This paper reviews algebraic structures like shelves, racks, and quandles that satisfy self-distributivity and lead to solutions of the set-theoretic Yang-Baxter equation, connecting them to quantum algebra and Hopf algebras.

Contribution

It introduces the universal algebras associated with racks and set-theoretic solutions, showing they are quasi-triangular Hopf algebras and deriving the universal set-theoretic R-matrix.

Findings

Racks and quandles satisfy self-distributivity and produce Yang-Baxter solutions.

Universal algebras related to set-theoretic solutions are quasi-triangular Hopf algebras.

Derivation of the universal set-theoretic R-matrix via admissible twist.

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.