Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices

Abstract

We present connections between left non-degenerate solutions of the set-theoretic braid equation and left shelves using Drinfel'd homomorphisms. We generalize the notion of affine quandle, by using heap endomorphisms and metahomomorphisms, and identify the underlying Yang-Baxter algebra for solutions of the braid equation associated to a given quandle. We introduce the notion of the pre-Lie skew brace and identify certain affine quandles that give rise to pre-Lie skew braces. Generalisations of the braiding of a group, associated to set-theoretic solutions of the braid equation are also presented. These generalized structures encode part of the underlying Hopf algebra. We then introduce the quasi-triangular (quasi) Hopf algebras and universal R-matrices for rack and set-theoretic algebras. Generic set-theoretic solutions coming from heap endomorphisms are also identified.

36 pages LaTex. Some general comments added in the introduction section. Minor typos corrected

References in corpus (7)

• Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
• Set theoretic Yang-Baxter equation, braces and Drinfeld twists
• Heaps of modules and affine spaces
• Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses
• 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

Cited by in corpus (3)

• Reflections to set-theoretic solutions of the Yang-Baxter equation
• Generalized digroups, di-skew braces, and solutions of the set-theoretic Yang-Baxter equation
• Self-distributive structures, braces & the Yang-Baxter equation