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.