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

We present connections between left non-degenerate solutions of set-theoretic Yang-Baxter equation and left shelves using certain maps called Drinfel'd homomorphisms. We further generalise the notion of affine quandle, by using heap endomorphisms and metahomomorphisms, and identify the 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 is also presented. These generalized structures encode part of the underlying Hopf algebra. Indeed, we also introduce the quasi-triangular Hopf algebras and the universal R-matrices for quandle algebras and for set-theoretic Yang-Baxter algebras. In fact, we obtain the universal R-matrix for the set-theoretic Yang-Baxter algebras after identifying the associated admissible Drinfel'd twist. Generic set-theoretic solutions coming from heap endomorphisms are also identified.