Parametric set-theoretic Yang–Baxter equation: p-racks, solutions and quantum algebras

The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. The first step toward this objective is the introduction of certain generalizations of the familiar shelves and racks called parametric (p)-shelves and racks. These objects satisfy a parametric self-distributivity condition and lead to solutions of the Yang-Baxter equation. Novel, non-reversible solutions are obtained from p-shelf/rack solutions by a suitable parametric twist, whereas all reversible set-theoretic solutions are reduced to the identity map via a parametric twist. The universal algebras associated to both p-rack and generic parametric, set-theoretic solutions are next presented and the corresponding universal ℛ-matrices are derived. The admissible universal Drinfel’d twist is constructed allowing the derivation of the general set-theoretic universal ℛ-matrix. By introducing the concept of a parametric coproduct we prove the existence of a parametric co-associativity. We show that the parametric coproduct is an algebra homomorphism and the universal ℛ-matrices satisfy intertwining relations with the algebra coproducts.