The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. We introduce generalizations of the familiar shelves and racks named 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. 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 intertwine with the algebra coproducts.
This video is part of the European Non-Associative Algebra Seminar series.
