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

Abstract: 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-shelve/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 homomorphsim and the universal R-matrices intertwine with the algebra coproducts.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar