Yang-Baxter Solutions from Categorical Augmented Racks

Abstract: An augmented rack is a set with a self-distributive binary operation induced by a group action, and has been extensively used in knot theory. Solutions to the Yang-Baxter equation (YBE) have been also used for knots, since the discovery of the Jones polynomial. In this paper, an interpretation of augmented racks in tensor categories is given for coalgebras that are Hopf algebra modules, and associated solutions to the YBE are constructed. Explicit constructions are given using quantum heaps and the adjoint of Hopf algebras. Furthermore, an inductive construction of Yang-Baxter solutions is given by means of the categorical augmented racks, yielding infinite families of solutions. Constructions of braided monoidal categories are also provided using categorical augmented racks.