Higher Conjugations and the Yang–Baxter Equation

Let H be a Hopf algebra with invertible antipode. Two different actions of the braid group Bn on H n are described. These are representations of the braid group associated to solutions of the Yang–Baxter equation. By passing to a submodule for one action and to a quotient module for the other, we obtain two actions of Bn on H n − 1. In this way we recover the actions described by the author in [Comm. Algebra29 (2001) 3351–3363] that arise naturally in stable homotopy theory.