Representations for certain crossed simplicial groups generated by braided Hopf algebras

Abstract

We find solutions of a nonlinear equation which provide representations for the new groups R(n) defined in [1]. The groups are crossed simplicial groups in the sense of Loday [8] (Sec.6.3). These solutions are based on the Bulacu and Beattie construction of braided Hopf algebras in the category of Yetter-Drinfeld modules [2]. We discuss the connection of these solutions with Hopf equation in a braided monoidal category (a system of mixed Yang-Baxter type equations presented in [1]). Generalized quantum doubles for pairs of Hopf algebras can afford weak projections.