### A nonlinear equation that unifies the quantum Yang-Baxter equation and the Hopf equation.Solutions and applications

#### Abstract

We present a nonlinear equation , called (UE),which unify QYBE and Hopf equations, and which provide representations for the new crossed-simplicial groups R(n) defined in ~\cite{barad3}. We continue the study of a system of mixed Yang-Baxter type equations presented in ~\cite{barad3}, which provide solutions for UE. Similar to the study of Hopf, Long or QYBE, we find sufficient conditions for a bilinear on a Hopf algebra to provide canonical solutions on any H-comodule, and also sufficient conditions for an entwining structure, such that the canonical application for an entwined module $R(m \otimes n)= m_1 n \otimes m _0$ on $M \otimes M$ verify the equation.\\ Any solution of the equation generates a twisted factorisation structure on a tensor algebra. Theory of Hopf algebras with a weak projection satisfying certain properties provides solutions. A new nonlinear equation is proposed.