A class of non-symmetric solutions for the integrability condition of the Knizhnik-Zamolodchikov equation: a Hopf algebra approach

This paper studied what shall be called the Long equation: that is the system of nonlinear equation $R^{12}R^{13}=R^{13}R^{12}$ and $R^{12}R^{23}= R^{23}R^{12}$. Any solution of this system supplies us a solution for the integrability solution of Knizhnic-Zamolodchikov equation. We shall approach this equation by introducing a new class of bialgebras, which we call Long bialgebras. A FRT type theorem is given: in the finite case any solution of the above equation is a co-homotety where the vector spase is a comodule over a Long bialgebra.