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

Let M be a vector space over a field k and R is an element of End (k)(M x M). This paper studies what shall be called the Long equation: that is, the system of nonlinear equations (RR13)-R-12 = (RR12)-R-13 and (RR23)-R-12 = (RR12)-R-23 in, End (k)(M x M x M). Any symmetric solution of this system supplies us a solution of the integrability condition of the Knizhnik-Zamolodchikov equation: [R-12, R-13+R-23] = 0 ([4] Or [10]). We shall approach this equation by introducing a new class of bialgebras, which we call Long bialgebras: these are pairs (H,sigma), where H is a bialgebra and sigma: H x H --> k is a k-bilinear map satisfying certain properties. The main theorem of this paper is a FRT type theorem: if M is finite dimensional, any solution R of the Long equation has the form R = R,, where M has a structure of a right comodule over a Long bialgebra (L(R),sigma), and R-sigma is the special map R-sigma(m x n) = Sigma sigma(m([1]) x n([1]))m([0]) x n([0]).