New types of bialgebras arising from the hopf equation

Let M be a k-vector space and R ∈ End k (M ⊗ M)In [13] we introduced and studied what we called the Hopf equation:R 12 R 23 = R 23 R 13 R 12.By means of a FRT type theorem, we have proven that the category of H-Hopf modules is deeply involved in solving this equation. In the present paper, we continue to study the Hopf equation from another perspective:having in mind the quantum Yang-Baxter equation, in the solution of which the co-quasitriangular (or braided) bialgebras play an important role (see [8]), we introduce and study what we call bialgebras with Hopf functions. The main theorem of this paper shows that, if M is finite dimensional, any solution R of the Hopf equation has the form R = Rσ. where M is a right comodule over a bialgebra with a Hopf function (B{R),C,σ) and R σ is the special map .