Let (A,△) be a Hopf-von Neumann algebra andRbe the unitary fundamental operator onAdefined by Takesaki in [28]:R(a⊗b) = △(b)(a⊗ 1). ThenR12R23=R23R13R12(see lemma 4.9 of [28]). This operatorRplays a vital role in the theory of duality for von Neumann algebras (see [28] or [2]). IfVis a vector space over an arbitrary fieldk, we shall study what we have called the Hopf equation:R12R23=R23R13R12in Endk(V⊗V⊗V). TakingW=rRrthe Hopf equation is equivalent with the pentagonal equation:Wl2Wl3W23=W23W12from the theory of operator algebras (see [2]), whereWare viewed as map inL(K⊗K), for a Hilbert spaceK. For a bialgebraH, we shall prove that the classic category of Hopf modulesplays a decisive role in describing all solutions of the Hopf equation. More precisely, ifHis a bialgebra overkandis anH-Hopf module, then the natural mapR=R(M, ,ρ)a solution for the Hopf equation. Conversely, the main result of this paper is a FRT type theorem:ifMis a finite dimensional vector space andR∈ Endk-(M⊙M) is a solution for the Hopf equation, then there exists a bialgebraB(R) such thatBy applying this result, we construct now examples of noncommutative and noncocomimitative bialgebras which are different from the ones arising from quantum group theory. In particular, over a field of characteristic two, an example of five dimensional noncommutative and noiicocommutative bialgebra is given.