The hopf modules category and the hopf equation

Abstract

Let (A,△) be a Hopf-von Neumann algebra and R be the unitary fundamental operator on A defined by Takesaki in [28]:R(a⊗ b) = △(b)(a ⊗ 1). Then R 12 R 23= R 23 R 13 R 12 (see lemma 4.9 of [28]). This operator R plays a vital role in the theory of duality for von Neumann algebras (see [28] or [2]). If V is a vector space over an arbitrary field k, we shall study what we have called the Hopf equation: R 12 R 23= R 23 R 13 R 12 in End k (V ⊗ V ⊗ V). Taking W = rRr the Hopf equation is equivalent with the pentagonal equation:W l2 W l3 W 23 = W 23 W 12 from the theory of operator algebras (see [2]), where W are viewed as map in L (K ⊗ K), for a Hilbert space K. For a bialgebra H, we shall prove that the classic category of Hopf modules plays a decisive role in describing all solutions of the Hopf equation. More precisely, if H is a bialgebra over k and is an H-Hopf module, then the natural map R = R (M, ,ρ) a solution for the Hopf equation. Conversely, the main result of this paper is a FRT type theorem:if M is a finite dimensional vector space and R ∈ End k -(M⊙M) is a solution for the Hopf equation, then there exists a bialgebra B(R) such that By 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.