New types of bialgebras arising from the Hopf equation

Let M be a k-vector space and R ∈ End k(M ⊗M). In [10] we introduced and studied what we called the Hopf equation: R12R23 = R23R13R12. By means of a FRT type theorem, we have proven that the category HM H 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 [7]), 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 Rσ is the special map Rσ(m ⊗ n) = ∑ σ(m<1> ⊗ n<1>)m<0> ⊗ n<0>. 0 Introduction Let H be a Hopf algebra over a field k. The strong link between the category HM H of Hopf modules and the category HYD H of Yetter-Drinfel’d modules recently highlighted in [1] (namely, the fact that both are particular cases of the same general category AM(H) C of Doi-Hopf modules, defined by Doi in [4]) led us in [2] and [3] to study the implication of the category HYD H in the classic, non-quantic part of Hopf algebra theory. In [10] we called this technique ”quantisation”. For example, the theorem 4.2 of [2], stating that the forgetful functor HYD H → HM is Frobenius if and only if H is finite dimensional and unimodular, can be viewed as the ”quantum version” of the classical theorem saying that any finite dimensional Hopf algebra is Frobenius. In [10] we start to study the reverse problem (it was called ”dequantisation”): that is, we study the category HM H in connection with problems which so far were specific solely to the 1 category HYD H . The starting point is simple; it is enough to remember that the category HYD H is deeply involved in the quantum Yang-Baxter equation: RRR = RRR where R ∈ End k(M⊗M), M being a k-vector space. We evidence the fact that the category HM H can also be studied in connection with a certain non-linear equation. We call it the Hopf equation, and it is: RR = RRR The main result of [10] is a FRT type theorem which shows that in the finite dimensional case, any solution R of the Hopf equation has the form R = R(M,·,ρ), where (M, ·, ρ) is an object in B(R)M , for some bialgebra B(R). In this paper we shall continue to study the Hopf equation from another perspective. To begin with, we remind that in the quantum Yang-Baxter equation another important role is played by the co-quasitriangular bialgebras. These can be viewed as bialgebras H with a k-bilinear map σ : H ⊗ H → k satisfying properties which ensure that the special map Rσ : M ⊗M → M ⊗M, Rσ(m⊗ n) = ∑ σ(m<1> ⊗ n<1>)m<0> ⊗ n<0> is a solution for the quantum Yang-Baxter equation. Starting from here, we introduce new classes of bialgebras which will play for the Hopf equation the same role as the coquasitriangular bialgebras do for the quantum Yang-Baxter equation. We called them (H,C, σ) bialgebras with a Hopf function σ : C ⊗ H → k, where C is a subcoalgebra of H . The reason why the map σ is not defined for the entire H ⊗ H , but only relative to a subcoalgebra C of H is explained in Remarks 2.2 and 2.9. The main result of this paper is theorem 2.8: if M is a finite dimensional vector space and R is a solution of the Hopf equation, then for a special subcoalgebra C of B(R) there exists a unique Hopf function σ : C ⊗ B(R) → k such that R = Rσ. We apply the above results by presenting several examples of Hopf functions on bialgebras. In the last section, as an appendix, we also introduced the concept corresponding to quasitriangular bialgebras. 1 Preliminaries Throughout this paper, k will be a field. All vector spaces, algebras, coalgebras and bialgebras considered are over k. ⊗ and Hom will mean ⊗k and Hom k. For a coalgebra C, we will use Sweedler’s Σ-notation, that is, ∆(c) = ∑ c(1)⊗c(2), (I⊗∆)∆(c) = ∑ c(1)⊗c(2)⊗c(3), etc. We will also use Sweedler’s notation for right C-comodules: ρM(m) = ∑ m<0> ⊗m<1>, for any m ∈ M if (M, ρM ) is a right C-comodule. M C will be the category of right C-comodules and C-colinear maps and AM will be the category of left A-modules and A-linear maps, if A is a k-algebra. From now on, H will be a bialgebra. An element T ∈ H is called a right integral on H (see [11]) if Tf = f(1H)T 2 for all f ∈ H. This is equivalent to