Weak Bialgebras of fractions

Abstract

We construct the algebra of fractions of a Weak Bialgebra relative to a suitable denominator set of group-like elements that is almost central, a condition we introduce in the present article which is sufficient in order to guarantee existence of the algebra of fractions and to render it a Weak Bialgebra. The monoid of all group-like elements of a coquasi-triangular Weak Bialgebra, for example, forms a suitable set of denominators as does any monoid of central group-like elements of an arbitrary Weak Bialgebra. We use this technique in order to construct new Weak Bialgebras whose categories of finitedimensional comodules relate to SL 2 -fusion categories in the same way as GL(2) relates to SL(2).

Two FRT bialgebroids and their relations

“…Later (Hopf) face algebras were integrated to weak bialgebras (weak Hopf algebras) by Böhm, Nill, and Szláchanyi [3]. Bennoun and Pfeiffer mentioned to the Hopf closure (they called the Hopf envelope) of the coquasitriangular weak bialgebra in [2]. Schauenburg [16] showed that a weak bialgebra (weak Hopf algebra) is a left bialgebroid (× R -Hopf algebra) whose base algebra is Frobenius-separable.…”

“…for all l ∈ L and a, b ∈ A. Here we write ∆ L (a) = a [1] ⊗ a [2] , called Sweedler's notation. The right-hand-side of (2.5) is well defined because of (2.3).…”

“…Let us recall the notion of Frobenius-separable K-algebras to discuss relations between the left bialgebroid and the weak bialgebra. A Frobenius-separable Kalgebra is a K-algebra L equipped with a K-linear map ψ : L → K and an element e (1) ⊗ e (2) ∈ L ⊗ K L such that l = ψ(le (1) )e (2) = e (1) ψ(e (2) l), e (1)…”

“…Let A L = (A, L, s L , t L , ∆ L , π L ) be a left bialgebroid. If the base algebra L is a Frobenius-separable K-algebra with an idempotent Frobenius system (ψ, e (1) ⊗ e (2) ), then the total algebra A has the following weak bialgebra structure (A, ∆, ε):…”

“…Then the tensor product A ⊗ N A has two meanings depending on left actions N A and N A. In order to avoid misunderstandings, we specify these actions by α : [1] ⊗ S HAD (a [2] )b ∈ A N ⊗ N A.…”