On the Structure of Weak Hopf Algebras

Abstract

We study the group of group-like elements of a weak Hopf algebra and derive an analogue of Radford's formula for the fourth power of the antipode S; which implies that the antipode has a finite order modulo, a trivial automorphism. We find a sufficient condition in terms of TrðS 2 Þ for a weak Hopf algebra to be semisimple, discuss relation between semisimplicity and cosemisimplicity, and apply our results to show that a dynamical twisting deformation of a semisimple Hopf algebra is cosemisimple. # 2002 Elsevier Science (USA)

Yetter-Drinfel'd algebras and coideals of Weak Hopf $C^*$-Algebras

“…The antipode S is unique, invertible, and satisfies (S • * ) 2 = id B . Since it was mentioned in [14], Remark 3.7 that problems regarding general WHAs can be translated to problems regarding those with the property S 2 | Bt = id which are called regular, we will only consider such WHAs (see also [28]). In this case, there exists a canonical positive element H in the center of B t such that S 2 is an inner automorphism implemented by G = HS(H) −1 , i.e., S 2 (b) = GbG −1 for all b ∈ B.…”

“…Let Ĝ be the dual of a WHA G = (B, ∆, S, ε). Let Ĝmin = Bt Bs be the minimal WHA contained in Ĝ and i min : Ĝmin → Ĝ the corresponding inclusion of WHAs (see [2], [14]). Then the adjoint map π min : G → Ĝ * min given by < i min ( Bmin ), B >=< Bmin , π min (B) >, is an epimorphism of WHAs.…”

Kaplansky's Type Constructions for Weak Bialgebras and Weak Hopf Algebras

“…The automorphisms group of the weak bialgebras (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), is the group of order 6 given by:…”

“…The weak Hopf algebras, called also quantum groupoids, appeared also in dynamic deformation theory of quantum groups [4]. Weak bialgebras and weak Hopf algebras were developed from the algebraic point of view and have been considered by several authors in various settings [5][6][7][8][9][10][11][12][13][14][15][16]. The weak bialgebras (resp.…”

Weak Bialgebras of fractions

“…In this case, condition (C2) of Definition 3.4 is always satisfied, and (C1) implies for all g, h ∈ G that gh = I g (h)g = I g (h)I g (g) = I g (hg) = hg, and so any monoid of group-likes that is almost central with respect to this I, is in fact commutative. In a coquasi-triangular WBA this is not necessarily true as can be seen using the results of [12] as follows: Let H be a WHA. The source base algebra H s forms the monoidal unit of the category of finite-dimensional right H-comodules using its right-regular comodule structure β Hs :…”