Weak Hopf Algebras Research Articles

Semidirect products of weak multiplier Hopf algebras: Smash products and smash coproducts

ABSTRACTIn this paper, we will develop the smash product of weak multiplier Hopf algebras unifying the cases of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras. We will show that the smash product R#A has a regular weak multiplier Hopf algebra structure if R and A are regular weak multiplier Hopf algebras. We shall investigate integrals on R#A. We also consider the result in the ∗-situation and new examples. Dually, we consider the smash coproduct of weak multiplier Hopf algebras under an appropriate form and integrals on the smash coproduct and we obtain results in the ∗-situation.

Weak quasi-entwining structures

In this paper we introduce the notion of weak quasi-entwining structure as a generalization of quasi-entwining structures and weak entwining structures. Also, we formulate the notions of weak cleft extension, weak Galois extension, and weak Galois extension with normal basis associated to a weak quasientwining structure. Moreover, we prove that, under some suitable conditions, there exists an equivalence between weak Galois extensions with normal basis and weak cleft extensions. As particular instances, we recover some results previously proved for Hopf quasigroups, weak Hopf quasigroups and weak Hopf algebras.

Wedderburn–Malcev and Levi theorems for weak Hopf algebras

Wedderburn–Malcev and Levi theorems for weak Hopf algebras

Green rings of weak Hopf algebras based on generalized Taft algebras

In this paper, we first introduce two classes of weak Hopf algebras $$\mathfrak {w}^s_{n,d}$$ of type $$s ~(s=0, 1)$$ , which are corresponding to generalized Taft Hopf algebras $$H_{n,d}(q)$$ . Then we compute their Green rings, and describe the structure of their Green rings by the generators and the generating relations. These results extend some known ones about Taft algebras and generalized Taft algebras.

Multiplier Hopf algebroids: Basic theory and examples

ABSTRACTMultiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable base algebras. The main structure maps of a multiplier Hopf algebroid are a left and a right comultiplication. We show that bijectivity of two associated canonical maps is equivalent to the existence of an antipode, discusses invertibility of the antipode, and presents some examples and special cases.

The Larson–Sweedler theorem for weak multiplier Hopf algebras

ABSTRACTThe Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [44]. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by Böhm et al. in [4]. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. Weak multiplier Hopf algebras are introduced and studied in [40]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13] and [14]. Our treatment of this material is motivated by the prospect of such a theory.

Inner actions of weak Hopf algebras

Let [Formula: see text] be an associative ring and [Formula: see text] idempotent elements of [Formula: see text]. In this paper we introduce the notion of [Formula: see text]-invertibility for an element of [Formula: see text] and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra [Formula: see text] and an algebra [Formula: see text], we present sufficient conditions for [Formula: see text] to admit an inner action of [Formula: see text]. We also prove that if [Formula: see text] is a left [Formula: see text]-module algebra then [Formula: see text] acts innerly on the smash product [Formula: see text] if and only if [Formula: see text] is a quantum commutative weak Hopf algebra.

A duality theorem for weak multiplier Hopf algebra actions

The main purpose of this paper is to unify the theory of actions of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras introduced by Van Daele and Wang. Using such developed actions, we will define the notion of a module algebra over weak multiplier Hopf algebras and construct their smash products. The main result is the duality theorem for actions and their dual actions on the smash product of weak multiplier Hopf algebras. As an application, we recover the main results found in the literature for weak Hopf algebras, multiplier Hopf algebras and groupoids.

WEAK HOPF ALGEBRAS CORRESPONDING TO NON-STANDARD QUANTUM GROUPS

We construct a weak Hopf algebra <TEX>$wX_q(A_1)$</TEX> corresponding to non-standard quantum group <TEX>$X_q(A_1)$</TEX>. The PBW basis of <TEX>$wX_q(A_1)$</TEX> is described and all the highest weight modules of <TEX>$wX_q(A_1)$</TEX> are classified. Finally we give the Clebsch-Gordan decomposition of the tensor product of two highest weight modules of <TEX>$wX_q(A_1)$</TEX>.

On duality of algebraic quantum groupoids

Like quantum groups, quantum groupoids frequently appear in pairs of mutually dual objects. We develop a general Pontrjagin duality theory for quantum groupoids in the algebraic setting that extends Van Daele's duality theory for multiplier Hopf algebras and overcomes the finiteness restrictions of the approach of Kadison, Szlachányi, Böhm and Schauenburg. Our construction is based on the integration theory for multiplier Hopf algebroids and yields, as a corollary, a duality theory for weak multiplier Hopf algebras with integrals. We compute the duals in several examples and introduce morphisms of multiplier Hopf algebroids to succinctly describe their structure. Moreover, we show that such morphisms preserve the antipode.

H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS

Let H be a weak Hopf algebra and let A be an H-comodule algebra with subalgebra of coinvariants AH. In this paper we introduce the notion of H-Galois extension with normal basis and we prove that AH ,→ A is an H-Galois extension with normal basis if and only if AH ,→ A is an H-cleft extension which admits a convolution invertible total integral. As a consequence, if H is cocommutative and A commutative, we obtain a bijective correspondence between the second cohomology group H2 ϕAH (H, AH) and the set of isomorphism classes of H-Galois extensions with normal basis whose left action over AH is ϕAH .

The quantum double of a factorizable weak Hopf algebra

ABSTRACTLet (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k. We first construct a weak Hopf algebra [Δ(1)(H⊗H)Δ(1)]R, which is based on the subalgebra of the tensor product algebra H⊗H. Next we verify that if H is factorizable, then the Drinfeld’s quantum double of H is isomorphic to [Δ(1)(H⊗H)Δ(1)]R.

Weak rigid monoidal category

We define the right regular dual of an object X in a monoidal category C; and give several results regarding the weak rigid monoidal category. Based on the definition of the right regular dual, we construct a weak Hopf algebra structure of H = End(F) whenever (F; J) is a fiber functor from category C to Vec and every X ∈ C has a right regular dual. To conclude, we give a weak reconstruction theorem for a kind of weak Hopf algebra.

Morita equivalence for weak Hopf–Galois extensions

ABSTRACTLet H be a commutative weak Hopf algebra over a fixed field k, and A,B be weak H-Hopf Galois extensions. In this paper, we give a structure theorem for the category of weak relative Hopf bimodules. We also investigate the Morita equivalences between A and B based on the structure theorem.

Pseudotriangular weak Hopf algebras

In this paper, in looking for the weak Hopf algebraic counterpart of pseudosymmetric braidings, we introduce the concept of a pseudotriangular weak Hopf algebra, which is a quasitriangular weak Hopf algebra satisfying an extra condition. Then, we investigate the question, when a quasitriangular weak Hopf algebra is pseudotriangular. As an application, we study a special class of pseudotriangular weak Hopf algebras, under the name almost-triangular weak Hopf algebras and list some nontrivial examples. Finally, in order to construct more examples of pseudotriangular weak Hopf algebras, we show that the pseudosymmetry of the Yetter–Drinfeld category [Formula: see text] is determined by the commutativity and cocommutativity of [Formula: see text], where [Formula: see text] is a weak Hopf algebra with a bijective antipode.

Module categories of finite Hopf algebroids, and self-duality

We characterize the module categories of suitably finite Hopf algebroids (more precisely, $X_R$-bialgebras in the sense of Takeuchi (1977) that are Hopf and finite in the sense of a work by the author (2000)) as those $k$-linear abelian monoidal categories that are module categories of some algebra, and admit dual objects for sufficiently of their objects. Then we proceed to show that in many situations the Hopf algebroid can be chosen to be self-dual, in a sense to be made precise. This generalizes a result of Pfeiffer for pivotal fusion categories and the weak Hopf algebras associated to them.

Structure Theorems for Bicomodule Algebras Over Quasi-Hopf Algebras, Weak Hopf Algebras, and Braided Hopf Algebras

Let H be a quasi-Hopf algebra, a weak Hopf algebra, or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v: H → B. Then we can define an object Bco(H), which is a left-left Yetter–Drinfeld module over H, having extra properties that allow to make a smash product Bco(H)#H, which is an H-bicomodule algebra, isomorphic to B.

MORITA CONTEXT FOR WEAK DOI-KOPPINEN SMASH PRODUCTS AND ITS APPLICATIONS

The paper is concerned with the Morita context for weak Doi-Koppinen smash products and the surjectivity of Morita maps are studied. As an application, we obtain a Morita context for endomorphism algebras for weak Doi-Hopf modules induced by coquasitriangular weak Hopf algebras.

Hopf Categories

We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories.

A Characterization of Weak Hopf (co) Quasigroups

In this paper, we show that weak Hopf (co)quasigroups can be characterized by a Galois-type condition. Taking into account that this notion generalizes the ones of Hopf (co)quasigroup and weak Hopf algebra, we obtain as a consequence the first fundamental theorem for Hopf (co)quasigroups and a characterization of weak Hopf algebras in terms of bijectivity of a Galois-type morphism (also called fusion morphism).