Weak Hopf Algebras Research Articles

Homological dimension of weak Hopf–Galois extensions

Let H be a weak Hopf algebra, A a right weak H-comodule algebra and B the subalgebra of the H-coinvariant elements of A. Let A/B be a right weak H-Galois extension. We prove that A/B is a separable extension if H is semisimple. Using this, we show that the global dimension and weak dimension of A are less than those of B. As an application, we obtain Maschke-type theorems for weak Hopf–Galois extensions and weak smash products.

Weak Hopf algebras corresponding to quantum algebras U q (f (K, H))

In this paper, we investigate weak Hopf algebras introduced in Li (J Algebra 208:72–100, 1998; Commun Math Phys 225:191–217, 2002) corresponding to quantum algebras Uq(f (K, H)) (see Wang et al. in Commun Algebra 30:2191–2211, 2002). A new class of algebras is defined, which is denoted by \({\mathfrak{w}U^{d}_q.}\) For d = ((1, 1) | (1, 1)), denote \({\mathfrak{w}U^{d}_q}\) briefly by \({\mathfrak{w}_{1}U_q}\) ; for d = ((0, 0) | (0, 0)), denote \({\mathfrak{w}U^{d}_q}\) briefly by \({\mathfrak{w}_{2}U_q.}\) In some cases, the necessary and sufficient conditions for \({\mathfrak{w}_{1}U_q}\) and \({\mathfrak{w}_{2}U_q}\) to be weak Hopf algebras are given. The PBW bases of \({\mathfrak{w}_{1}U_q}\) and \({\mathfrak{w}_{2}U_q}\) are presented. Finally, representations and the center of \({\mathfrak{w}_{1}U_q}\) are characterized over \({\mathbb{C}}\) with \({q \in \mathbb{C}}\) which is not a root of unity. Open image in new window

Braided groups and quantum groupoids

Let H be a quasitriangular weak Hopf algebra. It is proved that the centralizer subalgebra of its source subalgebra in H is a braided group (or Hopf algebra in the category of left H-modules), which is cocommutative and also a left braided Lie algebra in the sense of Majid.

On Weak Crossed Products of Weak Hopf Algebras

Let H be a weak Hopf algebra in the sense of Bohm et al. (J Algebra 221:385–438, 1999) measuring an algebra A. Let A# σ H be a weak crossed product with σ invertible. Then in this paper we first give some conditions for A# σ H to be a weak Hopf algebra. Next the spectral sequence for Ext will be constructed which yields an estimate for the global dimension of A# σ H in terms of the corresponding data for H and A. Furthermore, we will investigate when A ⊂ A# σ H becomes a separable extension. Finally, we prove that if H and its dual H* are both semisimple, then the finitistic dimension of A# σ H is equal to that of A.

MASCHKE-TYPE THEOREM AND DUALITY THEOREM FOR WEAK TWISTED SMASH PRODUCTS

Let $H$ be a weak Hopf algebra in the sense of Böhm and Szlachányi [3] and $A$ a weak $H$-bimodule algebra. Then in this paper we first introduce the notion of a weak twisted smash product $A \star H$ and then find some sufficient and necessary conditions making it into a weak bialgebra. Furthermore, we give a Maschke-type theorem for the weak twisted smash product over semisimple weak Hopf algebra $H$, which generalizes the well-known Maschke-type theorem in [5, 15, 17]. Finally, we obtain an analogue of the duality theorem for the weak twisted smash products.

Fusion categories in terms of graphs and relations

Every fusion category \mathcal{C} that is k -linear over a suitable field k is the category of finite-dimensional comodules of a weak Hopf algebra H . This weak Hopf algebra is finite-dimensional, cosemisimple and has commutative bases. It arises as the universal coend with respect to the long canonical functor \omega\colon\mathcal{C}\to \mathbf{Vect}_k . We show that H is a quotient H=H[\mathcal{G}]/I of a weak bialgebra H[\mathcal{G}] which has a combinatorial description in terms of a finite directed graph \mathcal{G} that depends on the choice of a generator M of \mathcal{C} and on the fusion coefficients of \mathcal{C} . The algebra underlying H[\mathcal{G}] is the path algebra of the quiver \mathcal{G}\times\mathcal{G} , and so the composability of paths in \mathcal{G} parameterizes the truncation of the tensor product of \mathcal{C} . The ideal I is generated by two types of relations. The first type enforces that the tensor powers of the generator M have the appropriate endomorphism algebras, thus providing a Schur–Weyl dual description of \mathcal{C} . If \mathcal{C} is braided, this includes relations of the form ‘ RTT=TTR ’ where R contains the coefficients of the braiding on \omega M\otimes\omega M , a generalization of the construction of Faddeev–Reshetikhin–Takhtajan to weak bialgebras. The second type of relations removes a suitable set of group-like elements in order to make the category of finite-dimensional comodules equivalent to \mathcal{C} over all tensor powers of the generator M . As examples, we treat the modular categories associated with U_q(\mathfrak{sl}_2) .

Infinite-dimensional compact quantum semigroup

In this paper we construct a compact quantum semigroup structure on the Toeplitz algebra $\mathcal{T}$. The existence of a subalgebra, isomorphic to the algebra of regular Borel's measures on a circle with convolution product, in the dual algebra $\mathcal{T}^*$ is shown. The existence of Haar functionals in the dual algebra and in the above-mentioned subalgebra is proved. Also we show the connection between $\mathcal{T}$ and the structure of weak Hopf algebra.

A compact quantum semigroup generated by an isometry

In this paper we construct a compact quantum semigroup structure on a Toeplitz algebra. We prove the existence of a subalgebra in the dual algebra isomorphic to the algebra of regular Borel measures on a circle with the convolution product. We also prove the existence of Haar functionals in the dual algebra and in the mentioned subalgebra. We show that this compact quantum semigroup contains a dense subalgebra with the structure of a weak Hopf algebra.

Transmutation theory of a coquasitriangular weak Hopf algebra

Let H be a coquasitriangular quantum groupoid. In this paper, using a suitable idempotent element e in H, we prove that eH is a braided group (or a braided Hopf algebra in the category of right H-comodules), which generalizes Majid’s transmutation theory from a coquasitriangular Hopf algebra to a coquasitriangular weak Hopf algebra.

The braided monoidal structures on the category of vector spaces graded by the Klein group

AbstractLet k be a field, let k* = k \ {0} and let C2 be a cyclic group of order 2. We compute all of the braided monoidal structures on the category of k-vector spaces graded by the Klein group C2 × C2. For the monoidal structures we compute the explicit form of the 3-cocycles on C2 × C2 with coefficients in k*, while, for the braided monoidal structures, we compute the explicit form of the abelian 3-cocycles on C2 × C2 with coefficients in k*. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras with underlying vector space k[C2 × C2].

The Miyashita–Ulbrich Action for Weak Hopf Algebras

In this article we extend the Miyashita–Ulbrich action for weak H-Galois extensions associated to a weak bialgebra H. Also, if H is a weak Hopf algebra, we prove that this action induces a monoidal connection with the category of right-right Yetter–Drinfeld modules over H.

Weak Hopf Quivers, Clifford Monoids and Weak Hopf Algebras

The purpose of this paper is to use Clifford monoids to define weak Hopf quivers and to construct the corresponding class of weak Hopf algebras, i.e., the semilattice of graded weak Hopf algebras which were found to be useful in obtaining a regular R-matrix of a quantum double without a quasi-triangular structure. We define weak Hopf quivers and classify the semilattice of graded weak Hopf algebra structures over path coalgebras using weak Hopf quivers. This gives, precisely, a generalization of the classification of the graded weak Hopf algebra structures over path coalgebras using such quivers.

Morita context of weak Hopf algebras

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A H . We prove that the smash product A # H is an A-ring with a grouplike character, and give a criterion for A # H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context 〈 A H, A#H, A, A , τ, μ〉 connecting the smash product A # H and the invariant subalgebra A H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.

A Clifford algebra is a weak Hopf algebra in a suitable symmetric monoidal category

It is well known that Clifford algebras are group algebras deformed by a 2-cocycle. Furthermore, these algebras, which are not commutative in the usual sense, can be viewed as commutative algebras in certain symmetric monoidal categories of graded vector spaces. In this note we invent a Clifford process for coalgebras that will allow us to show that Clifford algebras have also cocommutative coalgebra structures, and consequently commutative and cocommutative weak braided Hopf algebras structures, within the same symmetric monoidal categories where they lie as commutative algebras. Also, we will show that they are selfdual weak braided Hopf algebras, monoidal Frobenius algebras and monoidal coFrobenius coalgebras.

Projections of weak braided Hopf algebras

In this paper we study the projections of weak braided Hopf algebras using the notion of Yetter-Drinfeld module associated with a weak braided Hopf algebra. As a consequence, we complete the study of the structure of weak Hopf algebras with a projection in a braiding setting obtaining a categorical equivalence between the category of weak Hopf algebra projections associated with a weak Hopf algebra H living in a braided monoidal category and the category of Hopf algebras in the non-strict braided monoidal category of left-left Yetter-Drinfeld modules over H.

Maschke’s theorem for smash products of quasitriangular weak Hopf algebras

The paper is concerned with the semisimplicity of smash products of quasitriangular weak Hopf algebras. Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k and A any semisimple and quantum commutative weak H-module algebra. Based on the work of Nikshych et al. (Topol. Appl. 127(1–2):91–123, 2003), we give Maschke’s theorem for smash products of quasitriangular weak Hopf algebras, stating that A#H is semisimple if and only if A is a projective left A#H-module, which extends the Theorem 3.2 given in Yang and Wang (Commun. Algebra 27(3):1165–1170, 1999).

Quantum Groupoids Acting on Semiprime Algebras

Following Linchenko and Montgomery′s arguments we show that the smash product of an involutive weak Hopf algebra and a semiprime module algebra, satisfying a polynomial identity, is semiprime.

The Duality Theorem for Weak Hopf Algebra (Co) Actions

In this article, we mainly study a new notion of a generalized smash product for weak Hopf comodule algebras and provide a new version of the duality theorem for weak smash products as an application.

On Braided Lie Structures of Algebras in the Categories of Weak Hopf Bimodules

Let H be a weak Hopf algebra. In this paper, it is proved that the monoidal category [Formula: see text] of weak Hopf bimodules studied in Wang [19] is equivalent to the monoidal category [Formula: see text] of weak Yetter–Drinfel'd modules introduced in Böhm [2]. When H has a bijective antipode, a braiding in the category [Formula: see text] is constructed by the braiding on [Formula: see text], generalizing the main result in Schauenburg [14]. Finally, the braided Lie structures of an algebra A in the category [Formula: see text] are investigated, by showing that if A is a sum of two braided commutative subalgebras, then the braided commutator ideal of A is nilpotent.

Hopf-Type Algebras

In this article, we provide an alternative approach to the definition of a weak Hopf algebra (WHA). For an associative unital algebra A with a coassociative comultiplication Δ ∈Alg u (A, A ⊗ A), the set of homomorphisms from A to A ⊗ A, which do not preserve the units. If the linear maps Ξ1, Ξ2 ∈ End(A ⊗ A), defined by Ξ1(a ⊗ b) = Δ(a)(1 ⊗ b), Ξ2(a ⊗ b) = (a ⊗ 1)Δ(b), are von Neumann regular elements in the ring End(A ⊗ A) of endomorphisms of A ⊗ A satisfying some appropriate assumptions, we call the A a Hopf-type algebra. We show the existence of a target, a source, a counit, and an antipode of A as in the usual WHA.