Constructing Quasitriangular Multiplier Hopf Algebras By Twisted Tensor Coproducts

Let [Formula: see text] be a groupoid and assume that [Formula: see text] is a family of algebras with identity. First, we introduce the notion of a weak Hopf (groupoid)[Formula: see text]-coalgebra by that if, for each pair [Formula: see text], there is given a unital homomorphism [Formula: see text] satisfying certain properties, generalizing the notion of Hopf group-coalgebras as introduced by Turaev from groups to groupoids and Hopf algebra structures to weak Hopf algebra structures. Then one considers now the direct sum [Formula: see text] of these algebras. It is an algebra, without identity, except when [Formula: see text] is a finite groupoid, but the product is non-degenerate. The maps [Formula: see text] can be used to define a coproduct [Formula: see text] on [Formula: see text] and the conditions imposed on these maps give that [Formula: see text] is a weak multiplier Hopf algebra. It is [Formula: see text]-cograded as explained in this paper. We study these so-called groupoid-cograded weak multiplier Hopf algebras. They are, as explained above, more general than the weak Hopf group-coalgebras (introduced by Van Daele and Wang), generalizing the Turaev’s Hopf group-coalgebras. Moreover, our point of view makes it possible to use results and techniques from the theory of weak multiplier Hopf algebras in the study of weak Hopf groupoid-coalgebras (and generalizations).

Twisted tensor coproduct of multiplier Hopf algebras

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

Let ⟨ A, B⟩ be a pairing of two regular multiplier Hopf algebras A and B. One method of constructing the Drinfel'd double 𝒟 = A⋈ B cop is by the use of an invertible twist map R: B⊗ A→ A⊗ B defining an associative product on A⊗ B. In Delvaux (2003) and Drabant and Van Daele (2001), the authors construct R by , where R 2 and is only related to the module actions between A and B. Another way is given in Delvaux and Van Daele (2004a) in which the authors also just consider the module actions and then construct the Drinfel'd double 𝒟 as an algebra of operators on the vector space B⊗ A. In this article we will give two different points of view of constructing the Drinfel'd double 𝒟 for multiplier Hopf algebras. The first is that the Drinfel'd double 𝒟 associated to the pairing ⟨ A, B⟩ is constructed by using not only the module actions but also the comodule coactions, i.e., the Drinfel'd double 𝒟 is given in the framework of a special twisted tensor product algebra structure A ⊡ A op ⊗ A B. The second is as follows: If P is a multiplier Hopf algebra and a reduced (A, B)-bicomodule algebra (an A-Long module algebra), then we present a twisting construction of the product of P via the coactions of A and B on P, and we show that the Drinfel'd double 𝒟 is isomorphic as a multiplier Hopf algebra to the opposite twisting of (A op, cop ⊗ B op, cop ). As an application of our theory, we consider the case of group-cograded multiplier Hopf algebras and the case of Hopf group-coalgebras.

We extend the Larson–Sweedler theorem to group-cograded multiplier Hopf algebras introduced in Abd El-hafez et al. (2004), by showing that a group-cograded multiplier bialgebra with finite-dimensional unital components is a group-cograded multiplier Hopf algebra if and only if it possesses a nondegenerate left cointegral. We also generalize the theory of multiplier Hopf algebras of discrete type in Van Daele and Zhang (1999) to group-cograded multiplier Hopf algebras. Our results are applicable to Hopf group-coalgebras in the sense of Turaev (2000). Finally, we study regular multiplier Hopf algebras of η -discrete type.

Discrete Quantum Groups

Let H be a Hopf algebra in a rigid symmetric monoidal category C then the evaluation map τis a convolution-invertible skew pairing. In the previous paper, we constructed a Hopf algebra D(H)=H ⋈ r H *cop in C. In this paper, we first show that D(H) is a quasitriangular Hopf algebra in C. Next, let H be an ordinary triangular finite-dimensional Hopf algebra. Then one can form quasitriangular Hopf algebras B(H,H) and B(H,D(H)) (in a rigid braided monoidal category) by Majid’s method associated to the ordinary Hopf algebra maps H →H and iH H → D(H), where D(H) is the Drin-fePd quantum double. We show that D (B(H,H)) and B(H,D(H)) are isomorphic Hopf algebras in the braided monoidal category.

We study algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field k k of characteristic 0 0 . The first class consists of those Hopf k k -algebras that are connected graded as algebras, and the second class are those Hopf k k -algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel’fand–Kirillov dimension. It is shown that if the Hopf algebra H H is a connected graded Hopf algebra of finite Gel’fand–Kirillov dimension n n , then H H is a noetherian domain which is Cohen–Macaulay, Artin–Schelter regular, and Auslander regular of global dimension n n . It has S 2 = I d H S^2 = \mathrm {Id}_H , and it is Calabi–Yau. Detailed information is also provided about the Hilbert series of H H . Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k k -algebras of finite Gel’fand–Kirillov dimension n n with connected coalgebra, the underlying coalgebra is shown to be Artin–Schelter regular of global dimension n n . Both these classes of Hopf algebras share many features in common with enveloping algebras of finite-dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k k -algebra H H of Gel’fand–Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U ( g ) U(\mathfrak {g}) for any Lie algebra g \mathfrak {g} .

Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals.We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

Double Quantum Groups

We study versions of the categories of Yetter-Drinfel'd mod- ules over a Hopf algebra H in a braided monoidal category C. Contrary- wise to Bespalov's approach, all our structures live in C. This forces H to be transparent or equivalently to lie in Muger's center Z2(C) of C. We prove that versions of the categories of Yetter-Drinfel'd modules in C are braided monoidally isomorphic to the categories of (left/right) modules over the Drinfel'd double D(H) > C for H nite. We obtain that these categories polarize into two disjoint groups of mutually iso- morphic braided monoidal categories. We conclude that if H > Z2(C), then D(H)C embeds as a subcategory into the braided center category Z1(HC) of the category HC of left H-modules in C. For C braided, rigid and cocomplete and a quasitriangular Hopf algebra H such that H > Z2(C) we prove that the whole center category of HC is monoidally isomorphic to the category of left modules over Aut(HC) # H - the bosonization of the braided Hopf algebra Aut(HC) which is the coend in HC. A family of examples of a transparent Hopf algebras is discussed.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Hopf algebras have important connections to quantum theory, Lie algebras, knot and braid theory, operator algebras and other areas of physics and mathematics. They have been intensely studied in the past; in particular, the solution of a number of conjectures of Kaplansky from the 1970s has led to progress on the classification of semisimple Hopf algebras and on the structure of pointed Hopf algebras. Among the topics covered are results toward the classification of finite-dimensional Hopf algebras (semisimple and non-semisimple), as well as what is known about the extension theory of Hopf algebras. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to work in quantum groups. The book also explores the connections and applications of Hopf algebras to other fields.

It has become clear over the years that quantum groups (i.e., quasitriangular Hopf algebras, see [D]) and their semiclassical counterpart, Poisson Lie groups, are an essential algebraic structure underlying three related subjects: integrable models of statistical mechanics, conformal field theory, and integrable models of quantum field theory in 1+1 dimensions. Still, some points remain obscure from the point of view of Hopf algebras. In particular, integrable models associated with elliptic curves are still poorly understood. We propose her an elliptic version of quantum groups, based on the relation to conformal field theory, which hopefully will be helpful to complete the picture.

A fundamental problem in the theory of Hopf algebras is the classification and construction of finite-dimensional (minimal) triangular Hopf algebras (A,R) introduced by Drinfeld. Only recently Etingof and the author completely solved this problem for semisimple A over algebraically closed fields of characteristics 0 and p>>dim(A) (any p if one assumes that A is also cosemisimple). In this paper we take the first step towards solving this problem for finite-dimensional pointed Hopf algebras over an algebraically closed field k of characteristic 0. We first prove that the fourth power of the antipode of any triangular pointed Hopf algebra A is the identity. We do that by focusing on minimal triangular pointed Hopf algebras (A,R) (every triangular Hopf algebra contains a minimal triangular sub Hopf algebra) and proving that the group algebra of the group of grouplike elements of A (which must be abelian) admits a minimal triangular structure and consequently that A has the structure of a biproduct. We also generalize our result on the order of the antipode to any finite-dimensional quasitriangular Hopf algebra A whose Drinfeld element u acts as a scalar in any irreducible representation of A (e.g. when A^* is pointed). Second, we describe a method of construction of finite-dimensional pointed Hopf algebras which admit a minimal triangular structure, and classify all their minimal triangular structures. We conclude the paper by proving that any minimal triangular Hopf algebra which is generated as an algebra by grouplike elements and skew primitive elements is isomorphic to a minimal triangular Hopf algebra constructed using our method.