Discrete Quantum Groups

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 and B be multiplier Hopf algebras, and let R ∈ M(B ⊗ A) be an anti-copairing multiplier, i.e, the inverse of R is a skew-copairing multiplier in the sense of Delvaux [5]. Then one can construct a twisted tensor coproduct multiplier Hopf algebra A ⊗ R B. Using this, we establish the correspondence between the existence of quasitriangular structures in A ⊗ R B and the existence of such structures in the factors A and B. We illustrate our theory with a profusion of examples which cannot be obtained by using classical Hopf algebras. Also, we study the class of minimal quasitriangular multiplier Hopf algebras and show that every minimal quasitriangular Hopf algebra is a quotient of a Drinfel’d double for some algebraic quantum group.

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).

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.

Twisted tensor coproduct of multiplier Hopf algebras

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.

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsion. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C ∗ \mathrm {C}^{*} -algebra of a countable discrete quantum group \mathbbl {\Gamma } implies that \mathbbl {\Gamma } is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition.

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.

We find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one‐dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non‐Kac versions can be realized as a glued free product of a (non‐Kac) free orthogonal quantum group with , the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by‐product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi‐Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.

International audience

Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory

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.

This is a survey of some aspects of the subject of approximation properties for locally compact quantum groups, based on lectures given at the {\it Topological Quantum Groups} Graduate School, 28 June - 11 July, 2015 in Bedlewo, Poland. We begin with a study of the dual notions of amenability and co-amenability, and then consider weakenings of these properties in the form of the Haagerup property and weak amenability. For discrete quantum groups, the interaction between these properties and various operator algebra approximation properties are investigated. We also study the connection between central approximation properties for discrete quantum groups and monoidal equivalence for their compact duals. We finish by discussing the central weak amenability and central Haagerup property for free quantum groups.

The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined.

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.