Discrete quantum groups and their duals

Discrete quantum groups were introduced as duals of compact quantum groups by Podleś and Woronowicz in 1990. Shortly after, they were defined and studied intrinsically by Effros and Ruan, and by this author. In 1998, with the introduction of the multiplier Hopf algebras with integrals (also called algebraic quantum groups), the duality between discrete and compact quantum groups became part of the more general duality in the self-dual category of these algebraic quantum groups. Again a few years later the duality was extended to all locally compact quantum groups. In these notes, we give a new and a somewhat updated approach of the theory of discrete quantum groups. In particular, we view them as special cases of algebraic quantum groups. The duality between the compact quantum groups and the discrete quantum groups is seen in this larger context. This has a number of advantages as we will explain. On the one hand, we provide quite a bit of information about how all of this fits into the more general theory of algebraic quantum groups and its duality. Occasionally, we even go one step further and look at the most general case of locally compact quantum groups. Also sometimes, we compare with known results in pure Hopf algebra theory. On the other hand however, we have tried to make these notes highly self-contained. The aim of these notes in the first place is not to give new results but rather to review known results in a more modern perspective, taking into account recent developments. We believe this may be helpful for people who want to work with compact and discrete quantum groups now.

💡 Research Summary

This paper, titled “Discrete quantum groups and their duals” by A. Van Daele, presents a modernized and unified approach to the theory of discrete quantum groups by embedding it within the broader framework of algebraic quantum groups (multiplier Hopf algebras with integrals).

The introduction reviews the historical context, tracing the operator algebraic approach to quantum groups from Pontryagin duality for abelian groups to the modern theory of locally compact quantum groups. The author notes the technical complexity of the full locally compact theory and advocates for the utility of the simpler, more algebraic intermediate theory of multiplier Hopf *-algebras with positive integrals (∗-algebraic quantum groups), which naturally contains both discrete and compact quantum groups and is self-dual.

In Section 1, the author introduces a general concept: a multiplier Hopf algebra of discrete type, defined by the existence of both a left and a right cointegral (Definition 1.3). Key properties of cointegrals are established. A significant focus is on the case where the cointegral h satisfies ε(h) ≠ 0, allowing it to be normalized to a self-adjoint idempotent. In this case, Δ(h) plays the role of a separability idempotent in M(A⊗A), which becomes fundamental for constructing integrals on A.

Section 2 specializes to the definition of a discrete quantum group. It is defined as a multiplier Hopf ∗-algebra of discrete type whose underlying ∗-algebra is a direct sum of full matrix algebras over C (Definition 2.2). Within this concrete setting, the cointegral can be constructed explicitly from the algebra structure. A major result is that the square of the antipode, S², leaves each matrix component invariant, is a homomorphism, and is implemented on each component A_α by a positive, invertible element q_α. Consequently, the left and right integrals on A are expressed using the standard traces on these components, weighted by these elements q_α or their inverses. This treatment offers a fresh perspective compared to earlier intrinsic definitions.

Section 3 addresses duality. The dual of a discrete quantum group A is shown to be a compact quantum group Â, with the duality realized within the general duality theory of algebraic quantum groups. The author emphasizes a reversed perspective: instead of starting from a compact quantum group and deriving its discrete dual, the paper takes the discrete quantum group as the primary object and derives properties of its compact dual. A one-to-one correspondence between representations of A and corepresentations of  is established, leading to various properties of compact quantum groups.

Section 4 illustrates the general theory with classical examples: discrete groups and duals of compact groups. Section 5 offers conclusions and suggestions for further research. An appendix discusses differing conventions and approaches used by various authors in the field.

Overall, the paper’s main achievement is providing a coherent, updated, and self-contained exposition of discrete quantum groups. By systematically presenting them as special cases of algebraic quantum groups, it clarifies their structure, simplifies proofs (often by appealing to more general results), and offers a modern viewpoint that integrates historical developments with recent advances in the general theory of quantum groups. This approach aims to make the subject more accessible and useful for researchers currently working with compact and discrete quantum groups.