Duality Theory Achieves Broader Scope with Discrete Quantum Group Frameworks

The mathematical relationship between compact and discrete groups has long been a subject of intense study, and now Alfons Van Daele of KU Leuven, along with colleagues, presents a fresh perspective on this duality. Building on earlier work from the 1990s, this research views discrete groups not as isolated entities, but as specific examples within a broader class of algebraic groups, thereby illuminating their connection to compact groups within a more comprehensive framework. This approach offers significant advantages, providing a unified understanding of the duality and incorporating recent advances in the field, while also remaining accessible to researchers working with these fundamental mathematical structures. By synthesising established results with a modern outlook, the team aims to provide a valuable resource for those seeking to explore the interplay between compact and discrete groups.

By viewing discrete groups as specific instances within a broader class of algebraic groups, this research illuminates their connection to compact groups within a more comprehensive framework, providing a unified understanding of duality and remaining accessible to researchers working with these fundamental mathematical structures.

Quantum Groups, Duality and Multiplier Hopf Algebras

This collection represents an extensive compilation of references concerning algebraic quantum groups, duality, and related mathematical topics, focusing on algebraic quantum groups, their definition, structure, and properties. A central theme is duality, particularly as it relates to the multiplier Hopf algebra approach, which serves as a key algebraic tool throughout the research. The list also covers locally compact quantum groups, compact quantum groups, and the broader mathematical context of duality for both classical and non-unital algebras.

This is a remarkably thorough list, covering a wide range of papers and books from foundational works to current research, with a significant portion of the references to publications by A. The numerous references to preprints on arXiv indicate ongoing research activity, and S. Woronowicz is another prominent figure with key papers on compact quantum groups. The list traces the historical development of the field, and the inclusion of many arXiv preprints indicates that this is a very active area of research, with new results constantly being published, suggesting a strong emphasis on the technical and algebraic aspects of quantum groups.

Discrete Quantum Groups and Cointegral Properties

Recent research presents a refined approach to the theory of discrete quantum groups, positioning them within the broader context of algebraic and locally compact quantum groups. Researchers define a multiplier Hopf algebra as being of discrete type if it possesses both left and right cointegrals, fundamental elements used to construct integrals, meticulously examining their properties and demonstrating how their application to the coproduct yields a separability idempotent within the multiplier algebra.

Scientists then applied these results to the specific case of discrete quantum groups, building upon earlier work by Podleś and Woronowicz, who initially introduced these groups as duals of compact quantum groups. The study demonstrates that viewing discrete quantum groups as duals is historically motivated, but that a more general approach, where they are considered special cases within the framework of locally compact quantum groups, offers significant simplification, streamlining the analysis and providing a deeper understanding of the underlying mathematical structures.

Discrete Quantum Groups and General Duality

This work presents a modern review of discrete quantum groups, building upon decades of research originating with Podleś and Woronowicz in 1990, who initially introduced them as duals of compact quantum groups. The study revisits these foundational concepts, positioning discrete quantum groups within the broader framework of algebraic quantum groups and their duality, ultimately aiming for a simplified and more cohesive understanding, demonstrating that viewing discrete quantum groups as duals of compact quantum groups is a special case of a more general duality.

The investigation confirms that the underlying algebra of the dual of a compact quantum group is a direct sum of matrix algebras, a property derived from both compact quantum group theory and the existence of a cointegral within the dual structure. This work emphasizes that while historically motivated, viewing discrete quantum groups as duals offers a natural and potentially easier path for theoretical development compared to the study of compact quantum groups, building upon the development of locally compact quantum groups advanced by Masuda, Nakagami, and Woronowicz, as well as Kustermans and Vaes.

Discrete Quantum Groups and Hopf Algebras

This work presents a modern review of discrete quantum groups, positioning them within the broader context of multiplier Hopf algebras and locally compact quantum groups. Researchers successfully demonstrate that discrete quantum groups can be understood as special cases of these more general algebraic structures, offering a unified perspective on duality between compact and discrete groups, clarifying relationships within the field and providing a framework for understanding recent developments.

The study builds upon established theory, synthesizing results from diverse areas of Hopf algebra and quantum group research, providing a highly self-contained and updated perspective, and addressing existing nuances in conventions within the field. Researchers indicate that future work may explore further refinements of this unified approach and potentially expand the understanding of duality within locally compact quantum groups.