Proceedings - Mathematical Sciences

, Volume 127, Issue 5, pp 881–933 | Cite as

Introduction to compact (matrix) quantum groups and Banica–Speicher (easy) quantum groups



This is a transcript of a series of eight lectures, 90 min each, held at IMSc Chennai, India from 5–24 January 2015. We give basic definitions, properties and examples of compact quantum groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz’s quantum version of the Tannaka–Krein theorem. Building on this, we define Banica–Speicher quantum groups (also called easy quantum groups), a class of compact matrix quantum groups determined by the combinatorics of set partitions. We sketch the classification of Banica–Speicher quantum groups and we list some applications. We review the state-of-the-art regarding Banica–Speicher quantum groups and we list some open problems.


Compact quantum groups compact matrix quantum groups easy quantum groups Banica–Speicher quantum groups noncrossing partitions categories of partitions tensor categories Tannaka–Krein duality 

Mathematics Subject Classification

20G42 05A18 46LXX 



The author would like to thank V. S. Sunder for his invitation to IMSc, Chennai, for his great hospitality and his vision and support to get these lecture notes produced. He would like to wish him all the best for his retirement, in deepest admiration for all of his mathematical achievements. Furthermore, he thanks Soumya for all his great work regarding the contents and the design of these notes and for providing all the figures. The author thanks Pierre Fima and Issan Patri for their various and valuable critical remarks throughout the lectures at IMSc. He also thanks Felix Leid and Simon Schmidt for improvements regarding the exposition in the chapter on Tannaka–Krein theory. The author was partially supported by the ERC Advanced Grant NCDFP, held by Roland Speicher.


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Copyright information

© Indian Academy of Sciences 2017

Authors and Affiliations

  1. 1.Saarland UniversitySaarbrückenGermany

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