On square roots of the Haar state on compact quantum groups

Uwe Franz, Adam Skalski*, Reiji Tomatsu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


The paper is concerned with the extension of the classical study of probability measures on a compact group which are square roots of the Haar measure, due to Diaconis and Shahshahani, to the context of compact quantum groups. We provide a simple characterisation for compact quantum groups which admit no non-trivial square roots of the Haar state in terms of their corepresentation theory. In particular it is shown that such compact quantum groups are necessarily of Kac type and their subalgebras generated by the coefficients of a fixed two-dimensional irreducible corepresentation are isomorphic (as finite quantum groups) to the algebra of functions on the group of unit quaternions. An example of a quantum group whose Haar state admits no nontrivial square root and which is neither commutative nor cocommutative is given.

Original languageEnglish
Pages (from-to)2079-2093
Number of pages15
JournalJournal of Pure and Applied Algebra
Issue number10
Publication statusPublished - 2012 Oct
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory


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