On Square Roots of the Haar State on Compact Quantum Groups

Details On Square Roots of the Haar State on Compact Quantum Groups On Square Roots of the Haar State on Compact Quantum Groups

2010-12-09

arxiv.org/abs/1012.2004v2

Mathematics

Operator Algebras

23 pages, v2 simplifies the constructions of examples in Section 6 and corrects some typos. The paper will appear in the Journ

Scientific paper

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.

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