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On square roots of the Haar state on compact quantum groups
Authors:Uwe Franz  Adam Skalski  Reiji Tomatsu
Institution:1. Département de Mathématiques de Besançon, Université de Franche-Comté 16, route de Gray, 25 030 Besançon cedex, France;2. Institute of Mathematics of the Polish Academy of Sciences, ul.?niadeckich 8, 00-956 Warszawa, Poland;3. Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan
Abstract: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.
Keywords:
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