TY - JOUR

T1 - On square roots of the Haar state on compact quantum groups

AU - Franz, Uwe

AU - Skalski, Adam

AU - Tomatsu, Reiji

N1 - Funding Information:
U.F. was supported by a Marie Curie Outgoing International Fellowship of the EU (Contract Q-MALL MOIF-CT-2006-022137) and an ANR Project (Number 2011 BS01 008 01). A.S. was partly supported by the National Science Centre (NCN) grant no. 2011/01/B/ST1/05011.

PY - 2012/10

Y1 - 2012/10

N2 - 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.

AB - 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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U2 - 10.1016/j.jpaa.2012.01.020

DO - 10.1016/j.jpaa.2012.01.020

M3 - Article

AN - SCOPUS:84861345535

VL - 216

SP - 2079

EP - 2093

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 10

ER -