On Square Roots of the Haar State on Compact Quantum Groups

TL;DR

This paper extends the classical concept of square roots of the Haar measure to compact quantum groups, characterizing those with no non-trivial roots and exploring their corepresentation structures.

Contribution

It provides a simple characterization of compact quantum groups without non-trivial Haar state square roots, linking to their corepresentation theory and Kac type properties.

Findings

Such quantum groups are necessarily of Kac type.

Subalgebras generated by two-dimensional irreducible corepresentations resemble quaternionic function algebras.

An example of a non-commutative, non-cocommutative quantum group with no non-trivial Haar square root is provided.

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.