unimodularity

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Unimodularity

Let $G$ be a compact quantum group. Denote by $S$ the antipode on $\Pol G$ , by $h$ the Haar state of $G$ . Then the following are equivalent

$h$ is a trace,
$S^2=\id$ ,
$S$ is $*$ -preserving.

If those conditions are satisfied, we say $G$ is of Kac type or that $\hat G$ is unimodular.

Further reading

Sergey Neshveyev, Lars Tuset, Compact Quantum Groups and Their Representation Categories, Société Mathématique de France, 2013. Available online

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