====== Unimodularity ======

Let $G$ be a [[compact quantum group]]. Denote by $S$ the antipode on $\Pol G$, by $h$ the [[compact_quantum_group#haar_state|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. [[https://www.sciencesmaths-paris.fr/upload/Contenu/Fichiers_chaire/livre%20Sergey%20Neyshveyev.pdf|Available online]]