====== Half-liberated orthogonal easy quantum group  ======

**Half-liberated easy orthogonal quantum groups** are a particular class of [[easy_quantum_group|easy orthogonal quantum groups]] each of which can be seen as interpolating a [[classical matrix group]] and its free, "liberated"  version (without being classical or free itself). There is also a corresponding notion of [[half-liberated unitary easy quantum group]].

===== Definition =====

An [[easy_quantum_group|orthogonal easy quantum group]] $G\cong (C(G),u)$ associated partition category $\Cscr\subseteq \Pscr$ is called **half-liberated** if $\Cscr\cap \{\Pabab, \Pabcabc\}=\{\Pabcabc\}$.  Equivalently, the  corepresentation category $\FundRep(G)$ of $G$ contains the [[category_of_all_partitions#linear_maps_associated_to_partitions|intertwinter]] $T_p$ for $p=\Pabcabc$ but //not// for $p=\Pabab$.