Free orthogonal easy quantum group

Free easy orthogonal quantum groups are a particular class of easy orthogonal quantum groups each of which can be seen as a free, “liberated”, i.e., non-commutative, counterpart of a classical matrix group. There is also a corresponding notion of free unitary easy quantum group.

Definition

An orthogonal easy quantum group $G\cong (C(G),u)$ associated partition category $\Cscr\subseteq \Pscr$ is called free if $\Cscr$ is non-crossing, i.e., if $\Cscr\subseteq \langle \fourpart, \singleton \rangle$ , where the right hand side of this inclusion is the category of all non-crossing partitions.