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.

An orthogonal easy quantum group associated partition category is called free if is non-crossing, i.e., if , where the right hand side of this inclusion is the category of all non-crossing partitions.