When ever I find definition of the quantum plane it says $A_q^2 = C\langle x,y \rangle/I$, where $I = C\langle xy-qyx \rangle$. What I want to know is whether they mean the unital free algebra or just the free algebra. In brief, is the quantum plane unital?
Moreover, when people write $A_q^N$, they mean the free (unital) algebra with $N$ generaterators, where every generator just commutes with every other generator?