Definition of the Quantum Plane
When ever before I locate definition of the quantum aircraft it claims $A_q^2 = C\langle x,y \rangle/I$, where $I = C\langle xy-qyx \rangle$. What I need to know is whether they suggest the unital free algebra or simply the free algebra. In short, is the quantum aircraft unital?
In addition, when individuals write $A_q^N$, they suggest the free (unital) algebra with $N$ generaterators, where every generator simply commutes with every various other generator?
Kassel, p. 3 ($k$ is the ground area) :
An algebra is a ring along with a ring map $\eta_A : k \to A$ whose photo is had in the facility of $A$. ... A morphism of algebras or an algebra morphism _ is a ring map $f : A \to B$ such that $f \circ \eta_A = \eta_B.$ (1.1) As an effect of (1.1 ), $f$ maintains the devices , i.e., we have $f(1) = 1$.
There is additionally a definition of free algebras on p. 7 where he clearly mentions that they have a device.