# Definition of the Quantum Plane

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?

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Yes. At least, this is the definition of algebra given in Kassel. – Qiaochu Yuan Dec 1 '10 at 20:34
Where exactly does it say this in Kassel? – MikhailMatrix Dec 1 '10 at 20:44
... and please write your answer as an answer and not a comment so I can mark it as accepted. – MikhailMatrix Dec 1 '10 at 20:59
Why would one use the notation $A^N_q$ for the «free (unital) algebra with N generaterators, where every generator just commutes with every other generator»?! What people mean when they write that depends obviously on what they mean... This kind of question becomes sensible if you tell us what people wrote that and where. – Mariano Suárez-Alvarez Dec 31 '11 at 5:04

Kassel, p. 3 ($k$ is the ground field):
An algebra is a ring together with a ring map $\eta_A : k \to A$ whose image is contained in the center 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 a consequence of (1.1), $f$ preserves the units, i.e., we have $f(1) = 1$.
... and is my definition of $A^N_q$ correct? – MikhailMatrix Dec 1 '10 at 22:01