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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.2010-12-01
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    Where exactly does it say this in Kassel?2010-12-01
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    ... and please write your answer as an answer and not a comment so I can mark it as accepted.2010-12-01
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    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*.2011-12-31

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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 $f(1) = 1$.

There is also a definition of free algebras on p. 7 where he explicitly states that they have a unit.

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    Perfect. Thanks alot.2010-12-01
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    ... and is my definition of $A^N_q$ correct?2010-12-01
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    Unfortunately, I don't know.2010-12-01
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    Ok. Maybe I'll pose it as another question.2010-12-01