In the usual (fomal) construction of the quantum general linear group $GL_q(N)$, an Ore extension is used. See for example Kassel. Why is this necessary? Surely one can just augment the set of generators with an element det$^{-1}$ and the set of relations with the relations det.det$^{-1} = 1$ and det$^{-1}$.det$=1$ and get the same thing without going to all the trouble of considering Ore extensions?
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asked Mar 3, 2011 at 18:21
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I have hard time understanding what is your notion of usual; most references indeed define the quantum linear group $M_q(N)$ by generators and relations (or by the universal property as defined by Manin) and then localize at $det_q$. So far as the construction goes. But it is of course very useful to know that one can start with the polynomial ring and then do the iterated Ore extension $N^2-1$ times to obtain $M_q(N)$. This is useful to infer many useful ring-theoretic properties. For example, it follows that the matrix bialgebra $M_q(N)$ and its Hopf envelope $GL_q(N)$ are Ore domains (the same O. Ore!), hence they are contained in the Ore quotient ring in which we can do many useful computations. Various intermediate Ore localizations are also useful.
answered Mar 3, 2011 at 19:17
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1$\begingroup$ Well said. Another example of one of those useful ring-theoretic properties is that every prime ideal is completely prime (for $q$ not a root of unity). This would seem to go without saying, but since the question was asked--the study of any mathematical object is generally advanced by being able to view the object as a special case of something with a well-developed theory. $\endgroup$ Commented Mar 3, 2011 at 20:36