Studying quantized algebras, what motivates the choice of base ring?

Question

In the fields of representation theory and quantum algebra, we often start with, for example, some $\mathbf{C}$-algebra $A$ and study a quantization of $A$ by adjoining an indeterminate $q$, or sometimes $v$, to twist the multiplication in $A$. Now, in different contexts I've seen this quantized algebra regarded as an algebra over any of $$ \mathbf{Z}[q] \quad \mathbf{Z}\left[q,q^{-1}\right] \quad \mathbf{Q}\left[q,q^{-1}\right] \quad \mathbf{Q}\left(q\right) \quad \mathbf{C}\left[q,q^{-1}\right] \quad \mathbf{C}\left(q\right) . $$ Is there any overarching motivation/intuition behind which ring to consider as the base ring of our quantized version of $A$? Like, is there anything clever to be said here that I'm not picking up on, or do researchers/authors simply use the smallest one they need at the time?

Answer 1

I'm not an expert but here is what I understood :

First I believe there is not much difference for $\Bbb Q(q)$ vs $\Bbb C(q)$.

$\Bbb C(q)$ vs $\Bbb C[q,q^{-1}]$ : the nice thing with $\Bbb C(q)$ is that it's a field, and the nice feature with $\Bbb C[q,q^{-1}]$ is that there is an "evaluation morphism" $ev_a : \Bbb C[q,q^{-1}] \to \Bbb C$ for any complex number $a \in \Bbb C \backslash \{0,1,-1\}$. ($ev_a$ is not defined at $f \in \Bbb C(q)$ if $f$ has a pole at $a$.

$\Bbb Z[q,q^{-1}]$ is especially useful when you are dealing with root of unity. In this case Lusztig (but I believe it was introduced by Kostant in the context of Lie algebra of positive characteristic) introduced a nice form for quantum group involving divided powers, which totally disappear when extending to e.g $\Bbb Q(q)$. With divided power the representation theory of quantum groups at root of $1$ is easier and look a bit more familiar.

There are also other coefficients, for example $\Bbb Q[q]_{(q)}$ (localisation) is used a lot in the context of crystal basis, or Andersen-Polo-Kexin uses $\Bbb Z[v]_{\mathfrak m}$ where $\mathfrak m = (v,p)$ to be able to relate quantum groups with hyperalgebra of algebraic groups.