Take $\Bbbk$ to be a field, $q \in \Bbbk$ a nonroot of unity, and $U = U_q(\mathfrak g)$ the quantized enveloping algebra of a complex finite dimensional simple Lie algebra, and write $U^-$ for its negative part.

SHORT VERSION: can you get a Global basis of $U^-$ in the sense of Kashiwara and Lusztig in these conditions (i.e. not just $q$ transcendental over $q$), say by specialization, or do you lose some of the nice properties that make this basis a distinguished one?

LONG VERSION: Kashiwara and Lusztig discovered a very nice basis for $U^-$, called the global basis, but in order to show it exists they need $\Bbbk$ to be of characteristic $0$ and $q$ transcendental over $\mathbb Q$. On the other hand, they prove that this basis is in fact a basis of a suitable $\mathcal A = \mathbb Z[v,v^{-1}]$-form of $U^-$, denoted by $U^-_\mathcal A$. So letting $v$ act as multiplication by $q$ on $\Bbbk$ we get a map $U^- \to \Bbbk \otimes_{\mathcal A} U^-_{\mathcal A}$ that is (a) surjective and (b) compatible with the weight decomposition of $U^-$, hence an isomorphism (this is the part that breaks down if $q$ is a root of unity). So if I got this correctly, the global basis in fact survives in the case where $\Bbbk$ is any old field containing a nonroot of unity $q$.

Now the global basis has some other nice properties related to highest weight representations of $U$. In that case, I am less sure but I think similar (rather roundabout) arguments apply to show that they also survive from the $\mathbb Q(v)$-context to the $\Bbbk$ context, but maybe I am missing some obvious reason for which things break down quickly in the second case. I have searched the literature for references for global bases for $q$ nonroot of unity but could not find any. Is this simply because no one has found a need for them, or is it because they are not useful in this context?


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