Drinfeld-Jimbo quantum groups for $q=0$

Question

In the Wikipedia page of Drinfeld--Jimbo quantum groups the values of $q=0,1$ are excluded so as to avoid dividing by zero. The $q=1$ case is discussed in this old question. What about the $q=0$ case? Is it also possible to define the dual quantum coordinate algebra $\mathcal{O}_q(G)$ for $q=0$?

Answer 1

Hong & Kang's book on Crystal bases (Introduction to quantum groups and crystal bases, AMS Graduate Studies in Mathematics, vol 42) which describes the work of Kashiwara and Lusztig is a good place to start. In short, $U_q(\mathfrak{g})$ does not exist as a Hopf algebra at $q=0$, but one can talk about $U_q(\mathfrak{g})$-modules at $q=0$. Kashiwara had also dealt with the algebra of coordinate functions $\mathcal{O}_q(G)$ at $q=0$ in his paper Global crystal bases of quantum groups. Duke Math. J., 69(2):455–485, 1993. For a treatment of the coordinate function algebras at the $C^*$-algebra level, you can take a look at the following two papers:

1. Manabendra Giri and Arup Kumar Pal. Quantized function algebras at $q = 0$: type $A_{n}$ case. arXiv:2203.14665 [math.QA], 2022.
2. Marco Matassa and Robert Yuncken. Crystal limits of compact semisimple quantum groups as higher-rank graph algebras, 2022. arXiv:2208.13201 [math.QA], 2022.