I am reading the book. I am trying to understand the natural map $U_{A} \otimes_A \mathbb{Q}(q) \to U_q$ and the algebra $U_{\epsilon}$ on page 288 (Section 9.2).
What is the natural map $U_{A} \otimes_A \mathbb{Q}(q) \to U_q$ and what are the elements in $U_{\epsilon}$? Thank you very much.
To construct the natural map, note that $U_q$ is a $\mathbb{Q}(q)$-module. Hence we can use this module structure to define $$ X\otimes q\mapsto q\cdot X, \forall X\in U_{\mathcal{A}}.$$ This extends to the required map. Note that the authors denote by $\mathbb{Q}(q)$ the field of rational functions in one variable over $\mathbb{Q}$, but by $\mathbb{Q}(\epsilon)$ the field obtained by adjoining a concrete complex number $\epsilon$ to $\mathbb{Q}$ within $\mathbb{C}$.