A filtration on Drinfeld-Jimbo quantum enveloping algebras

Question

For the universal enveloping algebra $U(\frak{g})$ of a Lie algebra $\frak{g}$, one can define in a natural way an increasing $\mathbb{N}_{0}$-filtration. By the Poincaré-Birkhoff–Witt theorem, the associated graded algebra is the polynomial algebra. Can we define a similar filtration on the Drinfeld-Jimbo quantized enveloping algebra? If we can, then what is the associated graded algebra?

Answer 1

This question is answered here in Section 9. It states that the associated graded ring is a skew polynomial ring.

The skew polynomial ring to which the theorem refers is generated by the images of the $E_{\alpha}$ (respectively, $F_{\alpha}$), with relations $E_{\alpha}E_{\beta} = q_{\alpha \beta} E_{\beta}E_{\alpha}$ (respectively, $F_{\alpha}F_{\beta} = q_{\alpha \beta} F_{\beta}F_{\alpha}$) where each $q_{\alpha \beta}$ is a scalar determined by $q$ and by the positive roots $\alpha, \beta \in \Delta^+$.