For any lie algebra $\mathfrak g$, there is a natural filtration on $U(\mathfrak g)$ by "degree": the filtered piece $U^{\leq n}(\mathfrak g)$ is just the image in $U(\mathfrak g)$ of $\bigoplus_{k=0}^n\mathfrak g^{\otimes k}$.

Is there a natural filtration of the quantum group $U_q(\mathfrak g)$ which reduces to this filtration on $U(\mathfrak g)$ when we set $q=1$?

(of course, we restrict to the lie algebras $\mathfrak g$ for which one has a definition of $U_q(\mathfrak g)$, e.g. $\mathfrak g=\mathfrak s\mathfrak l_n$).

To increase the likelihood of a positive answer to this question, let's use the model for $U_q(\mathfrak g)$ which is often denoted $U_h(\mathfrak g)$, that is, the model over the complete power series ring $\mathbb C[[h]]$.

One possible motivation for this question is the following. One definition of the universal enveloping algebra $U(\mathfrak g)$ is that it is a dual of the complete local ring of $G$ at the identity $e\in G$ (where $\operatorname{Lie}(G)=\mathfrak g$). The complete local ring $\hat{\mathcal O}_{G,e}$ comes with a canonical set of quotients $\mathcal O_{G,e}/\mathfrak m_{G,e}^{n+1}$, and corresponding to this is the filtration $U^{\leq n}(\mathfrak g)$ by "degree". Thus we can interpret $U^{\leq n}(\mathfrak g)$ as the part of $U(\mathfrak g)$ consisting of differential operators of order $n$ at $e\in G$. I want to know whether there is a similar filtration/interpretation for quantum groups.