  We provide $\mathbb{N}$-filtrations on the negative part
$U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional
simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a
skew-polynomial algebra on $\mathfrak{n}^-$. The filtration is obtained by
assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees
can be described as lattice points in certain polyhedral cones. In the
classical limit, such a degree induces an $\mathbb{N}$-filtration on any finite
dimensional simple $\mathfrak{g}$-module. We prove for type $\tt{A}_n$,
$\tt{C}_n$, $\tt{B}_3$, $\tt{D}_4$ and $\tt{G}_2$ that a degree can be chosen
such that the associated graded modules are defined by monomial ideals, and
conjecture that this is true for any $\mathfrak{g}$.
