  We will introduce an $\mathbb{N}$-filtration on the negative part of a
quantum group of type $A_n$, such that the associated graded algebra is a
q-commutative polynomial algebra. This filtration is given in terms of the
representation theory of quivers, by realizing the quantum group as the Hall
algebra of a quiver. We show that the induced associated graded module of any
simple finite-dimensional module (of type 1) is isomorphic to a quotient of
this polynomial algebra by a monomial ideal, and we provide a monomial basis
for this associated graded module. This construction can be viewed as a quantum
analog of the classical PBW framework, and in fact, by considering the
classical limit, this basis is the monomial basis provided by Feigin,
Littelmann and the second author in the classical setup.
