  The discussions in the present paper arise from exploring intrinsically the
structure nature of the quantum $n$-space. A kind of braided category $\Cal
{GB}$ of $\La$-graded $\th$-commutative associative algebras over a field $k$
is established. The quantum divided power algebra over $k$ related to the
quantum $n$-space is introduced and described as a braided Hopf algebra in
$\Cal {GB}$ (in terms of its 2-cocycle structure), over which the so called
special $q$-derivatives are defined so that several new interesting quantum
groups, especially, the quantized polynomial algebra in $n$ variables (as the
quantized universal enveloping algebra of the abelian Lie algebra of dimension
$n$), and the quantum group associated to the quantum $n$-space, are derived
from our approach independently of using the $R$-matrix. As a verification of
its validity of our discussion, the quantum divided power algebra is equipped
with a structure of $U_q(\frak {sl}_n)$-module algebra via a certain
$q$-differential operators realization. Particularly, one of the four kinds of
roots vectors of $U_q(\frak {sl}_n)$ in the sense of Lusztig can be specified
precisely under the realization.
