  Let ${U}_q(sl_2)$ be the quantized enveloping algebra associated to the
simple Lie algebra $sl_2$. In this paper, we study the quantum double $D_q$ of
the Borel subalgebra ${U}_q((sl_2)^{\leq 0})$ of ${U}_q(sl_2)$. We construct an
analogue of Kostant--Lusztig ${Z}[v,v^{-1}]$-form for $D_q$ and show that it is
a Hopf subalgebra. We prove that, over an algebraically closed field, every
simple $D_q$-module is the pullback of a simple ${U}_q(sl_2)$-module through
certain surjection from $D_q$ onto ${U}_q(sl_2)$, and the category of finite
dimensional weight $D_q$-modules is equivalent to a direct sum of
$|k^{\times}|$ copies of the category of finite dimensional weight
${U}_q(sl_2)$-modules. As an application, we recover (in a conceptual way)
Chen's results as well as Radford's results on the quantum double of Taft
algebra. Our main results allow a direct generalization to the quantum double
of the Borel subalgebra of the quantized enveloping algebra associated to
arbitrary Cartan matrix.
