  We study the tensor product decomposition of the split real quantum group
$U_{q\tilde{q}}(sl(2,R))$ from the perspective of finite dimensional
representation theory of compact quantum groups. It is known that the class of
positive representations of $U_{q\tilde{q}}(sl(2,R))$ is closed under taking
tensor product. In this paper, we show that one can derive the corresponding
Hilbert space decomposition, given explicitly by quantum dilogarithm
transformations, from the Clebsch-Gordan coefficients of the tensor product
decomposition of finite dimensional representations of the compact quantum
group $U_q(sl_2)$ by solving certain functional equations and using
normalization arising from tensor products of canonical basis. We propose a
general strategy to deal with the tensor product decomposition for the higher
rank split real quantum group $U_{q\tilde{q}}(g_R)$
