  We study infinite dimensional generalisations of the Heisenberg doubles of
the Borel half of $U_q(sl(2))$ and of $U_q(osp(1|2))$ and find associated
canonical elements which satisfy pentagon equation. The former reproduces the
canonical element, expressed using the Faddeev's quantum dilogarithm, which has
been found by Kashaev to be realised within quantised Teichm\"uller theory,
while for the latter we show that it corresponds to an operator from quantised
super Teichm\"uller theory. We study infinite dimensional representations of
those two Heisenberg doubles and, using an algebra homomorphism between
Heisenberg doubles and Drinfeld doubles, we find associated representations of
Drinfeld doubles of the Borel half of $U_q(sl(2))$ and of $U_q(osp(1|2))$.
Moreover, we reproduce the previously obtained $R$-matrix for the former and
derive a novel $R$-matrix for the latter representation.
