Heisenberg double and Drinfeld double of the quantum superplane

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.