  We show that the stated skein algebra of a disc with two punctures on its
boundary and one inner puncture has a natural structure of Hopf algebra closely
related to a construction of Bigelow. Its image through the quantum trace map
is a Hopf algebra very similar to Uqsl2. We classify the weight indecomposable
modules of this quantum group and show that the change of coordinates in
quantum Teichm\"uller theory provides natural braiding operators between these
representations, including cyclic and semi-cyclic ones, and relate them to the
Kashaev-Reshetikhin braidings. For the Kashaev module, we show that this
braiding operator coincides with Drinfel'd R-matrix and deduce an alternative
proof of a theorem of Murakami and Murakami relating the Kashaev invariant to a
colored Jones polynomial. We eventually provide explicit formulas for generic
braiding operators depending on some quantum shape parameters familiar in
quantum hyperbolic geometry and explain how these braiding operators lead to
new links invariants.
