Quantum SL(2,R) and its irreducible representations

We define for real q a unital ∗-algebra Uq(sl(2,R)) quantizing the universal enveloping ∗-algebra of sl(2,R). The ∗-algebra Uq(sl(2,R)) is realized as a ∗-subalgebra of the Drinfeld double of Uq(su(2)) and its dual Hopf ∗-algebra Oq(SU(2)), generated by the equatorial Podleś sphere coideal ∗-subalgebra Oq(K∖SU(2)) of Oq(SU(2)) and its associated orthogonal coideal ∗-subalgebra Uq(k)⊆Uq(su(2)). We then classify all the irreducible ∗-representations of Uq(sl(2,R))