Quantum $SL(2,\mathbb{R})$ and its irreducible representations

Abstract: We define for real $q$ a unital $$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ quantizing the universal enveloping $$-algebra of $\mathfrak{sl}(2,\mathbb{R})$. The $$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ is realized as a $$-subalgebra of the Drinfeld double of $U_q(\mathfrak{su}(2))$ and its dual Hopf $$-algebra $\mathcal{O}_q(SU(2))$, generated by the equatorial Podle\'s sphere coideal $$-subalgebra $\mathcal{O}_q(K\backslash SU(2))$ of $\mathcal{O}_q(SU(2))$ and its associated orthogonal coideal $$-subalgebra $U_q(\mathfrak{k}) \subseteq U_q(\mathfrak{su}(2))$. We then classify all the irreducible $$-representations of $U_q(\mathfrak{sl}(2,\mathbb{R}))$.