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

Abstract

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

22 pages; author accepted manuscript

References in corpus (2)

• Compact and discrete subgroups of algebraic quantum groups I
• Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik-Zamolodchikov equations and Letzter-Kolb coideals