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 Podleś 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 $U_q(\mathfrak{sl}(2,\mathbb{R}))$.

Figures (3)

• Figure 1: Case $a=0$
• Figure 2: Case $a$ half integer
• Figure 4: Case of classical $SL(2,\mathbb{R})$ (parametrisation by values of $\Omega$)

Theorems & Definitions (52)

• Definition 1.1
• Lemma 1.2
• proof
• Definition 1.3
• Lemma 1.4
• proof
• Theorem 2.1
• proof
• Remark 2.2
• Definition 2.3