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

TL;DR

This paper constructs a quantum algebra related to $SL(2,\mathbb{R})$ and classifies all its irreducible $*$-representations, advancing the understanding of quantum groups associated with real Lie groups.

Contribution

It introduces a new quantum algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ and provides a complete classification of its irreducible $*$-representations.

Findings

Defined a unital $*$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ for real $q$

Realized the algebra as a subalgebra of the Drinfeld double of $U_q(\mathfrak{su}(2))$ and its dual

Classified all irreducible $*$-representations of the algebra

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