Journal of Operator Theory

Volume 91, Issue 2, Spring 2024 pp. 443-470.

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

Authors: Kenny De Commer (1), Joel Right Dzokou Talla (2)
Author institution: (1) Vakgroep Wiskunde en Data Science, Vrije Universiteit Brussel, Brussels, 1050, Belgium
(2) Vakgroep Wiskunde en Data Science, Vrije Universiteit Brussel, Brussels, 1050, Belgium

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

DOI: http://dx.doi.org/10.7900/jot.2022jun05.2380
Keywords: quantum groups, coideals, Drinfeld double, irreducible representations, Podle\'{s} spheres

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