Journal of Operator Theory
Volume 91, Issue 2, Spring 2024 pp. 443-470.
Quantum SL(2,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 Uq(sl(2,R)) quantizing the universal enveloping ∗-algebra of sl(2,R). We realize this ∗-algebra Uq(sl(2,R)) as a ∗-subalgebra of the Drinfeld double of Uq(su(2)) and its dual Hopf ∗-algebra Oq(SU(2)). More precisely, Uq(sl(2,R)) is generated by the coideal ∗-subalgebra Oq(K∖SU(2))⊆Oq(SU(2)) associated to the equatorial Podle\'s sphere, and by the associated orthogonal coideal ∗-subalgebra Uq(k)⊆Uq(su(2)). We then classify all the irreducible ∗-representations of Uq(sl(2,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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