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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(KSU(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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