QUANTIZED SEMISIMPLE LIE GROUPS

The goal of this expository paper is to give a quick introduction to q-deformations of semisimple Lie groups. We discuss principally the rank one examples of Uq(sl2), O(SUq(2)), D(SLq(2, ℂ)) and related algebras. We treat quantized enveloping algebras, representations of Uq(sl2), generalities on Hopf algebras and quantum groups, ∗-structures, quantized algebras of functions on q-deformed compact semisimple groups, the Peter-Weyl theorem, ∗-Hopf algebras associated to complex semisimple Lie groups and the Drinfeld double, representations of SLq(2, ℂ), the Plancherel formula for SLq(2, ℂ). This exposition is expanding the material treated in a series of lectures given by the second author at the CaLISTA CA 21100 Training School, “Quantum Groups and Noncommutative Geometry in Prague” in 2023.