Quantized semisimple Lie groups

These notes present a quick introduction to the q-deformations of semisimple Lie groups from the point of view of unitary representation theory. In order to remain concrete, we concentrate entirely on the case of the lie algebra $\mathrm{s}\mathrm{l}(2,\mathbb{C})$ and its associated compact and complex semisimple Lie groups $\mathrm{S}\mathrm{U}(2)$ and $\mathrm{S}\mathrm{L}(2,\mathbb{C})$. We treat the following topics: The quantized enveloping algebra and its representations; Hopf algebras and the various notions of quantum groups; real structures; quantized algebras of functions on a compact semisimple group; quantized convolution algebras; the Peter-Weyl theorem; quantized complex semisimple Lie groups as quantum doubles; representations of quantized complex semisimple Lie groups; the quantum analogue of Harish-Chandra’s Plancherel formula.