Address:
1Fabit, via S. Donato 15, University of Bologna, 40126 Bologna, Italy
2Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, 40126 Bologna, Italy
3Université de Lorraine, CNRS, IECL, F-57000 Metz, France
E-mail: rita.fioresi@unibo.it
Abstract: 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 $\mathcal{U}_q(\mathfrak{sl}_2)$, $\mathcal{O}(\mathrm{SU}_q(2))$, $\mathcal{D}(\mathrm{SL}_q(2,\mathbb{C}))$ and related algebras. We treat quantized enveloping algebras, representations of $\mathcal{U}_q(\mathfrak{sl}_2)$, 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 $\mathrm{SL}_q(2,\mathbb{C})$, the Plancherel formula for $\mathrm{SL}_q(2,\mathbb{C})$. 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.
AMSclassification: primary 17B37; secondary 16T20, 46L67.
Keywords: quantum groups, representation theory, semisimple Lie algebras.
DOI: 10.5817/AM2024-5-311