| Title:
|
Quantized semisimple Lie groups (English) |
| Author:
|
Fioresi, Rita |
| Author:
|
Yuncken, Robert |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
60 |
| Issue:
|
5 |
| Year:
|
2024 |
| Pages:
|
311-349 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
quantum groups |
| Keyword:
|
representation theory |
| Keyword:
|
semisimple Lie algebras |
| MSC:
|
16T20 |
| MSC:
|
17B37 |
| MSC:
|
46L67 |
| DOI:
|
10.5817/AM2024-5-311 |
| . |
| Date available:
|
2024-12-13T18:47:55Z |
| Last updated:
|
2024-12-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152655 |
| . |
| Reference:
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