Quantized semisimple Lie groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910382858174464 |
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| author | Fioresi, Rita Yuncken, Robert |
| author_facet | Fioresi, Rita Yuncken, Robert |
| contents | 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 $\mathrm{sl}(2,\mathbb{C})$ and its associated compact and complex semisimple Lie groups $\mathrm{SU}(2)$ and $\mathrm{SL}(2,\mathbb{C})$.
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17180 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantized semisimple Lie groups Fioresi, Rita Yuncken, Robert Quantum Algebra Operator Algebras Representation Theory 17B37, 20G42, 46L67, 16T05 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 $\mathrm{sl}(2,\mathbb{C})$ and its associated compact and complex semisimple Lie groups $\mathrm{SU}(2)$ and $\mathrm{SL}(2,\mathbb{C})$. 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. |
| title | Quantized semisimple Lie groups |
| topic | Quantum Algebra Operator Algebras Representation Theory 17B37, 20G42, 46L67, 16T05 |
| url | https://arxiv.org/abs/2403.17180 |