The smallest quantum Mackey deformation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914354104893440 |
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| author | Gaudillot-Estrada, Yvann |
| author_facet | Gaudillot-Estrada, Yvann |
| contents | When $G$ is a real semisimple group, there is a surprising interplay between its representation theory and that of its motion group $G_0$, known as the Mackey analogy. The present paper extends this analogy to the framework of $q$-deformations, for $G = \mathrm{SL}(2,\mathbb{R})$. In fact, we construct a deformation of $\mathrm{SL}(2,\mathbb{R})$ parametrized by $(q,t) \in \mathbb{R}_+^* \times \mathbb{R}$, where $q$ is the quantization parameter and $t$ is the Mackey parameter. We show how the representation theory varies along this deformation and we prove an analogue of the Connes-Kasparov isomorphism for the $q$-deformed reduced group C*-algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_23222 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The smallest quantum Mackey deformation Gaudillot-Estrada, Yvann Operator Algebras 46L65, 46L67 (Primary) 46L80, 22E46 (Secondary) When $G$ is a real semisimple group, there is a surprising interplay between its representation theory and that of its motion group $G_0$, known as the Mackey analogy. The present paper extends this analogy to the framework of $q$-deformations, for $G = \mathrm{SL}(2,\mathbb{R})$. In fact, we construct a deformation of $\mathrm{SL}(2,\mathbb{R})$ parametrized by $(q,t) \in \mathbb{R}_+^* \times \mathbb{R}$, where $q$ is the quantization parameter and $t$ is the Mackey parameter. We show how the representation theory varies along this deformation and we prove an analogue of the Connes-Kasparov isomorphism for the $q$-deformed reduced group C*-algebra. |
| title | The smallest quantum Mackey deformation |
| topic | Operator Algebras 46L65, 46L67 (Primary) 46L80, 22E46 (Secondary) |
| url | https://arxiv.org/abs/2602.23222 |