Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups
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arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911925340733440 |
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| author | Aniello, Paolo L'Innocente, Sonia Mancini, Stefano Parisi, Vincenzo Svampa, Ilaria Winter, Andreas |
| author_facet | Aniello, Paolo L'Innocente, Sonia Mancini, Stefano Parisi, Vincenzo Svampa, Ilaria Winter, Andreas |
| contents | We provide a general expression of the Haar measure $-$ that is, the essentially unique translation-invariant measure $-$ on a $p$-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the $p$-adic special orthogonal groups in dimension two, three and four (for every prime number $p$). In particular, the Haar measure on $\mathrm{SO}(2,\mathbb{Q}_p)$ is obtained by a direct application of our general formula. As for $\mathrm{SO}(3,\mathbb{Q}_p)$ and $\mathrm{SO}(4,\mathbb{Q}_p)$, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain $p$-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field $\mathbb{Q}_p$ and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the $p$-adic special orthogonal groups, with potential applications in $p$-adic quantum mechanics and in the recently proposed $p$-adic quantum information theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07110 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups Aniello, Paolo L'Innocente, Sonia Mancini, Stefano Parisi, Vincenzo Svampa, Ilaria Winter, Andreas Mathematical Physics Functional Analysis Number Theory We provide a general expression of the Haar measure $-$ that is, the essentially unique translation-invariant measure $-$ on a $p$-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the $p$-adic special orthogonal groups in dimension two, three and four (for every prime number $p$). In particular, the Haar measure on $\mathrm{SO}(2,\mathbb{Q}_p)$ is obtained by a direct application of our general formula. As for $\mathrm{SO}(3,\mathbb{Q}_p)$ and $\mathrm{SO}(4,\mathbb{Q}_p)$, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain $p$-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field $\mathbb{Q}_p$ and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the $p$-adic special orthogonal groups, with potential applications in $p$-adic quantum mechanics and in the recently proposed $p$-adic quantum information theory. |
| title | Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups |
| topic | Mathematical Physics Functional Analysis Number Theory |
| url | https://arxiv.org/abs/2306.07110 |