Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups
| Autor: | Aniello, Paolo, L'Innocente, Sonia, Mancini, Stefano, Parisi, Vincenzo, Svampa, Ilaria, Winter, Andreas |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Lett. Math. Phys. 114, 78 (2024) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s11005-024-01826-8 |
| Popis: | 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. Comment: 49 pages, minor changes |
| Databáze: | arXiv |
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