Characterising the Haar measure on the $p$-adic rotation groups via inverse limits of measure spaces
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917699173482496 |
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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 determine the Haar measure on the compact $p$-adic special orthogonal groups of rotations $\mathrm{SO}(d)_p$ in dimension $d=2,3$, by exploiting the machinery of inverse limits of measure spaces, for every prime $p>2$. We characterise $\mathrm{SO}(d)_p$ as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each $\mathrm{SO}(d)_p$. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on $\mathrm{SO}(d)_p$. Our results pave the way towards the study of the irreducible projective unitary representations of the $p$-adic rotation groups, with potential applications to the recently proposed $p$-adic quantum information theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_14298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterising the Haar measure on the $p$-adic rotation groups via inverse limits of measure spaces Aniello, Paolo L'Innocente, Sonia Mancini, Stefano Parisi, Vincenzo Svampa, Ilaria Winter, Andreas Mathematical Physics Functional Analysis Group Theory Number Theory We determine the Haar measure on the compact $p$-adic special orthogonal groups of rotations $\mathrm{SO}(d)_p$ in dimension $d=2,3$, by exploiting the machinery of inverse limits of measure spaces, for every prime $p>2$. We characterise $\mathrm{SO}(d)_p$ as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each $\mathrm{SO}(d)_p$. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on $\mathrm{SO}(d)_p$. Our results pave the way towards the study of the irreducible projective unitary representations of the $p$-adic rotation groups, with potential applications to the recently proposed $p$-adic quantum information theory. |
| title | Characterising the Haar measure on the $p$-adic rotation groups via inverse limits of measure spaces |
| topic | Mathematical Physics Functional Analysis Group Theory Number Theory |
| url | https://arxiv.org/abs/2401.14298 |