Geometry of the $p$-adic special orthogonal group $SO(3)_p$
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911759748562944 |
|---|---|
| author | Di Martino, Sara Mancini, Stefano Pigliapochi, Michele Svampa, Ilaria Winter, Andreas |
| author_facet | Di Martino, Sara Mancini, Stefano Pigliapochi, Michele Svampa, Ilaria Winter, Andreas |
| contents | We derive explicitly the structural properties of the $p$-adic special orthogonal groups in dimension three, for all primes $p$, and, along the way, the two-dimensional case. In particular, starting from the unique definite quadratic form in three dimensions (up to linear equivalence and rescaling), we show that every element of $SO(3)_p$ is a rotation around an axis. An important part of the analyis is the classification of all definite forms in two dimensions, yielding a description of the rotation subgroups around any fixed axis, which all turn out to be abelian and parametrised naturally by the projective line.
Furthermore, we find that for odd primes $p$, the entire group $SO(3)_p$ admits a representation in terms of Cardano angles of rotations around the reference axes, in close analogy to the real orthogonal case. However, this works only for certain orderings of the product of rotations around the coordinate axes, depending on the prime; furthermore, there is no general Euler angle decomposition. For $p=2$, no Euler or Cardano decomposition exists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_06228 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Geometry of the $p$-adic special orthogonal group $SO(3)_p$ Di Martino, Sara Mancini, Stefano Pigliapochi, Michele Svampa, Ilaria Winter, Andreas Number Theory Mathematical Physics Group Theory We derive explicitly the structural properties of the $p$-adic special orthogonal groups in dimension three, for all primes $p$, and, along the way, the two-dimensional case. In particular, starting from the unique definite quadratic form in three dimensions (up to linear equivalence and rescaling), we show that every element of $SO(3)_p$ is a rotation around an axis. An important part of the analyis is the classification of all definite forms in two dimensions, yielding a description of the rotation subgroups around any fixed axis, which all turn out to be abelian and parametrised naturally by the projective line. Furthermore, we find that for odd primes $p$, the entire group $SO(3)_p$ admits a representation in terms of Cardano angles of rotations around the reference axes, in close analogy to the real orthogonal case. However, this works only for certain orderings of the product of rotations around the coordinate axes, depending on the prime; furthermore, there is no general Euler angle decomposition. For $p=2$, no Euler or Cardano decomposition exists. |
| title | Geometry of the $p$-adic special orthogonal group $SO(3)_p$ |
| topic | Number Theory Mathematical Physics Group Theory |
| url | https://arxiv.org/abs/2104.06228 |