English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We determine the Haar measure on the compact $p$-adic special orthogonalgroups of rotations $\mathrm{SO}(d)_p$ in dimension $d=2,3$, by exploiting themachinery of inverse limits of measure spaces, for every prime $p>2$. Wecharacterise $\mathrm{SO}(d)_p$ as inverse limits of finite groups, of which weprovide parametrisations and orders, together with an equivalent descriptionthrough a multivariable Hensel lifting. Supplying these finite groups withtheir normalised counting measures, we get an inverse family of Haar measurespaces for each $\mathrm{SO}(d)_p$. Finally, we constructively prove theexistence 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 theirreducible projective unitary representations of the $p$-adic rotation groups,with potential applications to the recently proposed $p$-adic quantuminformation theory.

Characterising the Haar measure on the $p$-adic rotation groups via inverse limits of measure spaces