TY - JOUR

T1 - Characterising the Haar measure on the p-adic rotation groups via inverse limits of measure spaces

AU - Aniello, Paolo

AU - L'Innocente, Sonia

AU - Mancini, Stefano

AU - Parisi, Vincenzo

AU - Svampa, Ilaria

AU - Winter, Andreas

N1 - Publisher Copyright:
© 2024 Elsevier GmbH

PY - 2024

Y1 - 2024

N2 - We determine the Haar measure on the compact p-adic special orthogonal groups of rotations 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 the groups 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 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 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.

AB - We determine the Haar measure on the compact p-adic special orthogonal groups of rotations 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 the groups 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 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 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.

KW - Haar measure

KW - Inverse/projective limit

KW - p-adic rotation group

KW - Profinite group

UR - http://www.scopus.com/inward/record.url?scp=85200266054&partnerID=8YFLogxK

U2 - 10.1016/j.exmath.2024.125592

DO - 10.1016/j.exmath.2024.125592

M3 - Article

AN - SCOPUS:85200266054

SN - 0723-0869

JO - Expositiones Mathematicae

JF - Expositiones Mathematicae

M1 - 125592

ER -