Invariant measures on p-adic Lie groups : the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups
Aniello, Paolo (Istituto Nazionale di Fisica Nucleare Sezione di Napoli)
L'Innocente, Sonia (Università di Camerino)
Mancini, Stefano (Università di Camerino)
Parisi, Vicenzo (Istituto Nazionale di Fisica Nucleare Sezione di Perugia)
Svampa, Ilaria (Universitat Autònoma de Barcelona. Departament de Física)
Winter, Andreas (Universitat Autònoma de Barcelona. Departament de Física)

Data:	2024
Resum:	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 SO(2,Q) is obtained by a direct application of our general formula. As for SO(3,Q) and SO(4,Q), 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 Q 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.
Ajuts:	Agencia Estatal de Investigación PID2022-141283NB-I00 Agencia Estatal de Investigación PCI2022-132965
Nota:	Altres ajuts: acords transformatius de la UAB
Drets:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Locally compact group ; Haar measure ; P-adic Lie group ; Quaternion algebra
Publicat a:	Letters in mathematical physics, Vol. 114 (June 2024) , art. 78, ISSN 1573-0530

DOI: 10.1007/s11005-024-01826-8

59 p, 833.1 KB

El registre apareix a les col·leccions:
Articles > Articles de recerca
Articles > Articles publicats