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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access June 6, 2014

Semidirect Product of Groupoids and Associated Algebras

Leszek Pysiak EMAIL logo , Michał Eckstein , Michał Heller and Wiesław Sasin
From the journal Demonstratio Mathematica

Abstract

One of the pressing problems in mathematical physics is to find a generalized Poincaré symmetry that could be applied to nonflat space-times. As a step in this direction, we define the semidirect product of groupoids Γ0 ⋊ Γ1 and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid Γ1 and prove that it is isomorphic to the convolutive algebra of the groupoid Γ0 ⋊ Γ1. We show that families of unitary representations of semidirect product groupoids in a bundle of Hilbert spaces are random operators. An important example is the Poincaré groupoid defined as the semidirect product of the subgroupoid of generalized Lorentz transformations and the subgroupoid of generalized translations.

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Received: 2012-8-30
Revised: 2013-2-11
Published Online: 2014-6-6
Published in Print: 2014-6-1

© by Leszek Pysiak

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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