Semidirect Product of Groupoids, Its Representations and Random Operators

TL;DR

This paper introduces the semidirect product of groupoids and explores their representations as random operators, aiming to generalize Poincaré symmetry for nonflat space-times in mathematical physics.

Contribution

It defines the semidirect product of groupoids, studies its properties, and links its representations to random operators, with applications to generalized Poincaré symmetry.

Findings

Semidirect product of groupoids is well-defined and analyzed.

Crossed product of algebra bundles is isomorphic to convolution algebra.

Representations of these groupoids are shown to be random operators.

Abstract

One of pressing problems in mathematical physics is to find a generalized Poincar\'e symmetry that could be applied to nonflat space-times. As a step in this direction we define the semidirect product of groupoids $\Gamma_0 \rtimes \Gamma_1$ and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid $\Gamma_1$ and prove that it is isomorphic to the convolutive algebra of the groupoid $\Gamma_0 \rtimes \Gamma_1$. 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 Poincar\'e groupoid defined as the semidirect product of the subgroupoid of generalized Lorentz transformations and the subgroupoid of generalized translations.