SEMIDIRECT PRODUCT OF GROUPOIDS AND ASSOCIATED ALGEBRAS

One of the pressing problems in mathematical physics is to find a generalized Poincare symmetry that could be applied to nonflat space-times. As a step in this direction, we define the semidirect product of groupoids Gamma(0) x Gamma(1) and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid 1 and prove that it is isomorphic to the convolutive algebra of the groupoid Gamma(0) x 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 Poincare groupoid defined as the semidirect product of the subgroupoid of generalized Lorentz transformations and the subgroupoid of generalized translations.