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.
References
[1] R. Brown, Groupoids as coefficients, Proc. London Math. Soc. 25 (1972), 413-426.10.1112/plms/s3-25.3.413Search in Google Scholar
[2] A. Cannas da Silva, A. Weinstein, Geometric Models for Noncommutative Algebras, American Mathematical Society, Berkeley, 1999.Search in Google Scholar
[3] M. Chaichian, M. Oksanen, A. Tureanu, G. Zet, Gauging the twisted Poincaré symmetry as a noncommutative theory of gravitation, Phys. Rev. D 79 (2009), 044014-24.10.1103/PhysRevD.79.044014Search in Google Scholar
[4] A. Connes, Noncommutative Geometry, Academic Press, New York, 1994.Search in Google Scholar
[5] C. Fronsdal, Elementary particles in a curved space, Rev. Modern Phys. 37 (1965), 221-224. [6] G. Goehle, Groupoid crossed products, PhD Thesis, 2009, Dartmouth College, arXiv: 0905.4681v110.1103/RevModPhys.37.221Search in Google Scholar
[7] S. Hollands, R. W. Wald, Axiomatic quantum field theory in curved spacetime, Comm.Search in Google Scholar
Math. Phys. 293 (2010), 85-125.10.1007/s00220-009-0880-7Search in Google Scholar
[8] L. P. Landsman, Mathematical Topics between Classical and Quantum Mechanics, Springer, New York, 1998.10.1007/978-1-4612-1680-3Search in Google Scholar
[9] K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Society Lecture Notes Series, 124, Cambridge University Press, Cambridge, 1987.10.1017/CBO9780511661839Search in Google Scholar
[10] A. L. T. Paterson, Groupoids, Inverse Semigroups and their Operator Algebras, Birkhäuser, Boston-Basel-Berlin, 1999.10.1007/978-1-4612-1774-9Search in Google Scholar
[11] L. Pysiak, Imprimitivity theorem for groupoid representations, Demonstratio Math. 44 (2011), 29-48.Search in Google Scholar
[12] R. W. Wald, General Relativity, The University of Chicago Press, Chicago-London, 1984. Search in Google Scholar
© 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.
