Semidirect product of groupoids and associated algebras

• Autorzy: Pysiak L. , Eckstein M. , Heller M. , Sasin W.
• Języki publikacji: EN

Abstrakty

EN

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 x 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 x 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.

Słowa kluczowe

EN

groupoid; semidirect product; crossed product algebra

PL

produkt iloczynów; skrzyżowane algebry produktowej

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2014
• Tom: Vol. 47, nr 2
• Strony: 289--299
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Pysiak L.

• Technical University of Warsaw, Koszykowa 75, 00-662 Warszawa, Poland
• Copernicus Center For Interdisciplinary Studies, ul. Sławkowska 17, 31-016 Kraków, Poland

autor

Eckstein M.

• Faculty of Mathematics and Computer Science, Jagellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
• Copernicus Center For Interdisciplinary Studies, ul. Sławkowska 17, 31-016 Kraków, Poland

autor

Heller M.

• Copernicus Center For Interdisciplinary Studies, ul. Sławkowska 17, 31-016 Kraków, Poland
• Vatican Observatory, V-00120 Vatican City State

autor

Sasin W.

• Technical University of Warsaw, Koszykowa 75, 00-662 Warszawa, Poland
• Copernicus Center For Interdisciplinary Studies, ul. Sławkowska 17, 31-016 Kraków, Poland

Bibliografia

• [1] R. Brown, Groupoids as coefficients, Proc. London Math. Soc. 25 (1972), 413–426.
• [2] A. Cannas da Silva, A. Weinstein, Geometric Models for Noncommutative Algebras, American Mathematical Society, Berkeley, 1999.
• [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.
• [4] A. Connes, Noncommutative Geometry, Academic Press, New York, 1994.
• [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: 0905.4681v1
• [7] S. Hollands, R. W. Wald, Axiomatic quantum field theory in curved spacetime, Comm. Math. Phys. 293 (2010), 85–125.
• [8] L. P. Landsman, Mathematical Topics between Classical and Quantum Mechanics, Springer, New York, 1998.
• [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] A. L. T. Paterson, Groupoids, Inverse Semigroups and their Operator Algebras, Birkhäuser, Boston–Basel–Berlin, 1999.
• [11] L. Pysiak, Imprimitivity theorem for groupoid representations, Demonstratio Math. 44 (2011), 29–48.
• [12] R. W. Wald, General Relativity, The University of Chicago Press, Chicago–London, 1984.