Multiplicity formulas for representations of transformation groupoids

• Autorzy: Giżycki A. , Pysiak L.

Identyfikatory

DOI

10.1515/dema-2017-0004

• Języki publikacji: EN

Abstrakty

EN

We study the representations of transitive transformation groupoids with the aim of generalizing the Mackey theory. Using the Mackey theory and a bijective correspondence between the imprimitivity systems and the representations of a transformation groupoid we derive the irreducibility theory. Then we derive the direct sum decomposition for representations of a groupoid together with the formula for the multiplicity of subrepresentations. We discuss a physical interpretation of this formula. Finally, we prove the claim analogous to the Peter–Weyl theorem for a noncompact transformation groupoid. We show that the representation theory of a transitive transformation groupoids is closely related to the representation theory of a compact groups.

Słowa kluczowe

EN

groupoids; induced representations; imprimitivity systems

PL

grupoidy; reprezentacje indukowane

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2017
• Tom: Vol. 50, nr 1
• Strony: 42--50
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Giżycki A.

• Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland

autor

Pysiak L.

• Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland

Bibliografia

• [1] Westman J., Harmonic analysis on groupoids, Pacific J. Math., 1968, 27, 621-632
• [2] Boss R., Continuous representations of groupoids, 2007 [arXiv:math/0612639v3]
• [3] Buneci M. R., Groupoid C*-algebras, Surveys in Mathematics and its Applications, ISSN 1842-6298, 2006, 1, 71-98
• [4] Landsman K., Mathematical topics between classical and quantum mechanics, Springer, New York, 1998
• [5] Pysiak L., Groupoids their representations and imprimitivity systems, Demonstratio Mathematica, 2004, 37 (3), 661-670
• [6] Pysiak L., Imprimitivity theorem for groupoid representations, Demonstratio Mathematica, 2011, 44 (1), 29-48
• [7] Renault J. N., A groupoid approach to C*-algebras, Lecture Notes in Math. 793, Springer-Verlag, New York, 1980
• [8] Weinstein A., Groupoids: unifying internal and external geometry, Contemp. Math., 2001, 282, 1-19
• [9] Heller M., Pysiak L., Sasin W., Noncommutative unification of general relativity and quantum mechanics, J. Math. Phys., 2005, 46 (12), 122501
• [10] Heller M., Pysiak L., Sasin W., Noncommutative dynamics of random operators, Int. J. Theor. Phys., 2005, 44, 619-628
• [11] Heller M., Pysiak L., Sasin W., Conceptual unification of gravity and quanta, Int. J. Theor. Phys., 2007, 46, 2494-2512
• [12] Pysiak L., Time flow in a noncommutative regime, Internat. J. Theoret. Phys., 2007, 46 (1), 16-30
• [13] Riefel M., Unitary representations induced from compact subgroups, Studia Mathematica, 1972, 42, 145-175
• [14] Gilbert W., Modern algebra with applications, Wiley Interscience, 2003
• [15] Varadarajan V., Geometry of quantum theory, Springer, New York, 1998
• [16] Taylor M., Noncommutative harmonic analysis, Mathematical Surveys and Monographs, 1986, 22
• [17] Mackey G., Imprimitivity for representations of locally compact groups I, Proceedings National Academy of Sciences, 1949, 35, 537-545
• [18] Mackey G., The theory of unitary group representations, Chicago Lectures in Mathematics, Chicago, 1976
• [19] Dixmier J., Von Neumann algebras, North Holland Publ. Comp., Amsterdam, 1981
• [20] Amini M., Tannaka-Krein duality for compact groupoids I, Representation theory, 2003 [arXiv:math/0308259v1]
• [21] Mackey G., The relationship between classical mechanics and quantum mechanics, Contemporary Mathematics, 1998, 214, 91-109
• [22] Mackey G., Induced representations of locally compact groups I, Annals of Mathematics, 1952, 55, 101-139

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).