Abstract
We present a family of unitary representations of a group of diffeomorphisms of a finite-dimensional real Euclidean space using a family of quasi-invariant measures. In the one-dimensional case, for a special kind of group diffeomorphisms of the halfline, we prove the irreducibility of the representations thus obtained.
Similar content being viewed by others
References
E. D. Romanov, Family of Measures on a Space of Curves that are Quasi-Invariant with Respect to Some Action of Diffeomorphisms Group IDAQP, 19 (3), (2016) (to appear).
R. S. Ismagilov, “On Unitary Representations of the Group of Diffeomorphisms of a CompactManifold,” Izv. Akad. Nauk SSSR Ser. Mat. 36 (1), 180–208 (1972) [Math USSR-Izv. 6 (1), 181–209 (1972) (1973)].
R. S. Ismagilov, “On Unitary Representations of the Group of Diffeomorphisms of the Space Rn, n ≥ 2,” Mat. Sb. (N.S.) 98 (140):1(9), 55–71 (1975) [Math USSR-Sb, 27 (1), 51-65 (1975)].
A. M. Vershik, I. M. Gel’fand, and M. I. Graev, “Representations of the Group of Diffeomorphisms,” Uspekhi Mat. Nauk 30 (6), 3–50 (1975) [in: Representation Theory, London Mathematical Society Lecture Note Series, 69, Cambridge University Press, Cambridge–New York, 1982, pp. 61–110].
E. T. Shavgulidze, “Some Properties of Quasi-Invariant Measures on Groups of Diffeomorphisms of the Circle,” Russ. J. Math. Phys. 7 (4), 464–472 (2000).
E. T. Shavgulidze, “Properties of the Convolution Operation for Quasi-Invariant Measures on Groups of Diffeomorphisms of a Circle,” Russ. J. Math. Phys. 8 (4), 495–498 (2001).
A. A. Dosovitskii, “Quasi-Invariant Measures on Sets of Piecewise Smooth Homeomorphisms of Closed Intervals and Circles and Representations of Diffeomorphism Groups,” Russ. J. Math. Phys. 18 (3), 258–296 (2011).
Yu. A. Neretin, “Holomorphic Continuations of Representations of the Group of the Diffeomorphisms of the Circle,” Mat. Sb. 180 (5), 635–657 (1989) [Math. USSR-Sb. 67 (1), 75–97 (1990)].
Yu. A. Neretin, “Some Remarks on Quasi-Invariant Actions of Loop Groups and the Group of Diffeomorphisms of the Circle,” Communications in Mathematical Physics 164 (3), 599–626 (1994).
Yu. A. Neretin, Categories of Symmetries and Infinite-Dimensional Groups (The Clarendon Press, Oxford University Press, New York, 1996).
M. Libine, IntroductiontoRepresentationsofRealSemisimpleLieGroups (arXiv:1212.2578v2 [math.RT]).
R. H. Cameron and R. E. Graves, “Additive Functionals on a Space of Continuous Functions,” I. Trans. Amer. Math. Soc. 70, 160–176 (1951).
S. Bernshtein, “Sur la loi des grands nombres,” Communications de la Société mathématique de Kharkow. 2-me série 16 (1-2), 82–87 (1918).
V. V. Kozlov, T. Madsen, and A. A. Sorokin, “OnWeighted Mean Values ofWeakly Dependent Random Variables,” Moscow Univ. Math. Bull. 59 (5), 36–39 (2004).
M. Reed and B. Simon, Functional Analysis, Methods of Mathematical Physics, Vol. I (Academic Press, New York, 1972).
B. Oksendal, Stochastic Differentional Equations (5th Edition, 1998, Springer-Verlag–Heidelberg–New York).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Romanov, E.D. A series of irreducible unitary representations of a group of diffeomorphisms of the half-line. Russ. J. Math. Phys. 23, 369–381 (2016). https://doi.org/10.1134/S1061920816030079
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920816030079