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2010, vol. 14, br. 1, str. 35-46
Some remarks on the notion of contraction of lie group representations
(naslov ne postoji na srpskom)
Département de Mathématiques, Metz, France

e-adresacahen@univ-metz.fr
Ključne reči: contractions; lie groups; unitary representations; sequences of Hilbert spaces
Sažetak
(ne postoji na srpskom)
In the series of papers [1-4], L. Barker developed a general notion of convergence for sequences of Hilbert spaces and related objects (vectors, operators...). In this paper, we remark that Barker's convergence for sequences of operators provides a notion of contraction of Lie group (unitary) representations and we compare it to the usual one introduced by J. Mickelsson and J. Niederle. This allows us to illustrate Barker's convergence of operators by various examples taken from contraction theory.
Reference
Barker, L. (2001) Continuum Quantum Systems as Limits of Discrete Quantum Systems, I: State Vectors. Journal of Functional Analysis, 186(1): 153-166
Barker, L. (2001) Continuum Quantum Systems as Limits of Discrete Quantum Systems, II: State Functions. Journal of Physics A: Mathematical and General, 34(22): 4673-4682
Barker, L. (2001) Continuum quantum systems as limits of discrete quantum systems: III: Operators. Journal of Mathematical Physics, 42(10): 4653
Barker, L. (2003) Continuum quantum systems as limits of discrete quantum systems. IV. Affine canonical transforms. Journal of Mathematical Physics, 44(4): 1535
Cahen, B. (2001) Quantification d'orbites coadjointes et th\'eorie des contractions. J. Lie Theory, 11(2): 257
Cahen, B. (2003) Contraction de SU(2) vers le groupe de Heisenberg et calcul de Berezin. Beitrage Algebra Geom, 44, 2 581-603
Cahen, B. (2004) Contraction de SU(1; 1) vers le groupe de Heisenberg. u: Mathematical works, Luxembourg: Universite-Seminaire de Mathematique, Part XV, 19-43
Cahen, B. (2009) Contraction of discrete series via Berezin quantization. J. Lie Theory, 19(2): 291
Cahen, B. (2009) Contraction of compact semisimple Lie groups via Berezin quantization. Illinois J. Math., 53(1): 265
Cahen, B. (2005) Contractions ofSU(1, n) andSU(n+1) via Berezin quantization. J. Anal. Math., 97(1): 83
Cattaneo, U., Wreszinski, W.F. (1999) Contractions of Lie algebra representations. Reviews In Mathematical Physics, 11(10): 1179
Cishahayo, C., de Bievre, S. (1993) On the contraction of the discrete series of SU(1; 1). Ann. Inst. Fourier, 43, 551-567
Cotton, P., Dooley, A.H. (1997) Contraction of an adapted functional calculus. J. Lie Theory, 7(2): 147
Dooley, A.H. (1983) Contractions of Lie groups and applications to analysis. u: Topics in Modern Harmonic Analysis, Proc. Semin., Torino and Milano, 1982, Rome: Ist. di Alta Mat, vol. I, str. 483-515
Dooley, A.H., Rice, J.W. (1983) Contractions of rotation groups and their representations. Mathematical Proceedings of the Cambridge Philosophical Society, 94(03): 509
Dooley, A.H., Rice, J.W. (1985) On contractions of semisimple Lie groups. Transactions of the American Mathematical Society, 289(1): 185-185
Fialowski, A., Penkava, M. (2008) Formal Deformations, Contractions and Moduli Spaces of Lie Algebras. International Journal of Theoretical Physics, 47(2): 561-582
Gromov, N.A. (2002) From Wigner-Inonu Group Contraction to Contractions of Algebraic Structures. u: Proceedings of the Wigner Centennial Conference, Paper No 07, Wigner Centennial Conference, Pecs, Hungary, 8-12 July
Inonu, E., Wigner, E.P. (1953) On the Contraction of Groups and Their Representations. Proceedings of the National Academy of Sciences of the United States of America, 39(6): 510-24
Kirillov, A.A. (2004) Lectures on the orbit method. u: Graduate Studies in Mathematics, Providence, Rhode Island: American Mathematical Society, vol. 64
Knapp, A.W. (1986) Representation Theory of Semisimple Groups: An Overview Based on Examples. Princeton, NJ: Princeton University Press
Mickelsson, J., Niederle, J. (1972) Contractions of representations of de Sitter groups. Communications in Mathematical Physics, 27(3): 167-180
Ricci, F. (1986) A contraction of ${\rm SU}(2)$ to the Heisenberg group. Monatsh. Math., 101(3): 211
Varadarajan, V.S. (1986) Lie groups, Lie algebras, and their representations. New York: Springer-Verlag, Graduate Texts in Maths 102
 

O članku

jezik rada: engleski
vrsta rada: neklasifikovan
DOI: 10.5937/MatMor1001035C
objavljen u SCIndeksu: 08.02.2011.

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