Abstract
All irreducible* -representations of the involutive algebraA generated by the so-called Heisenberg relations for the quantum deformation of the group of motions of the Euclidean plane are described.
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Yu. M. Berezanskii, V. L. Ostrovskii, and Yu. S. Samoilenko, “Expansion of families of commuting operators in eigenfunctions and representations of commutation relations,”Ukr. Mat. Zh.,40, No. 1, 106–109 (1988).
Yu. M. Berezanskii and Yu. G. Kondrat'ev,Spectral Methods in Infinite-Dimensional Analysis [in Russian], Naukova Dumka, Kiev (1988).English translation: Kluwer AP, Dordrechtf. (1995)
V. L. Ostrovskii and Yu. S. Samoilenko, “Unbounded operators satisfying non-Lie commutation relations,”Repts. Math. Phys.,28, No. 1, 91–103 (1989).
V. L. Ostrovskii and Yu. S. Samoilenko, “On pairs of self-adjoint operators,”Seminar Sophus Lie,3, No. 2, 185–218 (1993).
E. E. Vaisleb and Yu. S. Samoilenko, “Representations of operator relations by unbounded operators and multi-dimensional dynamical systems,”Ukr. Math. Zh.,42, No. 9, 1011–1019 (1990).
V. L. Ostrovskii and L. B. Turovskaya, “Representations of* -algebras and multi-dimensional dynamical systems,”Ukr. Math. Zh.,47, No. 4, 488–497 (1995).
G. W. MacKey, “Induced representations of locally compact groups,”Ann. Math.,55, No. 1, 101–139 (1952).
S. L. Woronowicz, “Unbounded elements affiliated withC *-algebras and non-compact quantum groups,”Commun. Math. Phys.,136, 399–432 (1991).
K. Schmüdgen, “Integrable operator representations of ℝ 2q andSL q (2R),”Seminar Sophus Lie,2, 107–114 (1992).
L. L. Vaksman and L. I. Korogodskii, “Algebra of bounded functions on the quantum group of motions of the plane andq-analog of Bessel functions,”Dokl. Akad. Nauk SSSR,304, No. 5, 1036–1040 (1989).
A. M. Gavrilik and A. U. Klimyk, “q-Deformed polynomials and pseudo-orthogonal algebras and their representations,”Lett. Math. Phys.,21, 215–220 (1990).
P. Podleś, “Quantum spheres,”Lett. Math. Phys.,14, 193–202 (1987).
W. Pusz and S. L. Woronowicz, “Twisted second quantization,”Repts. Math. Phys.,27, 231–257 (1989).
S. Klimek and A. Lesniewski, “Quantum Riemann surfaces. I. The unit disc,”Commun. Math. Phys.,146, 103–122 (1992).
S. L. Woronowicz, “QuantumE(2) group and its Pontryagin dual,”Lett. Math. Phys.,23, 251–263 (1991).
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Dedicated to the 70th birthday of our teacher, Prof. Yu. M. Berezanskii
Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 689–692, May, 1995.
This research was partially supported by the Ukrainian State Committee on Science and Technology, the International Science Foundation, grant No. U6D000, and a grant from the American Mathematical Society.
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Ostrovs'kyi, V.L., Samoilenko, Y.S. On representations of the Heisenberg relations for the quantumE(2) group. Ukr Math J 47, 793–797 (1995). https://doi.org/10.1007/BF01059052
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DOI: https://doi.org/10.1007/BF01059052