caa a22 4500 477076998 CHVBK 20180405111441.0 cr unu---uuuuu 170330e19961101xx s 000 0 eng 10.1007/BF00047124 doi (NATIONALLICENCE)springer-10.1007/BF00047124 Classical and quantum Heisenberg groups, their representations and applications [Elektronische Daten] [A. Klimyk, W. Schempp] This paper is a survey on classical Heisenberg groups and algebras, q-deformed Heisenberg algebras, q-oscillator algebras, their representations and applications. Describing them, we tried, for the reader's convenience, to explain where the q-deformed case is close to the classical one, and where there are principal differences. Different realizations of classical Heisenberg groups, their geometrical aspects, and their representations are given. Moreover, relations of Heisenberg groups to other linear groups are described. Intertwining operators for different (Schrödinger, Fock, compact) realizations of unitary irreducible representations of Heisenberg groups are given in explicit form. Classification of irreducible representations and representations of the q-oscillator algebra is derived for the cases when q is not a root of unity and when q is a root of unity. The Fock representation of the q-oscillator algebra is studied in detail. In particular, q-coherent states are described. Spectral properties of some operators of the Fock representations of q-oscillator algebras are given. Some of applications of Heisenberg groups and algebras, q-Heisenberg algebras and q-oscillator algebras are briefly described. Kluwer Academic Publishers, 1996 Heisenberg groups nationallicence Heisenberg algebras nationallicence q -deformed Heisenberg algebra nationallicence oscillator algebra nationallicence q -oscillator algebra nationallicence group representations nationallicence algebra representations nationallicence coherent states nationallicence Klimyk A. Institute for Theoretical Physics, Ukrainian Academy of Sciences, 252143, Kiev, Ukraine aut Schempp W. Lehrstuhl für Mathematik I, Universität Siegen, D-57068, Siegen, Germany aut Acta Applicandae Mathematica Kluwer Academic Publishers 45/2(1996-11-01), 143-194 0167-8019 45:2<143 1996 45 10440 https://doi.org/10.1007/BF00047124 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF00047124 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Klimyk A. Institute for Theoretical Physics, Ukrainian Academy of Sciences, 252143, Kiev, Ukraine aut NATIONALLICENCE 700 1- Schempp W. Lehrstuhl für Mathematik I, Universität Siegen, D-57068, Siegen, Germany aut NATIONALLICENCE 773 0- Acta Applicandae Mathematica Kluwer Academic Publishers 45/2(1996-11-01), 143-194 0167-8019 45:2<143 1996 45 10440 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer