Abstract
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.
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Ahiezer, N. I. and Glazman, I. M.: The Theory of Linear Operators in Hilbert Space, Ungar, New York, 1961.
Arik, M. and Coon, D. D.: Operator algebra of dual resonance models, J. Math. Phys. 16 (1975), 1776–1779.
Arik, M. and Coon, D. D.: Hilbert spaces of analytical functions and generalized coherent states, J. Math. Phys. 17 (1976), 524–527.
Askey, R.: Continuous q-Hermite polynomials when q>1, in: D. Stanton (ed.), q-Series and Partitions, Springer, New York, 1989, pp. 151–158.
Askey, R. and Ismail, M.: A generalization of ultraspherical polynomials, in P. Erdös (ed.), Studies in Pure Mathematics, Birkhäuser, Basel, 1983, pp. 55–78.
Berezanskii, Ju. M.: Expansions in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc., Providence, R. I., 1968.
Biedenharn, L. C.: The quantum group SU q(2) and a q-analogue of the boson operators, J. Phys. A 22 (1989), L873-L878.
Boas, R. P. Jr.: Summation formulas and band-limited signals, Tohoku Math. J. 24 (1972), 121–125.
Burban, I. M.: Two-parameter deformation of the oscillator algebra, Phys. Lett. B 319 (1993), 485–489.
Burban, I. M. and Klimyk, A. U.: On spectral properties of q-oscillator operators, Lett. Math. Phys. 29 (1993), 13–18.
Burban, I. M. and Klimyk, A. U.: p, q-Differentiation, p, q-integration and p, q-hypergeometric functions related to quantum groups, Integral Transforms and Special Functions 2 (1994), 15–36.
Chakrabarti, R. and Jagannathan, R.: A (p, q)-oscillator realization of two-parameter quantum algebras, J. Phys. A 24 (1991), L711-L718.
Coon, D. D., Yu, S., and Baker, M. M.: Operator formulation of a dual multiparticle theory with nonlinear trajectories, Phys. Rev. D 5 (1972), 1429–1433.
Damaskinsky, E. V. and Kulish, P. P.: Deformed oscillators and their applications, Zap. Nauchn. Sem. LOMI 189 (1991), 37–74.
Damaskinsky, E. V. and Kulish, P. P.: q-Hermite polynomials and q-oscillators, Zap. Nauchn. Sem. LOMI 199 (1992), 82–90.
Dieudonné, J. A.: La géometrie des groupes classiques (Troisieme edition), Springer, Berlin, 1971.
Dunkl, C. F.: Boundary value problem for harmonic functions on the Heisenberg group, Canad. J. Math. 38 (1986), 478–512.
Feinsilver, P.: Commutators, anti-commutators and Eulerian calculus, Rocky Mountain J. Math. 12 (1982), 171–183.
Feinsilver, P.: Discrete analogues of the Heisenberg-Weyl algebra, Monatsh. Math. 104 (1987), 98–108.
Floreanini, R. and Vinet, L.: q-Orthogonal polynomials and q-oscillator quantum group, Lett. Math. Phys. 22 (1991), 45–54.
Floreanini, R. and Vinet, L.: Automorphisms of the q-oscillator algebra and basic orthogonal polynomials, Preprint UdeM-LPN-TH141, Montreal, 1993.
Geller, G.: Fourier analysis on the Heisenberg group. I, J. Funct. Anal. 36 (1980), 205–254.
Geller, D. and Stein, E. M.: Singular convolution operators on the Heisenberg group, Bull. Amer. Math. Soc. 6 (1982), 99–103.
Greiner, P. C.: Spherical harmonics on the Heisenberg group, Canad. J. Math. 23 (1980), 383–396.
Greiner, P. C. and Koornwinder, T. H.: Variations on the Heisenberg spherical harmonics, Preprint ZW 186/83, Amsterdam, 1983.
Gould, M. and Biedenharn, L. C.: The pattern calculus for tensor operators in quantum groups, J. Math. Phys. 33 (1992), 3613–3635.
Hayashi, T., Q-analogue of Clifford and Weyl algebras. Spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990), 129–144.
Hebecker, A., Schreckenberg, S., Schwenk, J., Weich, W. and Wess, J.: Representations of a q-deformed Heisenberg algebra, Preprint MPI-Ph/93-45, Munich, 1993.
Howe, R.: On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc. 3 (1980), 821–843.
Kalnins, E. G., Manocha, H. L. and Miller, W. Jr.: Models of q-algebra representations: Tensor products of special unitary and oscillator algebras, J. Math. Phys. 33 (1992), 2865–2883.
Kalnins, E. G., Miller, W. Jr. and Mukherjee, S.: Models of q-algebra representations: Matrix elements of the q-oscillator algebra, J. Math. Phys. 34 (1993), 5333–5356.
Kirillov, A. A.: Elements of the Theory of Representations, Springer, Berlin, 1970.
Kulish, P. P.: Contraction of quantum algebras and q-oscillator, Teor. i Mat. Fiz. 86 (1991), 157–160.
Lion, G. and Vergne, M.: The Weil Representation, Maslov Index and Theta Series, Birkhäuser, Basel, 1980.
Macfarlane, A. J.: On q-analogues of the quantum harmonic oscillator and the quantum group SU(2), J. Phys. A 22 (1989), 4581–4586.
Miller, W.: Lie Theory and Special Functions, Academic Press, New York, 1968.
Ostrovsky, V. L. and Samoilenko, U. S.: Structure theorem for a pair of unbounded selfadjoint operators satisfying a quadratic relation, Adv. Soviet Math. 9 (1992), 131–149.
Ostrovsky, V. L. and Samoilenko, U. S.: On pair of selfadjoint operators, Seminar S. Lie 3(2) (1993), 185–218.
Pusz, W. and Woronowicz, S. L.: Twisted second quantization, Rep. Math. Phys. 27 (1989), 231–257.
Quesne, C.: Raising and lowering operators for u q (n), J. Phys. A 26 (1993), 357–372.
Quesne, C.: Two-parameter versus one-parameter quantum deformation of su(2), Phys. Lett. A 174 (1993), 19–24.
Ray, R. and Nelson C. A.: Preprint SUNY-BING 7/11/90, State University of New York, 1990.
Rideau, G.: On the representations of quantum oscillator algebra, Lett. Math. Phys. 24 (1992), 147–153.
Schempp, W.: Radar ambiguity functions, the Heisenberg group, and holomorphic theta series, Proc. Amer. Math. Soc. 92 (1984), 103–110.
Schempp, W.: Harmonic Analysis on the Heisenberg Nilpotent Lie Group, Wiley, New York, 1986.
Schempp, W.: Wavelet interference, Fourier transform magnetic resonance imaging, and temporally encoded synchronized neural networks (to appear).
Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory, Birkhäuser, Basel, 1990.
Schmüdgen, K.: Operator representations of the real twisted commutation relations, J. Math. Phys. 35 (1994), 3211–3229.
Van Daele, A.: A quantum deformation of the Heisenberg group, Preprint, Leuven, 1989.
Van der Jeugt, J.: The q-boson operator algebra and q-Hermite polynomials, Lett. Math. Phys. 24 (1992), 267–274.
Vilenkin, N. Ja. and Klimyk, A. U.: Representation of Lie Groups and Special Functions, Kluwer, Dordrecht, Vol. 1, 1992; Vol. 2, 1993.
Vilenkin, N. Ja. and Klimyk, A. U.: Representation of Lie Groups and Special Functions. Recent Advances, Kluwer, Dordrecht, 1995.
Weil, A.: Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143–211.
Wolf, K. B., Integral Transforms in Science and Engineering, Plenum Press, New York, 1979.
Zhedanov, A. S.: Nonlinear shifts of q-Bose operators and q-coherent states, J. Phys. A 24 (1991), L1129-L1132.
Zhedanov, A. S.: Q-rotations and other Q-transformations as unitary nonlinear automorphisms of quantum algebras, J. Math. Phys. 35 (1994), 3756–3764.
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Klimyk, A.U., Schempp, W. Classical and quantum Heisenberg groups, their representations and applications. Acta Appl Math 45, 143–194 (1996). https://doi.org/10.1007/BF00047124
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DOI: https://doi.org/10.1007/BF00047124