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quantum groups


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1: 17.17 Physical Applications
See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys. Quantum groups also apply q -series extensively. Quantum groups are really not groups at all but certain Hopf algebras. They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. …
2: Tom H. Koornwinder
Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …
3: Bibliography V
  • N. Ja. Vilenkin and A. U. Klimyk (1992) Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.
  • 4: Bibliography K
  • C. Kassel (1995) Quantum Groups. Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York.
  • 5: Bibliography W
  • E. P. Wigner (1959) Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Pure and Applied Physics. Vol. 5, Academic Press, New York.
  • 6: 12.17 Physical Applications
    Miller (1974) treats separation of variables by group theoretic methods. Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. …
    7: Bibliography D
  • P. Dean (1966) The constrained quantum mechanical harmonic oscillator. Proc. Cambridge Philos. Soc. 62, pp. 277–286.
  • Delft Numerical Analysis Group (1973) On the computation of Mathieu functions. J. Engrg. Math. 7, pp. 39–61.
  • P. Deligne, P. Etingof, D. S. Freed, D. Kazhdan, J. W. Morgan, and D. R. Morrison (Eds.) (1999) Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2. American Mathematical Society, Providence, RI.
  • B. Dubrovin and M. Mazzocco (2000) Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141 (1), pp. 55–147.
  • 8: Bibliography G
  • GAP (website) The GAP Group, Centre for Interdisciplinary Research in Computational Algebra, University of St. Andrews, United Kingdom.
  • J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.
  • C. H. Greene, U. Fano, and G. Strinati (1979) General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.
  • W. Greiner, B. Müller, and J. Rafelski (1985) Quantum Electrodynamics of Strong Fields: With an Introduction into Modern Relativistic Quantum Mechanics. Texts and Monographs in Physics, Springer.
  • 9: Bibliography
  • J. V. Armitage (1989) The Riemann Hypothesis and the Hamiltonian of a Quantum Mechanical System. In Number Theory and Dynamical Systems (York, 1987), M. M. Dodson and J. A. G. Vickers (Eds.), London Math. Soc. Lecture Note Ser., Vol. 134, pp. 153–172.
  • V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups A k , D k , E k and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
  • R. A. Askey, T. H. Koornwinder, and W. Schempp (Eds.) (1984) Special Functions: Group Theoretical Aspects and Applications. D. Reidel Publishing Co., Dordrecht.
  • 10: Bibliography R
  • RISC Combinatorics Group (website) Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.
  • C. C. J. Roothaan and S. Lai (1997) Calculation of 3 n - j symbols by Labarthe’s method. International Journal of Quantum Chemistry 63 (1), pp. 57–64.