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A quantum algebra approach to discrete equations on uniform lattices

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Czechoslovak Journal of Physics Aims and scope

Abstract

A quantum algebra method for deducing the symmetries of discrete equations on uniform lattices is proposed. In principle, such a procedure can be applied to discretizations in a single coordinate (space or time) and the symmetries obtained in this way are indeed differential-difference operators. Firstly, the method is illustrated on two known examples that have been also analysed from the usual Lie symmetry approach: a uniform space lattice discretization of the (1+1) free heat-Schrödinger equation associated to a quantum Schrödinger algebra, and a discrete space (1+1) wave equation provided by a quantumso(2, 2) algebra. Furthermore, we construct a discrete space (2+1) wave equation from a new quantumso(3, 2) algebra, to show that this method is useful in higher dimensions. Time discretizations are also commented.

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References

  1. D. Levi and P. Winternitz: J. Math. Phys.37 (1996) 5551.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. D. Levi, L. Vinet, and P. Winternitz: J. Phys. A30 (1997) 633.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. R. Floreanini and L. Vinet: Lett. Math. Phys.32 (1994) 37.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Floreanini and L. Vinet: J. Math. Phys.36 (1995) 3134.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. V.K. Dobrev, H.D. Doebner, and C. Mrugalla: J. Phys. A29 (1996) 5909.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. R. Floreanini, J. Negro, L.M. Nieto, and L. Vinet: Lett. Math. Phys.36 (1996) 351.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Negro and L.M. Nieto: J. Phys. A29 (1996) 1107.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. C.R. Hagen: Phys. Rev. D5 (1972) 377.

    Article  ADS  Google Scholar 

  9. V. Chari and A. Pressley:A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1995.

    MATH  Google Scholar 

  10. A. Ballesteros, F.J. Herranz, and P. Parashar: J. Phys. A30 (1997) 8587.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. A. Ballesteros, F.J. Herranz, J. Negro, and L.M. Nieto: J. Phys. A33 (2000) 4859.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. F.J. Herranz: J. Phys. A33 (2000) 8217.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. F. Bonechi, E. Celeghini, R. Giachetti, E. Sorace, and M. Tarlini: Phys. Rev. Lett.68 (1992) 3718.

    Article  ADS  Google Scholar 

  14. F. Bonechi, E. Celeghini, R. Giachetti, E. Sorace, and M. Tarlini: Phys. Rev. B46 (1992) 5727.

    Article  ADS  Google Scholar 

  15. V.G. Drinfeld: Leningrad Math. J.1 (1990) 1419.

    MathSciNet  Google Scholar 

  16. A. Ballesteros, F.J. Herranz, and P. Parashar: Mod. Phys. Lett. A13 (1998) 1241.

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Física, Universidad de Burgos, E-09001, Burgos, Spain

    F. J. Herranz & A. Ballesteros

  2. Departamento de Física Teórica, Universidad de Valladolid, E-47011, Valladolid, Spain

    J. Negro & L. M. Nieto

Authors
  1. F. J. Herranz
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  2. A. Ballesteros
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  3. J. Negro
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  4. L. M. Nieto
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Additional information

This work was partially supported by the Ministerio de Ciencia y Tecnología, Spain (Project BFM2000-1055) and by Junta de Castilla y León, Spain (Projects CO1/399 and CO2/399).

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Herranz, F.J., Ballesteros, A., Negro, J. et al. A quantum algebra approach to discrete equations on uniform lattices. Czech J Phys 51, 321–330 (2001). https://doi.org/10.1023/A:1017537405763

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  • DOI: https://doi.org/10.1023/A:1017537405763

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