Foundations of Physics

, Volume 27, Issue 8, pp 1159–1177 | Cite as

Invariant Lie-admissible formulation of quantum deformations

  • Ruggero Maria Santilli
Part II. Invited Papers Dedicated to Lawrence Biedenharn


In this note we outline the history of q-deformations, indicate their physical shortcomings, suggest their apparent resolution via an invariant Lie-admissible formulation based on a new mathematics of genotopic type, and point out their expected physical significance.


Invariant Formulation Quantum Deformation Associative Product Problematic Aspect Nonassociative Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Albert,Trans. Am. Math. Soc. 64, 552 (1948).CrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Santilli,Nuovo Cimento 51, 570 (1967).MathSciNetGoogle Scholar
  3. 3.
    M. Santilli,Suppl. Nuovo Cimento 6, 1225 (1968).MathSciNetGoogle Scholar
  4. 4.
    M. Santilli,Meccanica 1 3 (1969).CrossRefGoogle Scholar
  5. 5.
    M. Santilli,Hadronic J. 1, 224 (5a), 574 (5b), and 1267 (5c) (1978);Phys. Rev. D 20, 555 (1979) (5d).Google Scholar
  6. 6.
    M. Santilli,Foundations of Theoretical Mechanics, Vol. II:Birkhoffian Generalization of Hamiltonian Mechanics (Springer, New York 1983).Google Scholar
  7. 7.
    C. Biedernharn,J. Phys. A 22, L873 (1989).CrossRefADSGoogle Scholar
  8. 8.
    J. Macfarlane,J. Phys. A 22, L4581 (1989).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Dobrev, inProceedings of the Second Wigner Symposium, Clausthal 1991 (World Scientific, Singapore, 1992). J. Lukierski, A. Novicki, H. Ruegg, and V. Tolstoy,Phys. Lett. B 264, 331 (1991). O. Ogivetski, W. B. Schmidke, J. Wess, and B. Zumino,Commun. Math. Phys. 50, 495 (1992). S. Giller, J. Kunz, P. Kosinky, M. MajewskiI, and P. Maslanka,Phys. Lett. B 286, 57 (1992).Google Scholar
  10. 10.
    J. Lukierski, A. Nowiski, and H. Ruegg,Phys. Lett. B 293, 344 (1992). J. Lukierski, H. Ruegg, and W. Rühl,Phys. Lett. B 313, 357 (1993). J. Lukierski and H. Ruegg,Phys. Lett. B 329, 189 (1994). S. Majid and H. Ruegg,Phys. Lett. B 334, 348 (1994).CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    L. Curtis, B. Fairlie, and Z. K. Zachos, eds.,Quantum Groups (World, Scientific, Singapore, 1991). Mo-Lin Ge and Bao Heng Zhao, eds.,Introduction to Quantum Groups and Integrable Massive Models of Quantum Field Theory (World Scientific, Singapore, 1991). Yu. F. Smirnov and R. M. Asherova, eds.,Proceedings of the Fifth Workshop Symmetry Methods in Physics (JINR, Dubna, Russia, 1992).Google Scholar
  12. 12.
    M. Santilli,Rendiconti Circolo Matematico Palermo, Suppl. 42, 7 (1996).Google Scholar
  13. 13.
    D. Schafer,An Introduction to Nonassociative Algebras (Academic Press, New York, 1966).zbMATHGoogle Scholar
  14. 14.
    M. Santilli, “Initiation of the representation theory of Lie adminssible algebras on bimodular Hilbert spaces,”Hadronic J. 3, 440 (1979).MathSciNetGoogle Scholar
  15. 15.
    S. Sourlas and G. T. Tsagas,Mathematical Foundations of the Lie-Santilli Theory (Ukraine Academy of Sciences, Kiev, 1993).zbMATHGoogle Scholar
  16. 16.
    Lôhmus, E. Paal, and L. Sorgsepp,Nonassociative Algebras in Physics (Hadronic Press, Palm Harbor, Florida, 1994).zbMATHGoogle Scholar
  17. 17.
    V. Kadeisvili,Santilli's Isotopies of contemporary Algebras, Geometries and Relativities (Hadronic Press, Florida, 1991), 2nd edn. (Ukraine Academy of Sciences, Kiev, 1997).Google Scholar
  18. 18.
    V. Kadeisvili,Math. Methods Appl. Sci. 19, 1349 (1996).CrossRefMathSciNetGoogle Scholar
  19. 19.
    M. Santilli, “Relativistic hadronic mechanics: Nonunitary axiom-preserving completion of relativistic quantum mechanics,”Found. Phys. 27 (5), 691 (1997).MathSciNetGoogle Scholar
  20. 20.
    M. Santilli,Elements of Hadronic Mechanics, Vols. I and II, 2nd edn. (Ukraine Academy of Sciences, Kiev, 1995).Google Scholar
  21. 21.
    M. Santilli,Algebras, Groups and Geometries 10, 273 (1993).MathSciNetGoogle Scholar
  22. 22.
    Ellis, N. E. Mavromatos, and D. V. Nanopoulos, inProceedings of the Erice Summer School, 31st Course: From Superstrings to the Origin of Space-Time (World Scientific, Singapore, 1996).Google Scholar
  23. 23.
    F. Lopez, inSymmetry Methods in Physics (Memorial Volume dedicated to Ya. S. Smorodinsky), A. N. Sissakian, G. S. Pogosyan, and S. I. Vinitsky, eds. (J.I.N.R., Dubna, Russia, 1994), p. 300;Hadronic J. 16, 429 (1993). Jannussis and D. Skaltzas,Ann. Fond. L. de Broglie 18, 1137 (1993). A. Jannussis, R. Mignani, and R. M. Santilli,Ann. Fond. L. de Broglie 18, 371 (1993). D. Schuch,Phys. Rev. A 55, 955 (1997). R. M. Santilli, “Problematic aspects of classical and quantum deformations,” preprint IBR-TH-97-S-037, submitted for publication.Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • Ruggero Maria Santilli
    • 1
  1. 1.Institute for Basic ResearchPalm Harbor

Personalised recommendations