88
K. K. Leelamma
4 Conclusion
Quantum algebras, being nonlinear extension of Lie algebras, are specially suited
for describing small perturbations in systems characterised by Lie symmetries. Their
use in physics became popular with the introduction of the q-deformed harmonic
oscillator as a tool for providing a boson realisation of the quantum algebra su q (2).
The q-oscillator algebra has been proved to be useful for the description of small
deviations in the behaviour of physical systems from that predicted by harmonic
approximation. Several examples for this have been mentioned. During the last three
decades, a great deal of work has been done in physics and mathematics, with different
approaches providing new view points leading to surprising possibilities such as qdeformed theories of gravitation.
References
1. L.D. Faddeev, Integrable Models in (1+1) Dimensional Quantum Field Theory, Les Houches
session XXXIX (1982), p. 563
2. L.D. Faddeev, E.K. Sklyanin, L.A. Takhtajan, Theor. Math. Phys. 40, 194 (1979)
3. L.D. Faddeev, L.A. Takhtajan, Russ. Math. Surv. 34(5), 11 (1979)
4. P.P. Kulish, N.Y. Reshetikhin, J. Soviet. Math. 23, 2435 (1983)
5. E.K. Sklyanin, Func. Anal. Appl. 16, 263 (1982)
6. E.K. Sklyanin, Func. Anal. Appl. 17, 263 (1982)
7. V.G. Drinfel’d, Sov. Math. Dokl. 32, 254 (1985)
8. V.G. Drinfel’d, Quantum groups, in Proceedings of the International Congress of Mathematicians, Berkeley (1986); Am. Math. Soc. 798 (1987)
9. V.G. Drinfel’d, J. Sov. Math. 41, 18 (1988)
10. M. Jimbo, Lett. Math. Phys. 10, 63 (1985)
11. M. Jimbo, Lett. Math. Phys. 11, 247 (1986)
12. Y.I. Manin, Quantum Groups and Non-commutative Geometry, Preprint Montreal Univ. CRM1561 (1988)
13. S.L. Woronowicz, Comm. Math. Phys. 111, 613 (1987)
14. S.L. Woronowicz, Invent. Math. 93, 35 (1988)
15. S.L. Woronowicz, Comm. Math. Phys. 122, 125 (1989)
16. L.D. Faddeev, N.Y. Reshetikhin, L.A. Takhtajan, Quantisation of Lie Groups and Lie Algebras,
LOMI Preprint E-14-87
17. K.K. Leelamma, V.C. Kuriakose, K. Babu Joseph, Int. J. Mod. Phys. B 7, 2697(1993)
18. V.C. Kuriakose, K.K. Leelamma, K. Babu Joseph, Pramana J. Phys. 39, 521 (1992)
19. K.K. Leelamma, Studies in Condensed Matter Physics Using q-Oscillator Algebra, chapter 5,
Ph.D. thesis submitted to CUSAT, Kochi (1997)
20. J. Wess, B. Zumino, Nucl. Phys. B Proc. Suppl. 18B, 302 (1991)
21. B. Zumino, Mod. Phys. Lett. A 6, 1225 (1991)
22. S. Majid, J. Class. Quantum Gravity 5, 1587 (1988)
23. F.M. Hoissen, J. Phys. A: Math. Gen. 25, 1703 (1992)
24. Y. Aref’eva, I.V. Volovich, Preprint CERN-TH 6137/91 (1991)
25. R. Chakrabarthi, R. Jagannathan, J. Phys. A 24, 5683 (1991)
26. J. Wess, B. Zumino, Nucl. Phys. B (Proc. Suppl.) 18B 302 (1990)
27. E. Heine, Handbuch der Kugelfunktionen (Reamer, Berlin, 1878) (reprinted by Physica-Verlag,
Wurzburg, vol. 1, 1961)
K. K. Leelamma
4 Conclusion
Quantum algebras, being nonlinear extension of Lie algebras, are specially suited
for describing small perturbations in systems characterised by Lie symmetries. Their
use in physics became popular with the introduction of the q-deformed harmonic
oscillator as a tool for providing a boson realisation of the quantum algebra su q (2).
The q-oscillator algebra has been proved to be useful for the description of small
deviations in the behaviour of physical systems from that predicted by harmonic
approximation. Several examples for this have been mentioned. During the last three
decades, a great deal of work has been done in physics and mathematics, with different
approaches providing new view points leading to surprising possibilities such as qdeformed theories of gravitation.
References
1. L.D. Faddeev, Integrable Models in (1+1) Dimensional Quantum Field Theory, Les Houches
session XXXIX (1982), p. 563
2. L.D. Faddeev, E.K. Sklyanin, L.A. Takhtajan, Theor. Math. Phys. 40, 194 (1979)
3. L.D. Faddeev, L.A. Takhtajan, Russ. Math. Surv. 34(5), 11 (1979)
4. P.P. Kulish, N.Y. Reshetikhin, J. Soviet. Math. 23, 2435 (1983)
5. E.K. Sklyanin, Func. Anal. Appl. 16, 263 (1982)
6. E.K. Sklyanin, Func. Anal. Appl. 17, 263 (1982)
7. V.G. Drinfel’d, Sov. Math. Dokl. 32, 254 (1985)
8. V.G. Drinfel’d, Quantum groups, in Proceedings of the International Congress of Mathematicians, Berkeley (1986); Am. Math. Soc. 798 (1987)
9. V.G. Drinfel’d, J. Sov. Math. 41, 18 (1988)
10. M. Jimbo, Lett. Math. Phys. 10, 63 (1985)
11. M. Jimbo, Lett. Math. Phys. 11, 247 (1986)
12. Y.I. Manin, Quantum Groups and Non-commutative Geometry, Preprint Montreal Univ. CRM1561 (1988)
13. S.L. Woronowicz, Comm. Math. Phys. 111, 613 (1987)
14. S.L. Woronowicz, Invent. Math. 93, 35 (1988)
15. S.L. Woronowicz, Comm. Math. Phys. 122, 125 (1989)
16. L.D. Faddeev, N.Y. Reshetikhin, L.A. Takhtajan, Quantisation of Lie Groups and Lie Algebras,
LOMI Preprint E-14-87
17. K.K. Leelamma, V.C. Kuriakose, K. Babu Joseph, Int. J. Mod. Phys. B 7, 2697(1993)
18. V.C. Kuriakose, K.K. Leelamma, K. Babu Joseph, Pramana J. Phys. 39, 521 (1992)
19. K.K. Leelamma, Studies in Condensed Matter Physics Using q-Oscillator Algebra, chapter 5,
Ph.D. thesis submitted to CUSAT, Kochi (1997)
20. J. Wess, B. Zumino, Nucl. Phys. B Proc. Suppl. 18B, 302 (1991)
21. B. Zumino, Mod. Phys. Lett. A 6, 1225 (1991)
22. S. Majid, J. Class. Quantum Gravity 5, 1587 (1988)
23. F.M. Hoissen, J. Phys. A: Math. Gen. 25, 1703 (1992)
24. Y. Aref’eva, I.V. Volovich, Preprint CERN-TH 6137/91 (1991)
25. R. Chakrabarthi, R. Jagannathan, J. Phys. A 24, 5683 (1991)
26. J. Wess, B. Zumino, Nucl. Phys. B (Proc. Suppl.) 18B 302 (1990)
27. E. Heine, Handbuch der Kugelfunktionen (Reamer, Berlin, 1878) (reprinted by Physica-Verlag,
Wurzburg, vol. 1, 1961)