Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 015, 22 pages      arXiv:0802.0744      https://doi.org/10.3842/SIGMA.2008.015
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Quasi-Linear Algebras and Integrability (the Heisenberg Picture)

Luc Vinet a and Alexei Zhedanov b
a) Université de Montréal PO Box 6128, Station Centre-ville, Montréal QC H3C 3J7, Canada
b) Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

Received November 16, 2007, in final form January 19, 2008; Published online February 06, 2008

Abstract
We study Poisson and operator algebras with the ''quasi-linear property'' from the Heisenberg picture point of view. This means that there exists a set of one-parameter groups yielding an explicit expression of dynamical variables (operators) as functions of ''time'' t. We show that many algebras with nonlinear commutation relations such as the Askey-Wilson, q-Dolan-Grady and others satisfy this property. This provides one more (explicit Heisenberg evolution) interpretation of the corresponding integrable systems.

Key words: Lie algebras; Poisson algebras; nonlinear algebras; Askey-Wilson algebra; Dolan-Grady relations.

pdf (293 kb)   ps (195 kb)   tex (23 kb)

References

  1. Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, 1-55.
  2. Baseilhac P., Koizumi K., A deformed analogue of Onsager's symmetry in the XXZ open spin chain, J. Stat. Mech. Theory Exp. 2005 (2005), P10005, hep-th/0507053.
  3. Baseilhac P., Deformed Dolan-Grady relations in quantum integrable models, Nuclear Phys. B 709 (2005), 491-521, hep-th/0404149.
  4. Davies B., Onsager's algebra and superintegrability, J. Phys. A: Math. Gen. 23 (1990), 2245-2261.
  5. Dolan L., Grady M., Conserved charges from self-duality, Phys. Rev. D (3) 25 (1982), 1587-1604.
  6. Fokas A.S., Gelfand I.M., Quadratic Poisson algebras and their infinite-dimensional extensions, J. Math. Phys. 35 (1994), 3117-3131.
  7. Gorsky A.S., Zabrodin A.V., Degenerations of Sklyanin algebra and Askey-Wilson polynomials, J. Phys. A: Math. Gen. 26 (1993), L635-L639, hep-th/9303026.
  8. Granovskii Ya.I., Lutzenko I.M., Zhedanov A.S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Phys. 217 (1992), 1-20.
  9. Ito T., Tanabe K., Terwilliger P., Some algebra related to P- and Q-polynomial association schemes, in Codes and Association Schemes (1999, Piscataway NJ), Amer. Math. Soc., Providence RI, 2001, 167-192, math.CO/0406556.
  10. Karasëv M.V., Maslov V.P., Nonlinear Poisson brackets. Geometry and quantization, Translations of Mathematical Monographs, Vol. 119, Amer. Math. Soc., Providence, RI, 1993.
  11. Korovnichenko A., Zhedanov A., Dual algebras with non-linear Poisson brackets, in Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory (2000, Kiev), NATO Sci. Ser. II Math. Phys. Chem., Vol. 35, Kluwer Acad. Publ., Dordrecht, 2001, 265-272.
  12. Korovnichenko A., Zhedanov A., Classical Leonard triples, in Proceedings of Workshop on Elliptic Integrable Systems (November 8-11, 2004, Kyoto), 2004, 71-84, available at http://www.math.kobe-u.ac.jp/publications/rlm18/6.pdf.
  13. Létourneau P., Vinet L., Quadratic algebras in quantum mechanics, in Symmetries in Science, VII (1992, Nakajo), Plenum, New York, 1993, 373-382.
  14. Nikiforov A.F., Suslov S.K., Uvarov V.B., Classical orthogonal polynomials of a discrete variable, Springer, 1991.
  15. Odake S., Sasaki R., Unified theory of annihilation-creation operators for solvable ("discrete") quantum mechanics, J. Math. Phys. 47 (2006), 102102, 33 pages, quant-ph/0605215.
  16. Odake S., Sasaki R., Exact solutions in the Heisenberg picture and annihilation-creation operators, Phys. Lett. B 641 (2006), 112-117, quant-ph/0605221.
  17. Odesskii A., Rubtsov V., Polynomial Poisson algebras with regular structure of symplectic leaves, Teoret. Mat. Fiz. 133 (2002), 1321-1337, math.QA/0110032.
  18. Perk J.H.H., Star-triangle equations, quantum Lax operators, and higher genus curves, in Proceedings 1987 Summer Research Institute on Theta functions, Proc. Symp. Pure. Math., Vol. 49, Part 1, Amer. Math. Soc., Providence, R.I., 1989, 341-354.
  19. Rosenberg A., Non-commutative algebraic geometry and representations of quantized algebras, Kluwer Academic Publishers, 1995.
  20. Rosengren H., An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic), Ramanujan J. 13 (2007), 131-166, math.CA/0312310.
  21. Sklyanin E.K., On some algebraic structures connected with the Yang-Baxter equation, Funct. Anal. Appl. 16 (1982), no. 4, 263-270.
    Sklyanin E.K., Some algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras, Funct. Anal. Appl. 17 (1983), no. 4, 273-284.
  22. Smith S.P., Bell A.D., Some 3-dimensional skew polynomial rings, Preprint, 1991.
  23. Spiridonov V., Zhedanov A., Poisson algebras for some generalized eigenvalue problems, J. Phys. A: Math. Gen. 37 (2004), 10429-10443.
  24. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001), 149-203.
  25. Terwilliger P., Two relations that generalize the q-Serre relations and the Dolan-Grady relations, math.QA/0307016.
  26. Vinet L., Zhedanov A., Generalized Bochner theorem: characterization of the Askey-Wilson polynomials, J. Comput. Appl. Math. 211 (2008), 45-56, arXiv:0712.0069.
  27. Wiegmann P.B., Zabrodin A.V., Algebraization of difference eigenvalue equations related to Uq(sl2), Nuclear Phys. B 451 (1995), 699-724, cond-mat/9501129.
  28. Zhedanov A.S., "Hidden symmetry" of Askey-Wilson polynomials, Teoret. Mat. Fiz. 89 (1991), 190-204 (English transl.: Theoret. and Math. Phys. 89 (1991), 1146-1157).
  29. Zhedanov A., Korovnichenko A., 'Leonard pairs' in classical mechanics, J. Phys. A: Math. Gen. 35 (2002), 5767-5780.

Previous article   Next article   Contents of Volume 4 (2008)