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

SIGMA 11 (2015), 043, 33 pages      arXiv:1502.00128      https://doi.org/10.3842/SIGMA.2015.043
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

### Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems

Robin Heinonen a, Ernest G. Kalnins b, Willard Miller Jr. a and Eyal Subag c
a) School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA
b) Department of Mathematics, University of Waikato, Hamilton, New Zealand
c) School of Mathematical Science, Tel Aviv University, Tel Aviv 69978, Israel

Received February 03, 2015, in final form May 30, 2015; Published online June 08, 2015

Abstract
Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Inönü-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as ${\hbar}\to 0$ and nonrelativistic phenomena from special relativistic as $c\to \infty$, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classified contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras $e(2,{\mathbb C})$ in flat space and $o(3,{\mathbb C})$ on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of Inönü-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. After constant curvature spaces, the 4 Darboux spaces are the 2D manifolds admitting the most 2nd order Killing tensors. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by Inönü-Wigner contractions. We present tables of the contraction results.

Key words: contractions; quadratic algebras; superintegrable systems; Darboux spaces; Askey scheme.

pdf (749 kb)   tex (495 kb)

References

1. Bôcher M., Über die Riehenentwickelungen der Potentialtheory, B.G. Teubner, Leipzig, 1894.
2. Bonatsos D., Daskaloyannis C., Kokkotas K., Deformed oscillator algebras for two-dimensional quantum superintegrable systems, Phys. Rev. A 50 (1994), 3700-3709, hep-th/9309088.
3. Daskaloyannis C., Tanoudis Y., Quantum superintegrable systems with quadratic integrals on a two dimensional manifold, J. Math. Phys. 48 (2007), 072108, 22 pages, math-ph/0607058.
4. Doebner H.D., Melsheimer O., On a class of generalized group contractions, Nuovo Cimento A 49 (1967), 306-311.
5. Eisenhart L.P., Transformations of surfaces, 2nd ed., Chelsea, New York, 1966.
6. Evans N.W., Super-integrability of the Winternitz system, Phys. Lett. A 147 (1990), 483-486.
7. Fordy A.P., Quantum super-integrable systems as exactly solvable models, SIGMA 3 (2007), 025, 10 pages, math-ph/0702048.
8. Grabowski J., Marmo G., Perelomov A.M., Poisson structures: towards a classification, Modern Phys. Lett. A 8 (1993), 1719-1733.
9. Granovskii Ya.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved space. I. Oscillator, Theoret. and Math. Phys. 91 (1992), 474-480.
10. Granovskii Ya.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved space. II. The Kepler problem, Theoret. and Math. Phys. 91 (1992), 604-612.
11. Gromov N.A., From Wigner-Inönü group contraction to contractions of algebraic structures, Acta Phys. Hung. A 19 (2004), 209-212, hep-th/0210304.
12. Huddleston P.L., Inönü-Wigner contractions of the real four-dimensional Lie algebras, J. Math. Phys. 19 (1978), 1645-1649.
13. Inönü E., Wigner E.P., On the contraction of groups and their representations, Proc. Nat. Acad. Sci. USA 39 (1953), 510-524.
14. Kalnins E.G., Kress J.M., Miller Jr. W., Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory, J. Math. Phys. 46 (2005), 053509, 28 pages.
15. Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages.
16. Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems, J. Math. Phys. 47 (2006), 093501, 25 pages.
17. Kalnins E.G., Kress J.M., Miller Jr. W., Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties, J. Phys. A: Math. Theor. 40 (2007), 3399-3411, arXiv:0708.3044.
18. Kalnins E.G., Kress J.M., Miller Jr. W., Post S., Structure theory for second order 2D superintegrable systems with 1-parameter potentials, SIGMA 5 (2009), 008, 24 pages, arXiv:0901.3081.
19. Kalnins E.G., Kress J.M., Miller Jr. W., Winternitz P., Superintegrable systems in Darboux spaces, J. Math. Phys. 44 (2003), 5811-5848, math-ph/0307039.
20. Kalnins E.G., Kress J.M., Winternitz P., Superintegrability in a two-dimensional space of nonconstant curvature, J. Math. Phys. 43 (2002), 970-983, math-ph/0108015.
21. Kalnins E.G., Miller Jr. W., Quadratic algebra contractions and second-order superintegrable systems, Anal. Appl. (Singap.) 12 (2014), 583-612, arXiv:1401.0830.
22. Kalnins E.G., Miller Jr. W., Post S., Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials, SIGMA 9 (2013), 057, 28 pages, arXiv:1212.4766.
23. Koenigs G., Sur les géodésiques a intégrales quadratiques, in Darboux G., Lecons sur la théorie générale des surfaces et les applications geométriques du calcul infinitesimal, Vol. 4, Chelsea, New York, 1972, 368-404.
24. Kress J.M., Equivalence of superintegrable systems in two dimensions, Phys. Atomic Nuclei 70 (2007), 560-566.
25. Létourneau P., Vinet L., Superintegrable systems: polynomial algebras and quasi-exactly solvable Hamiltonians, Ann. Physics 243 (1995), 144-168.
26. Maduemezia A., On hyper-relativistic quantum systems, J. Math. Phys. 28 (1987), 79-84.
27. Miller Jr. W., Symmetry and separation of variables, Encyclopedia of Mathematics and its Applications, Vol. 4, Addison-Wesley Publishing Co., Reading, Mass. - London - Amsterdam, 1977.
28. Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694.
29. Nesterenko M., Popovych R., Contractions of low-dimensional Lie algebras, J. Math. Phys. 47 (2006), 123515, 45 pages, math-ph/0608018.
30. Saletan E.J., Contraction of Lie groups, J. Math. Phys. 2 (1961), 1-21.
31. Talman J.D., Special functions: a group theoretic approach (based on lectures by Eugene P. Wigner), W. A. Benjamin, Inc., New York - Amsterdam, 1968.
32. Tempesta P., Turbiner A.V., Winternitz P., Exact solvability of superintegrable systems, J. Math. Phys. 42 (2001), 4248-4257, hep-th/0011209.
33. Tempesta P., Winternitz P., Harnad J., Miller W., Pogosyan G., Rodriguez M. (Editors), Superintegrability in classical and quantum systems, CRM Proceedings and Lecture Notes, Vol. 37, American Mathematical Society, Providence, RI, 2004.
34. Weimar-Woods E., The three-dimensional real Lie algebras and their contractions, J. Math. Phys. 32 (1991), 2028-2033.