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Bôcher and Abstract Contractions of $2$nd Order Quadratic Algebras
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of $2$nd 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 Bôcher contractions of the conformal Lie algebra ${\mathfrak{so}}(4,\mathbb {C})$ to itself. In this paper we give a precise definition of Bôcher contractions and show how they can be classified. They subsume well known contractions of ${\mathfrak{e}}(2,\mathbb {C})$ and ${\mathfrak{so}}(3,\mathbb {C})$ and have important physical and geometric meanings, such as the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. We also classify abstract nondegenerate quadratic algebras in terms of an invariant that we call a canonical form. We describe an algorithm for finding the canonical form of such algebras. We calculate explicitly all canonical forms arising from quadratic algebras of $2\mathrm{D}$ nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems.
Keywords: contractions; quadratic algebras; superintegrable systems; conformal superintegrability. 
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     author = {Mauricio A. Escobar Ruiz and Ernest G. Kalnins and Willard Miller Jr. and Eyal Subag},
     title = {B\^ocher and {Abstract} {Contractions} of $2$nd {Order} {Quadratic} {Algebras}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a12/}
}
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Mauricio A. Escobar Ruiz; Ernest G. Kalnins; Willard Miller Jr.; Eyal Subag. Bôcher and Abstract Contractions of $2$nd Order Quadratic Algebras. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a12/

[1] Bôcher M., Über die Riehenentwickelungen der Potentialtheory, B. G. Teubner, Leipzig, 1894

[2] Borel A., Linear algebraic groups, Graduate Texts in Mathematics, 126, 2nd ed., Springer-Verlag, New York, 1991 | DOI | MR | Zbl

[3] Capel J. J., Kress J. M., Post S., “Invariant classification and limits of maximally superintegrable systems in 3D”, SIGMA, 11 (2015), 038, 17 pp., arXiv: 1501.06601 | DOI | MR | Zbl

[4] Daskaloyannis C., Tanoudis Y., “Quantum superintegrable systems with quadratic integrals on a two dimensional manifold”, J. Math. Phys., 48 (2007), 072108, 22 pp., arXiv: math-ph/0607058 | DOI | MR | Zbl

[5] Escobar-Ruiz M. A., Kalnins E. G., Miller W. (Jr.), 2D 2nd order Laplace superintegrable systems, Heun equations, QES and Bôcher contractions, arXiv: 1609.03917

[6] Escobar-Ruiz M. A., Miller W. (Jr.), “Toward a classification of semidegenerate 3D superintegrable systems”, J. Phys. A: Math. Theor., 50 (2017), 095203, 22 pp., arXiv: 1611.02977 | DOI

[7] Evans N. W., “Super-integrability of the Winternitz system”, Phys. Lett. A, 147 (1990), 483–486 | DOI | MR

[8] Fordy A. P., “Quantum super-integrable systems as exactly solvable models”, SIGMA, 3 (2007), 025, 10 pp., arXiv: math-ph/0702048 | DOI | MR | Zbl

[9] Gantmacher F. R., The theory of matrices, v. II, Chelsea, New York, 1959 | MR

[10] Heinonen R., Kalnins E. G., Miller W. (Jr.), Subag E., “Structure relations and Darboux contractions for 2D 2nd order superintegrable systems”, SIGMA, 11 (2015), 043, 33 pp., arXiv: 1502.00128 | DOI | MR | Zbl

[11] Inönü E., Wigner E. P., “On the contraction of groups and their representations”, Proc. Nat. Acad. Sci. USA, 39 (1953), 510–524 | DOI | MR | Zbl

[12] Izmest'ev A. A., Pogosyan G. S., Sissakian A. N., Winternitz P., “Contractions of Lie algebras and separation of variables”, J. Phys. A: Math. Gen., 29 (1996), 5949–5962 | DOI | MR | Zbl

[13] Izmest'ev A.A., Pogosyan G. S., Sissakian A. N., Winternitz P., “Contractions of Lie algebras and the separation of variables: interbase expansions”, J. Phys. A: Math. Gen., 34 (2001), 521–554 | DOI | MR | Zbl

[14] Kalnins E. G., Kress J. M., Miller W. (Jr.), “Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory”, J. Math. Phys., 46 (2005), 053509, 28 pp. | DOI | MR | Zbl

[15] Kalnins E. G., Kress J. M., Miller W. (Jr.), “Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform”, J. Math. Phys., 46 (2005), 053510, 15 pp. | DOI | MR | Zbl

[16] Kalnins E. G., Kress J. M., Miller W. (Jr.), “Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory”, J. Math. Phys., 46 (2005), 103507, 28 pp. | DOI | MR | Zbl

[17] Kalnins E. G., Kress J. M., Miller W. (Jr.), “Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory”, J. Math. Phys., 47 (2006), 043514, 26 pp. | DOI | MR | Zbl

[18] Kalnins E. G., Kress J. M., Miller W. (Jr.), “Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems”, J. Math. Phys., 47 (2006), 093501, 25 pp. | DOI | MR | Zbl

[19] Kalnins E. G., Kress J. M., Miller W. (Jr.), “Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties”, J. Phys. A: Math. Theor., 40 (2007), 3399–3411, arXiv: 0708.3044 | DOI | MR | Zbl

[20] Kalnins E. G., Kress J. M., Miller W. (Jr.), Post S., “Laplace-type equations as conformal superintegrable systems”, Adv. in Appl. Math., 46 (2011), 396–416, arXiv: 0908.4316 | DOI | MR | Zbl

[21] Kalnins E. G., Kress J. M., Miller W. (Jr.), Winternitz P., “Superintegrable systems in Darboux spaces”, J. Math. Phys., 44 (2003), 5811–5848, arXiv: math-ph/0307039 | DOI | MR | Zbl

[22] Kalnins E. G., Kress J. M., Pogosyan G. S., Miller W. (Jr.), “Completeness of superintegrability in two-dimensional constant-curvature spaces”, J. Phys. A: Math. Gen., 34 (2001), 4705–4720, arXiv: math-ph/0102006 | DOI | MR | Zbl

[23] Kalnins E. G., Miller W. (Jr.), “Quadratic algebra contractions and second-order superintegrable systems”, Anal. Appl. (Singap.), 12 (2014), 583–612, arXiv: 1401.0830 | DOI | MR | Zbl

[24] Kalnins E. G., Miller W. (Jr.), Post S., “Coupling constant metamorphosis and $N$th-order symmetries in classical and quantum mechanics”, J. Phys. A: Math. Theor., 43 (2010), 035202, 20 pp., arXiv: 0908.4393 | DOI | Zbl

[25] Kalnins E. G., Miller W. (Jr.), Post S., “Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials”, SIGMA, 9 (2013), 057, 28 pp., arXiv: 1212.4766 | DOI | MR | Zbl

[26] Kalnins E. G., Miller W. (Jr.), Reid G. J., “Separation of variables for complex Riemannian spaces of constant curvature. I. Orthogonal separable coordinates for ${\rm S}_{nC}$ and ${\rm E}_{nC}$”, Proc. Roy. Soc. London Ser. A, 394 (1984), 183–206 | DOI | MR | Zbl

[27] Kalnins E. G., Miller W. (Jr.), Subag E., “Bôcher contractions of conformally superintegrable Laplace equations”, SIGMA, 12 (2016), 038, 31 pp., arXiv: 1512.09315 | DOI | MR | Zbl

[28] Koenigs G. X. P., “Sur les géodésiques a integrales quadratiques”, Le{ c}ons sur la théorie générale des surfaces, v. 4, ed. J. G. Darboux, Chelsea Publishing, 1972, 368–404

[29] Kress J. M., “Equivalence of superintegrable systems in two dimensions”, Phys. Atomic Nuclei, 70 (2007), 560–566 | DOI

[30] Miller W. (Jr.), Post S., Winternitz P., “Classical and quantum superintegrability with applications”, J. Phys. A: Math. Theor., 46 (2013), 423001, 97 pp., arXiv: 1309.2694 | DOI | MR | Zbl

[31] Nesterenko M., Popovych R., “Contractions of low-dimensional Lie algebras”, J. Math. Phys., 47 (2006), 123515, 45 pp., arXiv: math-ph/0608018 | DOI | MR | Zbl

[32] Post S., “Models of quadratic algebras generated by superintegrable systems in 2D”, SIGMA, 7 (2011), 036, 20 pp., arXiv: 1104.0734 | DOI | MR | Zbl

[33] Tempesta P., Turbiner A. V., Winternitz P., “Exact solvability of superintegrable systems”, J. Math. Phys., 42 (2001), 4248–4257, arXiv: hep-th/0011209 | DOI | MR | Zbl

[34] Tempesta P., Winternitz P., Harnad J., Miller W., Pogosyan G., Rodriguez M. (eds.), Superintegrability in classical and quantum systems, CRM Proceedings and Lecture Notes, 37, Amer. Math. Soc., Providence, RI, 2004 | MR | Zbl

[35] Turbiner A. V., “The Heun operator as a Hamiltonian”, J. Phys. A: Math. Theor., 49 (2016), 26LT01, 8 pp., arXiv: 1603.02053 | DOI | MR | Zbl

[36] Turbiner A. V., “One-dimensional quasi-exactly solvable Schrödinger equations”, Phys. Rep., 642 (2016), 1–71, arXiv: 1603.02992 | DOI | MR

[37] Weimar-Woods E., “The three-dimensional real Lie algebras and their contractions”, J. Math. Phys., 32 (1991), 2028–2033 | DOI | MR | Zbl