Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras

Quadratic algebras are generalizations of Lie algebras which 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...

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Дата:	2017
Автори:	Escobar Ruiz, M.A., Kalnins, E.G., Miller Jr., W., Subag, E.
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Опубліковано:	Інститут математики НАН України 2017
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras / M.A. Escobar Ruiz, E.G. Kalnins, W. Miller Jr., E. Suba // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 37 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1486172025-02-09T20:30:07Z Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras Escobar Ruiz, M.A. Kalnins, E.G. Miller Jr., W. Subag, E. Quadratic algebras are generalizations of Lie algebras which 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 Bôcher contractions of the conformal Lie algebra so(4,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 e(2,C) and so(3,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 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems. This work was partially supported by a grant from the Simons Foundation (# 208754 to Willard Miller Jr. and by CONACYT grant (# 250881 to M.A. Escobar-Ruiz). The author M.A. Escobar-Ruiz is grateful to ICN UNAM for the kind hospitality during his visit, where a part of the research was done, he was supported in part by DGAPA grant IN108815 (Mexico). 2017 Article Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras / M.A. Escobar Ruiz, E.G. Kalnins, W. Miller Jr., E. Suba // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 37 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E70; 16G99; 37J35; 37K10; 33C45; 17B60; 81R05; 33C45 DOI:10.3842/SIGMA.2017.013 https://nasplib.isofts.kiev.ua/handle/123456789/148617 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Quadratic algebras are generalizations of Lie algebras which 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 Bôcher contractions of the conformal Lie algebra so(4,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 e(2,C) and so(3,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 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems.
format	Article
author	Escobar Ruiz, M.A. Kalnins, E.G. Miller Jr., W. Subag, E.
spellingShingle	Escobar Ruiz, M.A. Kalnins, E.G. Miller Jr., W. Subag, E. Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Escobar Ruiz, M.A. Kalnins, E.G. Miller Jr., W. Subag, E.
author_sort	Escobar Ruiz, M.A.
title	Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
title_short	Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
title_full	Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
title_fullStr	Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
title_full_unstemmed	Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
title_sort	bôcher and abstract contractions of 2nd order quadratic algebras
publisher	Інститут математики НАН України
publishDate	2017
url	https://nasplib.isofts.kiev.ua/handle/123456789/148617
citation_txt	Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras / M.A. Escobar Ruiz, E.G. Kalnins, W. Miller Jr., E. Suba // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 37 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT escobarruizma bocherandabstractcontractionsof2ndorderquadraticalgebras AT kalninseg bocherandabstractcontractionsof2ndorderquadraticalgebras AT millerjrw bocherandabstractcontractionsof2ndorderquadraticalgebras AT subage bocherandabstractcontractionsof2ndorderquadraticalgebras
first_indexed	2025-11-30T12:16:58Z
last_indexed	2025-11-30T12:16:58Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 013, 38 pages Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras Mauricio A. ESCOBAR RUIZ † 1†2, Ernest G. KALNINS † 3 , Willard MILLER Jr. † 2 and Eyal SUBAG †4 †1 Instituto de Ciencias Nucleares, UNAM, Apartado Postal 70-543, 04510 Mexico D.F. Mexico E-mail: mauricio.escobar@nucleares.unam.mx †2 School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA E-mail: miller@ima.umn.edu URL: https://www.ima.umn.edu/~miller/ †3 Department of Mathematics, University of Waikato, Hamilton, New Zealand E-mail: math0236@waikato.ac.nz †4 Department of Mathematics, Pennsylvania State University, State College, Pennsylvania, 16802, USA E-mail: eyalsubag@gmail.com Received November 19, 2016, in final form February 27, 2017; Published online March 06, 2017 https://doi.org/10.3842/SIGMA.2017.013 Abstract. Quadratic algebras are generalizations of Lie algebras which include the sym- metry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum me- chanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Bôcher contractions of the conformal Lie algebra so(4,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 e(2,C) and so(3,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 polyno- mials. 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 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems. Key words: contractions; quadratic algebras; superintegrable systems; conformal superinte- grability 2010 Mathematics Subject Classification: 22E70; 16G99; 37J35; 37K10; 33C45; 17B60; 81R05; 33C45 1 Introduction Second order 2D superintegrable systems and their associated quadratic symmetry algebras are basic in mathematical physics. Among the simplest such solvable systems are the 2D Kepler and hydrogen atom and the isotropic and Higgs oscillators [30, 34]. All the systems are multi- separable, with the quantum separable solutions characterized as eigenfunctions of commuting operators in the quadratic algebras. The separation equations are the Gaussian hypergeometric equation and its various confluent forms in full generality, as well as the Heun equation and its confluent forms in full generality [5]. Solutions of the hypergeometric and Heun equations are mailto:mauricio.escobar@nucleares.unam.mx mailto:miller@ima.umn.edu https://www.ima.umn.edu/~miller/ mailto:math0236@waikato.ac.nz mailto:eyalsubag@gmail.com https://doi.org/10.3842/SIGMA.2017.013 2 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag linked through their solution of the same superintegrable system. The confluences are related to Bôcher contractions of the conformal algebra so(4,C) to itself [27]. The interbasis expansion coefficients relating distinct separable systems lead to other special functions, several of them functions of discrete variables, such as the Racah, Wilson and Hahn polynomials in full genera- lity [25]. The contractions also allow the derivation of the Askey scheme for the classification of hypergeometric orthogonal polynomials. The classification of quasi-exactly solvable (QES) systems based on the Heun operator coincide exactly with QES separation equations for these superintegrable systems [35, 36]. In short, the structure and classification of these quadratic algebras and their relations via contractions are matters of considerable significance in mathematical physics. Historically, the superintegrable systems have been classified and their associated quadratic algebras then com- puted. Here we are reversing the process: we first classify abstract quadratic algebras and then determine which of these correspond to 2nd order superintegrable systems. Also we determine how the abstract quadratic algebras are related via contractions and examine which of these con- tractions can be realized geometrically as Bôcher contractions. The eventual goal is to isolate the algebras and contractions that do not correspond to geometrical superintegrable systems and to determine their significance. Bôcher invented a recipe for a limit procedure which showed how to find what we now know are all R-separable coordinate systems for free Laplace and wave equations in n dimensions [1]. We have recently recognized that these limits can be interpreted as contractions of so(n+2,C) to itself and classified; we call them Bôcher contractions. In this paper we give for the first time the precise definition of these contractions and their properties and classification for the case n = 2. We start with some basic facts. We define a quantum (Helmholtz) superintegrable system as an integrable Hamiltonian system on an n-dimensional pseudo-Riemannian manifold with poten- tial: H = ∆n +V that admits 2n− 1 algebraically independent partial differential operators Lj commuting with H, the maximum possible: [H,Lj ] = 0, j = 1, 2, . . . , 2n−1. Similarly a classical superintegrable on such a manifold, with HamiltonianH = ∑ gijpipj+V , is an integrable system that admits 2n− 1 functionally independent constants of the motion Lj , polynomial in the mo- menta, in involution with H, the maximum possible. Superintegrability captures the properties of quantum Hamiltonian systems that allow the Schrödinger eigenvalue problem (or Helmholtz equation) HΨ = EΨ to be solved exactly, analytically and algebraically [7, 8, 30, 33, 34] and the classical trajectories to be computed algebraically. A system is of order K if the maximum order of the symmetry operators (or the polynomial order of the classical constants of the motion), other than H, is K. For n = 2, K = 1, 2 all systems are known, e.g., [4, 14, 15, 16, 17, 18, 19]. For K = 1 the symmetry algebras are just Lie algebras. We review briefly the facts for free 2nd order superintegrable systems (i.e., no potential, K = 2) in the case n = 2, 2n − 1 = 3. The complex spaces with Laplace–Beltrami operators admitting at least three 2nd order symmetries were classified by Koenigs (1896) [28]. They are: the two constant curvature spaces (flat space and the complex sphere), the four Darboux spaces (one of which, D4, contains a parameter) [21], and 5 families of 4-parameter Koenigs spaces, see Section 1.1. For 2nd order systems with non-constant potential the generating symmetry operators of each system close under commutation (or via Poisson brackets in the classical case) to determine a quadratic algebra, and the irreducible representations of the quantum algebra determine the eigenvalues of H and their multiplicities. More precisely, in the classical case, closedness means that the Poisson algebra generated by the constants of motion is finitely generated as an associative algebra. The quantum case is defined analogously. Here we consider only the nondegenerate superintegrable systems: Those with 4-parameter potentials (including the additive constant) (the maximum possible): V (x) = a1V(1)(x) + a2V(2)(x) + a3V(3)(x) + a4, (1.1) Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 3 where {V(1)(x), V(2)(x), V(3)(x), 1} is a linearly independent set. Here the possible classical and quantum potentials are identical and there is a 1-1 relationship between classical and quantum systems. The classical constants of the motion determine the quantum symmetry operators, modulo symmetrization. The classical symmetry algebra generated by H, L1, L2 always closes under commutation and gives the following nondegenerate quadratic algebra structure: Definition 1.1. An abstract nondegenerate (classical) quadratic algebra is a Poisson algebra with functionally independent generators H, L1, L2, and parameters a1, a2, a3, a4, such that all generators are in involution with H and the following relations hold: {Lj ,R} = ∑ 0≤e1+e2+e3≤2 M (j) e1,e2,e3L e1 1 L e2 2 H e3 , ek ≥ 0, L0k = 1, R2 = F ≡ ∑ 0≤e1+e2+e3≤3 Ne1,e2,e3L e1 1 L e2 2 H e3 . Here, R ≡ {L1,L2}. In both equations the constants M (j) e1,e2,e3 and Ne1,e2,e3 are polynomials in the parameters a1, a2, a3 of degree 2 − e1 − e2 − e3 and 3 − e1 − e2 − e3, respectively. The symmetry algebras obeyed by the quantum superintegrable systems have a similar structure, slightly more complicated due to the need for symmetrization of the noncommuting operators. In the case a1 = a2 = a3 = a4 = 0, the corresponding quadratics algebras are called free. Note that we can think of a nondegenerate (classical or quantum) quadratic algebra as a fam- ily of algebras parametrized by the constants ai. The algebra is called quadratic because the Poisson brackets {Lj ,R} are 2nd order polynomials in the generators Li, H, whereas for a Lie algebra they are 1st order. Nondegenerate 2D superintegrable systems always have a quadratic algebra structure in which the parameters aj are those of the potential; we call these quadratic algebras geometrical. Although the full sets of classical structure equations can be rather complicated, the func- tion F contains all of the structure information for nondegenerate systems. In particular, it is easy to show that, e.g., [23], {L1,R} = 1 2 ∂F ∂L2 , {L2,R} = −1 2 ∂F ∂L1 , (1.2) for any algebra satisfying Definition 1.1, so F determines the structure equations explicitly. For a nondegenerate superintegrable system with potential (1.1) the structure equations are determined by F(H,L1,L2, a1, a2, a3, a4) as defined above. The effect of a Stäckel transform [24] generated by the specific special choice of the potential function, say V(3) is to determine a new superintegrable system with Casimir R̃2 = F(−a3,L1,L2, a1, a2,−H, a4). Of course, the switch of a3 and H is only for illustration; there is a Stäckel transform that replaces any aj by −H and H by −aj and similar transforms that apply to any basis that we choose for the potential space. If we consider the free systems (zero potential which is the case with all parameters equal zero) on the spaces classified by Koenigs, then the vector space of 2nd order symmetries may be larger than 3: 6-dimensional for constant curvature spaces, 4-dimensional for Darboux spaces, and 3-dimensional for Koenigs spaces. In general the Poisson algebras generated by taking Poisson brackets of these 2nd order elements are infinite-dimensional; they do not close (in the sense that was explained above). However, in [23], the possible 3-dimensional subspaces of 2nd order free symmetries that generate quadratic algebras were classified, up to conjugacy by symmetry groups of these spaces: e(2,C) for flat space, o(3,C) for nonzero constant curvature spaces, and a 1-dimensional translation subalgebra for Darboux spaces. For Koenigs spaces the first order symmetry algebra is 0-dimensional and the space of 2nd order symmetries is 3-dimensional which always generates a unique quadratic algebra. 4 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag Theorem 1.2. For each of the spaces classified by Koenigs, there is a bijection between free quadratic algebras of 2nd order symmetries, classified up to conjugacy, and 2nd order nondege- nerate superintegrable systems on these spaces. The proof of this theorem is constructive [23]. Given a free quadratic algebra Q̃ one can com- pute the potential V and the symmetries of the quadratic algebra Q of the nondegenerate su- perintegrable system. (The quadratic algebra structure guarantees that the Bertrand–Darboux equations for the potential are satisfied identically. In this sense the free systems “know” the possible nondegenerate superintegrable systems they can support. Since there is a 1-1 relation- ship between quantum and classical nondegenerate systems, the information about all of these systems is encoded in the free quadratic algebras generated by 2nd order constants of the motion (Killing tensors) of constant curvature, Darboux and Koenigs spaces. Note that for flat space the generators for the free quadratic algebras can be expressed as 2nd order elements in the uni- versal enveloping algebra of e(2,C), and for nonzero constant curvature spaces the generators for the free quadratic algebras can be expressed as 2nd order elements in the universal enveloping algebra of so(3,C) [23]. All 2nd order 2D superintegrable systems with potential and their quadratic algebras are known. There are 33 nondegenerate systems, on a variety of manifolds classified up to conju- gacy, see Section 1.1 where the numbering for constant curvature systems is taken from [22], (the numbers are not always consecutive because the lists in [22] also include degenerate systems) and the numbering for Darboux spaces is taken from [21]. For each system we give the 4-parameter potential and the abstract free structure equation R2 −F = 0. Note that many of the abstract structure equations for the superintegrable systems are identical, even for superintegrable sys- tems on different manifolds. Of course the geometrical structure equations are distinct because the generators L1, L2, H are distinct for each geometrical superintegrable system. Under the Stäckel transform (we discuss this in Section 2.1) these systems divide into 6 equivalence classes with representatives on flat space and the 2-sphere, see [29] and Section 3.3. 1.1 The Helmholtz nondegenerate superintegrable systems Flat space systems: H ≡ p2x + p2y + V = E. 1. E1: V = α ( x2 + y2 ) + β x2 + γ y2 , R2 = L1L2(H+ L2), 2. E2: V = α ( 4x2 + y2 ) + βx+ γ y2 , R2 = L21(H+ L1), 3. E3′: V = α ( x2 + y2 ) + βx+ γy, R2 = 0, 4. E7: V = α(x+iy)√ (x+iy)2−b + β(x−iy)√ (x+iy)2−b ( x+iy+ √ (x+iy)2−b )2 + γ ( x2 + y2 ) , R2 = L1L22 + bL2H2, 5. E8: V = α(x−iy) (x+iy)3 + β (x+iy)2 + γ ( x2 + y2 ) , R2 = L1L22, 6. E9: V = α√ x+iy + βy + γ(x+2iy)√ x+iy , R2 = L1(L1 +H)2, 7. E10: V = α(x− iy) + β ( x+ iy − 3 2(x− iy)2 ) + γ ( x2 + y2 − 1 2(x− iy)3 ) , R2 = L31, 8. E11: V = α(x− iy) + β(x−iy)√ x+iy + γ√ x+iy , R2 = L1H2, 9. E15: V = f(x − iy), where f is arbitrary, R2 = L31 (the exceptional case, characterized by the fact that the symmetry generators are functionally linearly dependent [14, 15, 16, 17, 18, 19, 22]), 10. E16: V = 1√ x2+y2 ( α+ β y+ √ x2+y2 + γ y− √ x2+y2 ) , R2 = L1 ( L1H+ L22 ) , 11. E17: V = α√ x2+y2 + β (x+iy)2 + γ (x+iy) √ x2+y2 , R2 = L1L22, Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 5 12. E19: V = α(x+iy)√ (x+iy)2−4 + β√ (x−iy)(x+iy+2) + γ√ (x−iy)(x+iy−2) , R2 = L1 ( L22 +H2 ) , 13. E20: V = 1√ x2+y2 ( α+ β √ x+ √ x2 + y2 + γ √ x− √ x2 + y2 ) , R2 = H ( L21 + L22 ) . Systems on the complex 2-sphere: H ≡ J 2 23+J 2 13+J 2 12+V = E. Here, Jk` = skps`−s`psk and s21 + s22 + s23 = 1. 1. S1: V = α (s1+is2)2 + βs3 (s1+is2)2 + γ(1−4s23) (s1+is2)4 , R2 = L31, 2. S2: V = α s23 + β (s1+is2)2 + γ(s1−is2) (s1+is2)3 , R2 = L1L22, 3. S4: V = α (s1+is2)2 + βs3√ s21+s 2 2 + γ (s1+is2) √ s21+s 2 2 , R2 = L1L22, 4. S7: V = αs3√ s21+s 2 2 + βs1 s22 √ s21+s 2 2 + γ s22 , R2 = L21L2 + L22L1 − 1 16L 2 1H, 5. S8: V = αs2√ s21+s 2 3 + β(s2+is1+s3)√ (s2+is1)(s3+is1) + γ(s2+is1−s3)√ (s2+is1)(s3−is1) , R2 = L21L2 + L1L22 − 1 4L1L2H, 6. S9: V = α s21 + β s22 + γ s23 , R2 = L21L2 + L1L22 + 1 16L1L2H. Darboux 1 systems: H ≡ 1 4x ( p2x + p2y ) + V = E. 1. D1A: V = b1(2x−2b+iy) x √ x−b+iy + b2 x √ x−b+iy + b3 x + b4, R2 = L31 + L2L1H− bL21H− 2ibH2L2, 2. D1B: V = b1(4x2+y2) x + b2 x + b3 xy2 + b4, R2 = L31 + L2L1H, 3. D1C V = b1(x2+y2) x + b2 x + b3y x + b4, R2 = L2H2. Darboux 2 systems: H ≡ x2 x2+1 ( p2x + p2y ) + V = E. 1. D2A: V = x2 x2+1 ( b1 ( x2 + 4y2 ) + b2 x2 + b3y ) + b4, R2 = L31 + L21H+ 1 4L1H 2, 2. D2B: V = x2 x2+1 ( b1 ( x2 + y2 ) + b2 x2 + b3 y2 ) + b4, R2 = L1L22 + L1L2H− 1 16L2H 2, 3. D2C: V = x2√ x2+y2(x2+1) ( b1 + b2 y+ √ x2+y2 + b3 y− √ x2+y2 ) + b4, R2 = L1L22 + L21H− 1 4L1H 2. Darboux 3 systems: H ≡ 1 2 e2x ex+1 ( p2x + p2y ) + V = E. 1. D3A: V = b1 1+ex + b2ex√ 1+2ex+iy(1+ex) + b3ex+iy√ 1+2ex+iy(1+ex) + b4, R2 = H ( L21 + L22 −H2 ) , 2. D3B: V = ex ex+1 ( b1 + e− x 2 ( b2 cos y2 + b3 sin y 2 )) + b4, R2 = L1L22 +HL21 − 1 4H 2L1, 3. D3C: V = ex ex+1 ( b1 + ex( b2 cos2 y 2 + b3 sin2 y 2 ) ) + b4, R2 = L1L22 + L21H− 1 8L1H 2, 4. D3D: V = e2x 1+ex ( b1e −iy + b2e −2iy)+ b3 1+ex + b4, R2 = L1L22 + L1L2H+ L2H2 −H3. Darboux 4 systems: H ≡ − sin2 2x 2 cos 2x+b ( p2x + p2y ) + V = E. 1. D4(b)A: V = sin2 2x 2 cos 2x+b ( b1 sinh2 y + b2 sinh2 2y ) + b3 2 cos 2x+b + b4, R2 = L1L22, 2. D4(b)B: V = sin2 2x 2 cos 2x+b ( b1 sin2 2x + b2e 4y + b3e 2y ) + b4, R2 = L1L22 +L21L2 + bHL22 − 4H2L2, 3. D4(b)C: V = e2y b+2 sin2 x + b−2 cos2 x ( b1 Z+(1−e2y) √ Z + b2 Z+(1+e2y) √ Z + b3 e−2y cos2 x ) + b4, R2 = − b 163 H3 + L21L2 + L1L22 − b 16L1L2H− b 16L 2 2H+ 1 256L1H 2. 6 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag Note: Systems D4(b)A, D4(b)B, D4(b)C are in fact families of distinct systems parametrized by b, and E15 is a family of systems parametrized by the function f . The parameters b can be normalized away in systems E7, D1A, but it is convenient to keep them. Generic Koenigs spaces: (We do not list the relatively unenlightening expressions of R2 for the Koenigs spaces. Each involves 4 arbitrary parameters obtained via a generic Stäckel transformation from a constant curvature system.) 1. K[1, 1, 1, 1]: H ≡ 1 V (b1,b2,b3,b4) ( p2x + p2y + V (a1, a2, a3, a4) ) = E, V (a1, a2, a3, a4) = a1 x2 + a2 y2 + 4a3 (x2+y2−1)2 − 4a4 (x2+y2+1)2 , 2. K[2, 1, 1]: H ≡ 1 V (b1,b2,b3,b4) ( p2x + p2y + V (a1, a2, a3, a4) ) = E, V (a1, a2, a3, a4) = a1 x2 + a2 y2 − a3 ( x2 + y2 ) + a4, 3. K[2, 2]: H ≡ 1 V (b1,b2,b3,b4) ( p2x + p2y + V (a1, a2, a3, a4) ) = E, V (a1, a2, a3, a4) = a1 (x+iy)2 + a2(x−iy) (x+iy)3 + a3 − a4 ( x2 + y2 ) , 4. K[3, 1]: H ≡ 1 V (b1,b2,b3,b4) ( p2x + p2y + V (a1, a2, a3, a4) ) = E, V (a1, a2, a3, a4) = a1 − a2x+ a3 ( 4x2 + y2 ) + a4 y2 , 5. K[4]: H ≡= 1 V (b1,b2,b3,b4) ( p2x + p2y + V (a1, a2, a3, a4) ) = E, V (a1, a2, a3, a4) = a1−a2(x+ iy)+a3 ( 3(x+ iy)2 +2(x− iy) ) −a4 ( 4 ( x2 +y2 ) +2(x+ iy)3 ) , 6. K[0]: H ≡= 1 V (b1,b2,b3,b4) ( p2x + p2y + V (a1, a2, a3, a4) ) = E, V (a1, a2, a3, a4) = a1 − (a2x+ a3y) + a4 ( x2 + y2 ) . 1.2 Contractions In [23] it has been shown that all the 2nd order superintegrable systems are obtained by taking coordinate limits of the generic system S9 [22], or are obtained from these limits by a Stäckel transform (an invertible structure preserving mapping of superintegrable systems [14, 15, 16, 17, 18, 19]). Analogously all quadratic symmetry algebras of these systems are limits of that of S9. These coordinate limits induce limit relations between the special functions associated as eigenfunctions of the quantum superintegrable systems. The limits also induce contractions of the associated quadratic algebras, and via the models of the irreducible representations of these algebras, limit relations between the associated special functions. The Askey scheme for ortho- gonal functions of hypergeometric type is an example of this [25]. For constant curvature systems the required limits are all induced by Inönü–Wigner-type Lie algebra contractions of o(3,C) and e(2,C) [11, 31, 37]. Inönü–Wigner-type Lie algebra contractions have long been applied to relate separable coordinate systems and their associated special functions, see, e.g., [12, 13] for some more recent examples, but the application to quadratic algebras is due to the authors and their collaborators. Recall the definition of (natural) Lie algebra contractions: Let (A; [ ; ]A), (B; [ ; ]B) be two complex Lie algebras. We say that B is a contraction of A if for every ε ∈ (0, 1] there exists a linear invertible map tε : B → A such that for every X,Y ∈ B, lim ε→0 t−1ε [tεX, tεY ]A = [X,Y ]B. Thus, as ε → 0 the 1-parameter family of basis transformations can become singular but the structure constants of the Lie algebra go to a finite limit, necessarily that of another Lie algebra. The contractions of the symmetry algebras of 2D constant curvature spaces have long since been classified [23]. There are 6 nontrivial contractions of e(2,C) and 4 of o(3,C). They are each induced by coordinate limits. Just as for Lie algebras we can define a contraction of a quadratic algebra in terms of 1-parameter families of basis changes in the algebra. As ε → 0 the 1- parameter family of basis transformations becomes singular but the structure constants go to a finite limit [23]. Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 7 Theorem 1.3. Every Lie algebra contraction of A = e(2,C) or A = o(3,C) induces a contraction of a free (zero potential) quadratic algebra Q̃ based on A, which in turn induces a contraction of the quadratic algebra Q with potential. This is true for both classical and quantum algebras. Similarly the coordinate limit associated with each contraction takes H to a new superin- tegrable system with the contracted quadratic algebra. This relationship between coordinate limits, Lie algebra contractions and quadratic algebra contractions for superintegrable systems on constant curvature spaces breaks down for Darboux and Koenigs spaces. For Darboux spaces the Lie symmetry algebra is only 1-dimensional, and there is no Lie symmetry algebra at all for Koenigs spaces. Furthermore, there is the issue of finding a more systematic way of classifying the 44 distinct Helmholtz superintegrable systems on different manifolds, and their relations. These issues can be clarified by considering the Helmholtz systems as Laplace equations (with potential) on flat space. As announced in [27], the proper object to study is the conformal symmetry algebra so(4,C) of the flat space Laplacian and its contractions. The basic idea is that families of (Stäckel-equivalent) Helmholtz superintegrable systems on a variety of manifolds correspond to a single conformally superintegrable Laplace equation on flat space. We exploit this here in the case n = 2, but it generalizes easily to all dimensions n ≥ 2. The conformal symmetry algebra for Laplace equations with constant potential on flat space is the conformal algebra so(n+ 2,C). In his 1894 thesis [1] Bôcher introduced a limit procedure based on the roots of quadratic forms to find families of R-separable solutions of the ordinary (zero potential) flat space Laplace equation in n dimensions. An important feature of his work was the introduction of special projective coordinates in which the action of the conformal group so(n + 2,C) on solutions of the Laplace equation can be linearized. For n = 2 these are tetraspherical coordinates. In Sections 3 and 4 we describe in detail the Laplace equation mechanism and how it can be applied to systematize the classification of Helmholtz superintegrable systems and their relations via limits. We show that Bôcher’s limit procedure can be interpreted as constructing generalized Inönü–Wigner Lie algebra contractions of so(4,C) to itself. We call these Bôcher contractions and show that they induce contractions of the conformal quadratic algebras associated with Laplace superintegrable systems. All of the limits of the Helmholtz systems classified before for n = 2 [10, 23] are induced by the larger class of Bôcher contractions [27]. In this paper we replace Bôcher’s prescription by a precise definition of Bôcher contractions and introduce special Bôcher contractions, which are simpler and more easily classified. 2 2D conformal superintegrability of the 2nd order Classical nD systems of Laplace type are of the form H ≡ n∑ i,j=1 gijpipj + V = 0. A conformal symmetry of this equation is a function S(x,p) in the variables x = (x1, . . . , xn), polynomial in the momenta p = (p1, . . . , pn), such that {S,H} = RSH for some function RS(x,p), polynomial in the momenta. Two conformal symmetries S,S ′ are identified if S = S ′ + RH for some function R(x,p), polynomial in the momenta. (For short we will say that S = S ′, mod H and that S is a conformal constant of the motion (or conformal symmetry) if {S,H} = 0, mod (H).) The system is conformally superintegrable for n > 2 if there are 2n− 1 functionally independent conformal symmetries, S1, . . . ,S2n−1 with S1 = H. It is second order conformally superintegrable if each symmetry Si can be chosen to be a polynomial of at most second order in the momenta. There are obvious operator counterparts to these definitions for the operator Laplace equation HΨ ≡ (∆n + V )ψ = 0. 8 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag For n = 2 the definition must be restricted, since for a potential V = 0 there will be an infinite-dimensional space of conformal symmetries. We assume V 6= 0, possibly a constant. Every 2D Riemannian manifold is conformally flat, so we can always find a Cartesian-like coordinate system with coordinates x ≡ (x, y) ≡ (x1, x2) such that the Laplace equation takes the form H̃ = 1 λ(x, y) ( p2x + p2y ) + Ṽ (x) = 0. (2.1) However, this equation is equivalent to the flat space equation H ≡ p2x + p2y + V (x) = 0, V (x) = λ(x)Ṽ (x). (2.2) In particular, the conformal symmetries of (2.1) are identical with the conformal symmetries of (2.2). Thus without loss of generality we can assume the manifold is flat space with λ ≡ 1. In general the space of 2nd order conformal symmetries could be infinite-dimensional. How- ever, the requirement that H have a multiparameter potential reduces the possible symmetries to a finite-dimensional space. The result, from the Bertrand–Darboux conditions, is that the pure 2nd order polynomial terms in conformal symmetries belong to the space spanned by symmetrized products of the conformal Killing vectors P1 = px, P2 = py, J = xpy − ypx, D = xpx + ypy, K1 = ( x2 − y2 ) px + 2xypy, K2 = ( y2 − x2 ) py + 2xypx. (2.3) For a given multiparameter potential only a subspace of these conformal tensors occurs. 2.1 The conformal Stäckel transform We review briefly the concept of the conformal Stäckel transform [24]. Suppose we have a second order conformal superintegrable system H ≡ 1 λ(x, y) ( p2x + p2y ) + V (x, y) = 0, H ≡ H0 + V (2.4) with V the general potential solution for this system, and suppose U(x, y) is a particular potential solution, nonzero in an open set. The conformal Stäckel transform induced by U is the system H̃ = E, H̃ ≡ 1 λ̃ ( p2x + p2y ) + Ṽ , (2.5) where λ̃ = λU , Ṽ = V U . In [20, 27] we proved Theorem 2.1. The transformed (Helmholtz) system H̃ is superintegrable (in the nonconformal sense). This result shows that any second order conformal Laplace superintegrable system admitting a nonconstant potential U can be Stäckel transformed to a Helmholtz superintegrable system. This operation is invertible, but the inverse is not a Stäckel transform. By choosing all possible special potentials U associated with the fixed Laplace system (2.4) we generate the equivalence class of all Helmholtz superintegrable systems (2.5) obtainable through this process. As is easy to check, any two Helmholtz superintegrable systems lie in the same equivalence class if and only if they are Stäckel equivalent in the standard sense, see [27, Theorem 4]. All Helmholtz superintegrable systems are related to conformal Laplace systems in this way, so the study of all Helmholtz superintegrability on conformally flat manifolds can be reduced to the study of all Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 9 conformal Laplace superintegrable systems on flat space. All of these results have direct analogs for operator Laplace systems. The basic structure of quadratic algebras for nondegenerate Helmholtz superintegrable sys- tems is preserved under the transformation to Laplace equations, except that all identities hold mod H: Theorem 2.2 ([27]). The symmetries S1, S2 of the 2D nondegenerate conformal superintegrable Hamiltonian H generate a quadratic algebra {R,S1} = f (1)(S1,S2, α1, α2, α3, α4), {R,S2} = f (2)(S1,S2, α1, α2, α3, α4), R2 = f (3)(S1,S2, α1, α2, α3, α4), where R = {S1,S2} and all identities hold mod H. Here the αj are the parameters in the nondegenerate potential. A crucial observation now is that the free parts (those parts that one obtains by setting all the ai to zero) of the generators for 2nd order conformal superintegrable systems lie in the universal enveloping algebra of the conformal Lie algebra, mod H. Thus for the 2D case it follows that contractions of so(4,C) induce contractions of the conformal quadratic algebras of 2nd order superintegrable systems with nondegenerate potentials, and contractions of one system into another. In [27] it is shown how these Laplace contractions then induce contractions of Helmholtz superintegrable systems. 3 Tetraspherical coordinates and Laplace systems As already mentioned, the free parts of the 2nd order conformal symmetries of the Laplace equation H ≡ p2x + p2y + V (x) = 0 lie in the universal enveloping algebra of so(4,C) with generators (2.3). To linearize the action of these so(n + 2,C) operators on Laplace equations in n dimensions, Bôcher introduced a family of projective coordinates on the null cone in n+ 2 dimensions. In our case n = 2 these are the tetraspherical coordinates (x1, . . . , x4). They satisfy x21 +x22 +x23 +x24 = 0 (the null cone) and 4∑ k=1 xk∂xk = 0. They are projective coordinates on the null cone and have 2 degrees of freedom. Their principal advantage over flat space Cartesian coordinates is that the action of the conformal algebra (2.3) and of the conformal group SO(4,C) is linearized in tetraspherical coordinates. 3.1 Relation to Cartesian coordinates (x, y) and coordinates on the 2-sphere (s1, s2, s3) x1 = 2XT, x2 = 2Y T, x3 = X2 + Y 2 − T 2, x4 = i ( X2 + Y 2 + T 2 ) , x = X T = − x1 x3 + ix4 , y = Y T = − x2 x3 + ix4 , x = s1 1 + s3 , y = s2 1 + s3 . The projective variables X, Y , T are defined by these relations s1 = 2x x2 + y2 + 1 , s2 = 2y x2 + y2 + 1 , s3 = 1− x2 − y2 x2 + y2 + 1 , H ≡ p2x + p2y + Ṽ = (x3 + ix4) 2 ( 4∑ k=1 p2xk + V ) = (1 + s3) 2  3∑ j=1 p2sj + V  , Ṽ = (x3 + ix4) 2V, (1 + s3) = −i(x3 + ix4) x4 , s1 = ix1 x4 , s2 = ix2 x4 , s3 = − ix3 x4 . 10 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag Thus the Laplace equation H ≡ p2x + p2y + Ṽ in Cartesian coordinates becomes 4∑ k=1 p2xk + V = 0 in tetraspherical coordinates. 3.2 Relation to flat space and 2-sphere 1st order conformal constants of the motion We define Ljk = xj∂xk − xk∂xj , 1 ≤ j, k ≤ 4, j 6= k, where Ljk = −Lkj . The generators for flat space conformal symmetries (2.3) are related to these via P1 = L13 + iL14, P2 = L23 + iL24, D = iL34, J = L12, Kj = Lj3 − iLj4, j = 1, 2. The generators for 2-sphere conformal symmetries are related to the Ljk via L12 = J12 = s1ps2 − s2ps1 , L31 = J31, L23 = J23, Lj4 = −ipsj , j = 1, 2, 3. (3.1) In identifying tetraspherical coordinates we can always permute the parameters 1, . . . , 4. Also, we can apply an arbitrary SO(4,C) transformation to the tetraspherical coordinates, so the above relations between Euclidean and tetraspherical coordinates are far from being unique. 3.3 The 6 Laplace superintegrable systems with nondegenerate potentials The systems are all of the form 4∑ j=1 ∂2xj + V (x) Ψ = 0 in tetraspherical coordinates, or ( ∂2x + ∂2y + Ṽ ) Ψ = 0 as a flat space system in Cartesian coordi- nates. Each Laplace system is an equivalence class of Stäckel equivalent Helmholtz systems. In each case the expression for R2 in the conformal symmetry algebra can be put in a normal form which is a polynomial in Lj , ak of order ≤ 3. We show the terms of order ≥ 2 in the Lj alone. The parameter α is linear in the aj . The remaining terms are of lower order in the Lj : LOT. The potentials are V[1,1,1,1] = a1 x21 + a2 x22 + a3 x23 + a4 x24 , Ṽ[1,1,1,1] = a1 x2 + a2 y2 + 4a3 (x2 + y2 − 1)2 − 4a4 (x2 + y2 + 1)2 , R2 = L1L2(L1 + L2) + αL1L2 + LOT. Stäckel equivalent systems: S9, S8, S7, D4B(b), D4C(b), K[1, 1, 1, 1]. V[2,1,1] = a1 x21 + a2 x22 + a3(x3 − ix4) (x3 + ix4)3 + a4 (x3 + ix4)2 , Ṽ[2,1,1] = a1 x2 + a2 y2 − a3 ( x2 + y2 ) + a4, R2 = L21L2 + αL22 + LOT. Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 11 Stäckel equivalent systems: S4, S2, E1, E16, D4A(b), D3B, D2B, D2C, K[2, 1, 1]. V[2,2] = a1 (x1 + ix2)2 + a2(x1 − ix2) (x1 + ix2)3 + a3 (x3 + ix4)2 + a4(x3 − ix4) (x3 + ix4)3 , (3.2) Ṽ[2,2] = a1 (x+ iy)2 + a2(x− iy) (x+ iy)3 + a3 − a4 ( x2 + y2 ) , R2 = L21L2 + LOT. Stäckel equivalent systems: E8, E17, E7, E19, D3C, D3D, K[2, 2]. V[3,1] = a1 (x3 + ix4)2 + a2x1 (x3 + ix4)3 + a3(4x1 2 + x2 2) (x3 + ix4)4 + a4 x22 , Ṽ[3,1] = a1 − a2x+ a3 ( 4x2 + y2 ) + a4 y2 , R2 = L31 + αL22 + LOT. Stäckel equivalent systems: S1, E2, D1B, D2A, K[3, 1]. V[4] = a1 (x3 + ix4)2 + a2 x1 + ix2 (x3 + ix4)3 + a3 3(x1 + ix2) 2 − 2(x3 + ix4)(x1 − ix2) (x3 + ix4)4 + a4 4(x3 + ix4)(x 2 3 + x24) + 2(x1 + ix2) 3 (x3 + ix4)5 , Ṽ[4] = a1 − a2(x+ iy) + a3 ( 3(x+ iy)2 + 2(x− iy) ) − a4 ( 4 ( x2 + y2 ) + 2(x+ iy)3 ) , R2 = L31 + αL1L2 + LOT. Stäckel equivalent systems: E10, E9, D1A, K[4]. V[0] = a1 (x3 + ix4)2 + a2x1 + a3x2 (x3 + ix4)3 + a4 x21 + x22 (x3 + ix4)4 , Ṽ[0] = a1 − (a2x+ a3y) + a4 ( x2 + y2 ) , R2 = αL1L2 + LOT. Stäckel equivalent systems: E20, E11, E3′, D1C, D3A, K[0]. 4 Definition and composition of Bôcher contractions Before introducing precise definitions, let us note that all geometrical contractions of e(2,C)→ e(2,C) and so(3,C)→ so(3,C), e(2,C), i.e., pointwise coordinate limits of functions on flat space or the sphere as classified in [23], induce geometrical contractions of so(4,C)→ so(4,C). Recall that a basis for so(4,C) is (2.3) where the subset P1, P2, J forms a basis for e(2,C). As an example, consider the coordinate limit x = εx′, y = εy′. This induces the contraction εP1 = P ′1, εP2 = P ′2, J = J ′ of e(2,C) and, further, the contraction D = D′, K1 = εK ′1, K2 = εK ′2 of so(4, C). The other contractions of e(2,C) work similarly. For so(3,C) we have the basis J23, J31, J12, where s21 + s22 + s23 = 1, s1ps1 + s2ps2 + s3ps3 = 0. The generators for the conformal symmetry algebra of the so(3,C) Laplace equation are related to the Ljk basis for so(4,C) via (3.1). Now consider the example limit s1 = εx′, s2 = εy′. It induces the contraction εJ23 = −py′ , εJ31 = px′ , J12 = x′py′ − y′px′ 12 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag of so(3,C) to e(2,C) and the contraction L12 = x′py′ − y′px′ = J ′, iεL14 = px′ = P ′1, iεL24 = py′ = P ′2, −2 ε (iL14 + L13) = ( x′ 2 − y′2 ) px′ + 2x′y′py′ +O(ε) = K ′1 +O(ε), −2 ε (iL24 + L23) = ( y′ 2 − x′2 ) py′ + 2x′y′px′ +O(ε) = K ′2 +O(ε), of so(4,C) to itself. The other contractions of so(3,C) work similarly. We now present a general definition of Bôcher contractions of so(4,C) to itself and demon- strate that the above induced contractions can be reformulated as Bôcher contractions. Let x = A(ε)y, and x = (x1, . . . , x4), y = (y1, . . . , y4) be column vectors, and A = (Ajk(ε)), be a 4× 4 matrix with matrix elements Akj(ε) = N∑ `=−N a`kjε `, (4.1) where N is a nonnegative integer and the a`kj are complex constants. (Here, N can be arbitrarily large, but it must be finite in any particular case.) We say that the matrix A defines a Bôcher contraction of the conformal algebra so(4,C) to itself provided 1) det(A) = ±1, constant for all ε 6= 0, (4.2) 2) x · x ≡ 4∑ j=1 xi(ε) 2 = y · y +O(ε). (4.3) If, in addition, A ∈ O(4,C) for all ε 6= 0 the matrix A defines a special Bôcher contraction. For a special Bôcher contraction x · x = y · y, with no error term. We explain why this is a contraction in the generalized Inönü–Wigner sense. Let Lts = xt∂xs − xs∂xt , s 6= t be a generator of so(4,C) and Ã(ε) = A−1(ε) be the matrix inverse. (Note that Ã also has an expansion of the form (4.1) in ε.) We have the expansion Lts = ∑ k,` (AtkÃ`s −AskÃ`t)yk∂y` = εαts (∑ k` Fk` yk∂y` +O(ε) ) , (4.4) where F is a constant nonzero matrix. Thus the integer αts is the smallest power of ε occurring in the expansion of Lts. Now consider the product Lts(x · x). On one hand it is obvious that Lts(x · x) ≡ 0, but on the other hand the expansions (4.3), (4.4) yield Lts(x · x) = εαts (∑ k` Fk` yk∂y` )∑ j y2j +O ( εαts ) . Thus, (∑ k` Fk` yk∂y` )(∑ j y 2 j ) ≡ 0 for F a constant nonzero matrix. However, the only differential operators of the form ∑ k` Fk`yk∂y` that map y · y to zero are elements of so(4,C):∑ k` Fk`yk∂y` = ∑ j>k bjkL ′ jk, L′jk = yj∂yk − yk∂yj . Thus lim ε−→0 ε−αtsLts = ∑ j>k bjkL ′ jk ≡ L′ (4.5) Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 13 and this determines a limit of Lts to L′. Similarly, if we apply this same procedure to the operator L = ∑ t>s c(ε)tsLts for any rational polynomials cts(ε) we will obtain an operator L′ =∑ j>k bjkL ′ jk in the limit. Further, due to condition (4.2), by choosing the c(ε)ts appropriately we can obtain any L′ ∈ so(4,C) in the limit. (Indeed, modulo rational functions of ε, this is just the adjoint action of O(4,C) on so(4,C). In this sense the mapping L→ L′ is onto.) Theorem 4.1. Suppose the matrix A(ε) defines a Bôcher contraction of so(4,C). Let {Ltisi , i = 1, . . . , 6} be an ordered linearly independent for so(4,C) such that αt1s1 ≤ αt2s2 ≤ · · · ≤ αt6s6. Then there is an ordered linearly independent set {Lj , j = 1, . . . , 6} for so(4,C) such that 1) Lj ∈ span{Ltisi , i = 1, . . . , j}, 2) there are integers α1 ≤ α2 ≤ · · · ≤ α6 such that lim ε→0 Lj εαj = L′j , 1 ≤ j ≤ 6, and {L′j , j = 1, . . . , 6} forms a basis for so(4,C) in the yk variables. Proof. The proof is by induction on j. For j = 1 the result follows from (4.5). Assume the assertion is true for j ≤ j0 < 6. Then, due to the nonsingularity condition (4.2), we can always find polynomials in ε, {a1(ε), a2(ε), . . . , aj0(ε)} such that Lj0+1 = Ltj0+1,sj0+1 − j0∑ i=1 aiLi = εαj0+1L′j0+1 +O ( εαj0+2 ) , where L′j0+1 is linearly independent of {L′i, 1 ≤ i ≤ j0} and αj0+1 ≥ αj0 . � In [27] we have used this theorem to compute explicitly the bases for the basic Bôcher contractions. 4.1 Composition of Bôcher contractions Let A and B define Bôcher contractions of so(4,C) to itself. Thus there exist expansions x(ε1) · x(ε1) = y · y +O ( εa1 ) , y(ε2) · y(ε2) = z · z +O ( εb2 ) , where x = A(ε1)y, y(ε2) = B(ε2)z. Now let x(ε1, ε2) = A(ε1)y(ε2) = A(ε1)B(ε2)z. Then x(ε1, ε2) · x(ε1, ε2) = y(ε2) · y(ε2) +Oε2 ( εa1 ) = z · z +O ( εb2 ) + εa1f(ε1, ε2,y). Now set ε1 = εm, ε2 = ε. It follows from these expansions that we can always find an m > 0 such that x ( εm, ε ) · x ( εm, ε ) = z · z +O ( εq ) and lim ε→0 ε−αtsLts = ∑ j>k cjkL ′′ jk ≡ L′′ for some q > 0, with L′′ in the so(4,C) Lie algebra of operators such that L′′(z · z) = 0. Thus this composition of the A and B contractions yields a new Bôcher contraction. For special Bôcher contractions the composition is defined without restriction and the resulting contraction is uniquely determined for ε1, ε2 going to 0 independently. However, if we set ε2 = εm1 , in general the resulting contraction will depend on m. 14 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag 4.2 Special Bôcher contractions Special Bôcher contractions are much easier to understand and manipulate than general Bôcher contractions: composition is merely matrix multiplication. The contractions that arise from the Bôcher recipe are not “special”. However, we shall show that we can associate a special Bôcher contraction with each contraction obtained from Bôcher’s recipe, such that the special contraction contains the same basic geometrical information. The (projective) tetraspherical coordinates are associated with points (x, y) in 2D flat space via the relation (x, y) ≡ (x1, x2, x3, x4) = [x3 + ix4] ( −x,−y, 1 2 ( 1− x2 − y2 ) ,− i 2 ( 1 + x2 + y2 )) . (4.6) In particular, x = − x1 x3 + ix4 , y = − x2 x3 + ix4 , x3 + ix4 x3 − ix4 = −1 x2 + y2 . (4.7) For coordinates on the 2-sphere we have (s1, s2, s3) ≡ (x1, x2, x3, x4) = x4(−is1,−is2, is3, 1). The action of Bôcher contractions on the flat space coordinates (x, y) is an affine mapping and this affine action carries all of the geometrical information about the contraction. For example, the [1, 1, 1, 1] ↓ [2, 1, 1] contraction x3 = − i√ 2 ε x′3 − i√ 2ε x′4, x4 = i√ 2 ( 1 ε − ε ) x′3 − 1√ 2 ( 1 ε + ε ) x′4, and x1 = x′1, x2 = x′2, gives x = − x1 x3 + ix4 = εx′1√ 2(x′3 + ix′4) +O ( ε2 ) = ε′x′ +O ( ε′ 2) , y = ε′y′ +O ( ε′ 2) , for ε′ = ε/ √ (2). Thus the geometric content of the action of this contraction in flat space is x = ε′x′, y = ε′y′. The terms of order ε′2 disappear in the limit. On the complex sphere we have s1 = ix1 x4 = − √ 2iεx′1 x′3 + ix′4 +O ( ε2 ) = ε′x′ +O ( ε′ 2) , s2 = ε′y′ +O ( ε′ 2) , s3 = − ix3 x4 = 1 +O ( ε2 ) , where ε′ = √ 2iε and x′, y′ are flat space coordinates. Thus the geometric content of the action of this contraction on the 2-sphere is s1 = ε′x′, s2 = ε′y′. Note that distinct contractions on flat space and the sphere are induced by the same Bôcher contraction. Using the fact that the contraction limits are completely determined by the geometric limits, we can derive special Bôcher contractions that produce the same geometric limits. We again consider the example discussed above. We will design a special Bôcher contraction with the property x = εx′, y = εy′ such that equations (4.6), (4.7) hold. In this case we require x = x1/(x3 + ix4) = εx′ = ε x′1/(x ′ 3 + ix′4), y = x2/(x3 + ix4) = x′2/(x ′ 3 + ix′4). The solution is, essentially unique up to conformal transformation: x1 = x′1, x3 = x′3(ε+ 1/ε)/2 + ix′4(−ε+ 1/ε)/2, x2 = x′2, x4 = ix′3(ε− 1/ε)/2 + x′4(ε+ 1/ε)/2. Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 15 This contraction satisfies x21 + x22 + x23 + x24 = x′1 2 + x′22 + x′3 2 + x′4 2 and agrees with [1, 1, 1, 1] ↓ [2, 1, 1] on Laplace equations. Similarly we can use each of the geometric contractions of flat space and the 2-sphere as classified in [23], to construct special Bôcher contractions that take V[1,1,1,1] to each of V[2,1,1], V[2,2], V[3,1], V[4]. For example V[1,1,1,1] → V[3,1] : x1 = x′1 + x′3 ε + ix′4 ε , x3 = −x ′ 1 ε + x′3 ( 1− 1 2ε2 ) − ix′4 2ε2 , x2 = x′2, x4 = − ix ′ 1 ε − ix′3 2ε2 + x′4 ( 1 + 1 2ε2 ) . A more general way to construct special Bôcher contractions is to make use of the normal forms for conjugacy classes of so(4,C) under the adjoint action of SO(4,C). They are derived in [9]: C1 =  0 λ 0 0 −λ 0 0 0 0 0 0 0 0 0 0 0  , C2 =  0 λ 0 0 −λ 0 0 0 0 0 0 µ 0 0 −µ 0  , C3 =  0 1 + i 0 0 −1− i 0 −1 + i 0 0 1− i 0 0 0 0 0 0  , C4 = 1 2  0 1 i 2λ −1 0 2λ i −i −2λ 0 −1 −2λ −i 1 0  . Every 1-parameter subgroup A(t) of SO(4,C) (i.e., A(t1 + t2) = A(t1)A(t2)), is conjugate to one of the forms Aj(t) = exp(tCj), j = 1, 2, 3, 4. By making an appropriate change of complex coordinate t = t(ε) we can obtain a special Bôcher contraction matrix A1(t) = 1 2  ε2+1 ε − i(ε2−1) ε 0 0 i(ε2−1) ε ε2+1 ε 0 0 0 0 0 0 0 0 0 0  , ε = eiλt, (4.8) A2(t) = 1 2  ε21+1 ε1 − i(ε21−1) ε1 0 0 i(ε21−1) ε1 ε21+1 ε1 0 0 0 0 ε22+1 ε2 − i(ε22−1) ε2 0 0 i(ε22−1) ε2 ε22+1 ε2  , ε1 = eiλt, ε2 = eiµt, (4.9) A3(t) =  1− 1 2ε2 1 ε i 2ε2 0 −1 ε 1 i ε 0 i 2ε2 − i ε 1 + 1 2ε2 0 0 0 0 1  , ε = 2 t(1 + i) , (4.10) A4(t) = 1 2  ε21+1 ε1 1 ε1ε2 i ε1ε2 i(ε21−1) ε1 − ε1 ε2 ε21+1 ε1 i(ε21−1) ε1 iε1 ε2 − iε1 ε2 i(ε21−1) ε1 ε21+1) ε1 − ε1 ε2 i(ε21−1) ε1 i ε1ε2 1 ε1ε2 ε21+1) ε1  , ε1 = eiλt, ε2 = 1 t . (4.11) The contraction (4.8) takes V[1,1,1,1] to V[2,11], (4.9) takes it to V[2,2], and (4.10) takes it to V[3,1]. The contractions (4.11), on the other hand, takes V[1,1,1,1] to V[2,2] again. Consider though the special case H(ε), of (4.11) where ε1 = 1, ε2 = ε. It, too, maps V[1,1,1,1] to V[2,2], but the 16 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag composition H(ε)H(ε2) takes V[1,1,1,1] to V[4]. (We note that the composition H(ε)H(ε3) takes V[1,1,1,1] to V[3,1], showing that, in general, the result of a composition A(ε1)B(ε2) depends on the relationship between ε1 and ε2.) If the matrix A(ε) defines a general Bôcher contraction, by transposing two rows if necessary, we can assume det A(ε) = 1 for all ε 6= 0. Thus, A(ε) is a curve on SL(4,C). We could use the results of [9] to list all the conjugacy classes of sl(4,C) to attempt a classification. However, it would be necessary to check condition (4.3) in every case, whereas for special Bôcher contractions this condition is satisfied automatically. Both Bôcher’s original recipes and the normal forms given above provide a generating basis for all Bôcher contractions in two dimensions; the general contractions are obtained by composing these generators. 5 Classif ication of free abstract nondegenerate quadratic algebras. Identification of those from free nondegenerate 2nd order superintegrable systems 5.1 Free nondegenerate classical quadratic algebras Recall from Definition 1.1 that the symmetry algebra of a free 2D superintegrable system on a constant curvature space, A, is a quadratic algebra which is completely determined by the function F . More specifically, it is a Poisson algebra generated by three linearly independent elements {L1,L2,H} where H generates the center of A and the structure equations of the algebra are given by (1.2) with R2 = F(H,L1,L2) for some third order homogeneous polynomial F . We call R2, which is the same as F(H,L1,L2), the Casimir of A in terms of {L1,L2,H}. Motivated by the superintegrable case we define an abstract free nondegenerate 2D classical quadratic algebra as follows. Definition 5.1. A free nondegenerate 2D classical quadratic algebra is a Poisson algebra A over C that is generated by {L1,L2,H} where H generates the center of A, {R,L1} = −1 2 ∂R2 ∂L2 , {R,L2} = 1 2 ∂R2 ∂L1 , R = {L1,L2}, and R2 = F(H,L1,L2) for some third order homogeneous polynomial F . Below we shall refer to free nondegenerate 2D classical quadratic algebras simply as (abstract) quadratic algebras. Remark 5.2. As an associative algebra A is the quotient of the free C-algebra generated by {L1,L2,H,R} and its two sided ideal generated by R2 − F . For any choice of a polynomial of degree three for F , the above equations define Lie brackets on A that make it a Poisson algebra, but higher order polynomials will not define Lie brackets on A. For any other generating set L̃1, L̃2, H̃, R̃ of the same Poisson algebra that satisfies: (i) The linear span over C of L̃1, L̃2, H̃ coincides with the linear span of L1, L2, H. (ii) H̃ is in the center of the Poisson algebra, i.e., Poisson commutes with everything. (iii) R̃ = {L̃1, L̃2}. Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 17 (iv) The generators L̃1, L̃2, H̃, R̃ satisfy the structure equations, i.e., {R̃, L̃1} = −1 2 ∂R̃2 ∂L̃2 , {R̃, L̃2} = 1 2 ∂R̃2 ∂L̃1 . It easy to see thatL̃1L̃2 H̃  = A1,1 A1,2 A1,3 A2,1 A2,2 A2,3 0 0 A3,3 L1L2 H  (5.1) for some A = A1,1 A1,2 A1,3 A2,1 A2,2 A2,3 0 0 A3,3  ∈ GL(3,C). (5.2) For a matrix as above we define A2 = ( A1,1 A1,2 A2,1 A2,2 ) ∈ GL(2,C). We denote the group of matrices of the form (5.2) by G, it is a complex algebraic group. Moreover, if R2 = F and R̃2 = F̃ then there is A ∈ G, such that F̃(L̃1, L̃2, H̃) = det(A2) 2F ( A−1 ( L̃1, L̃2, H̃ )) . (5.3) Obviously, two quadratic algebras are isomorphic if and only if their Casimirs are related by A ∈ G via equation (5.3). This fact is fundamental for the classification of quadratic algebras. Let C[3][x1, x2, x3] be the complex algebraic variety of homogeneous polynomials of degree three in the variables x1, x2, x3. The group G acts on C[3][x1, x2, x3] via equation (5.3). Obvi- ously there is a bijection between isomorphism classes of quadratic algebras and orbits of G in C[3][x1, x2, x3]. We will determine all isomorphism classes of quadratic algebras by classifying all orbits of G in C[3][x1, x2, x3]. We shall distinguish an element in each orbit that defines the Canonical form for the Casimir of a given quadratic algebra. Moreover we present an algorithm for finding the canonical form of the Casimir for a given quadratic algebra which gives a practical way to determine if two given quadratic algebras are isomorphic. 5.2 The algorithm for casting the Casimir to its the canonical form In this section we introduce the notation X1 = L1, X2 = L2, X3 = H and similarly, X̃1 = L̃1, X̃2 = L̃2, X̃3 = H̃. For any realization of the Casimir, R2 = F(X1, X2, X3), there are homogeneous polynomials in X1, X2 of order j, F (j), such that F(X1, X2, X3) = F (3)(X1, X2) +X3F (2)(X1, X2) +X3 2F (1)(X1, X2) +X3 3F (0). For any f ∈ C[3][X1, X2, X3] we shall denote the stabilizer of f in G by StabG{f}. We shall use the notation StabG{f + O(H)} for the subgroup of G consisting of all elements that do not change the part in f that is H independent. That is g ∈ StabG{f + O(H)} preserves the lowest order term in f as a polynomial of H = X3. Similarly StabG{f +O(H2)} stands for the subgroup of G consisting of all elements that preserves the part in f that is a polynomial of degree 1 in H. Similarly we define StabG{f +O(H3)}. For a given f ∈ C[3][X1, X2, X3] we shall denote by f (i)(X1, X2) it homogeneous component that are uniquely defined by f(X1, X2, X3) = f (3)(X1, X2) +X3f (2)(X1, X2) +X3 2f (1)(X1, X2) +X3 3f (0). 18 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag Note that StabG { f (3) +O(H) } ⊇ StabG { f (3) +Hf (2) +O ( H2 )} ⊇ StabG { f (3) +Hf (2) +H2f (1) +O ( H3 )} ⊇ StabG{f}. The algorithm for casting R2 = F(X1, X2, X3) into its canonical form is as follows: Step1 Using a certain g1 ∈ G we transform F(X1, X2, X3) to a form in which F (3) is in a canonical form, F (3) c . Step2 Using a certain g2 ∈ StabG{F (3) c + O(H)} we transform F(X1, X2, X3) (that we got in step 1) to a form in which F (3) +HF (2) is in a canonical form F (3) c +HF (2) c . Step3 Using a certain g3 ∈ StabG{F (3) c +HF (2) c + O(H2)} we transform F(X1, X2, X3) (that we got in step 2) to a form in which F (3) +HF (2) +H2F (1) is in a canonical form F (3) c + HF (2) c +H2F (1) c . Step4 Using a certain g4 ∈ StabG{F (3) c +HF (2) c +H2F (1) c +O(H3)} we transform F(X1, X2, X3) (that we got in step 3) to a form in which F (3)+HF (2)+H2F (1)+H3F (0) is in a canonical form F (3) c +HF (2) c +H2F (1) c +H3F (0) c . This is the canonical form of F . At the end of the section we list all possible canonical form of quadratic algebras in a table. 5.2.1 The four cases for F(3) Note that for two presentations of the Casimir of a given quadratic algebra: R2 = F(X1, X2, X3) and R̃2 = F̃(X̃1, X̃2, X̃) that are related by equation (5.1) with A = A1,1 A1,2 0 A2,1 A2,2 0 0 0 1  ∈ GL(3,C) and R̃2 = F̃ (3) ( X̃1, X̃2 ) + X̃3F̃ (2) ( X̃1, X̃2 ) + X̃2 3 F̃ (1) ( X̃1, X̃2 ) + X̃3 3 F̃ (0) we have F̃ (i) ( X̃1, X̃2 ) = det(A2) 2F (i) ( A−12 ( X̃1, X̃2 )) . From this we can deduce the following lemma. Lemma 5.3. Given F ∈ C[3][x1, x2, x3] we can find an explicit matrix A ∈ G such that for F̃ ( L̃1, L̃2, H̃ ) = det(A2) 2F ( A−1 ( L̃1, L̃2, H̃ )) we have F̃ (3)(X̃1, X̃2) = CI(X1, X2), where CI equal to exactly one of the following 0, C1(X1, X2) = X1X2(X1 +X2), C2(X1, X2) = X2 1X2, C3(X1, X2) = X3 1 . Proposition 5.4. StabG(C1 +O(H)) = {( A v 0 c ) \|A ∈ Ω(C1), v ∈ C2, c ∈ C∗ } , Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 19 where Ω(C1) = {( 0 1 1 0 ) , ( 0 1 −1 −1 ) , ( 1 0 0 1 )} ∐{( −1 −1 0 −1 ) , ( 1 0 −1 −1 ) , ( −1 −1 1 0 )} , StabG(C2 +O(H)) =  a 0 v1 0 1 v2 0 0 c  \| v1, v2 ∈ C, a, c ∈ C∗  , StabG(C3 +O(H)) =  d2 0 v1 b d v2 0 0 c  \| b, v1, v2 ∈ C, c, d ∈ C∗  . 5.3 First case: three distinct roots Suppose that F (3)(X1, X2) +HF (2)(X1, X2) = C1(X1, X2) +H ( c5X 2 1 + c6X 2 2 + c7X1X2 ) . Acting with A = 1 0 −c6 0 1 −c5 0 0 1 −1 ∈ StabG(C1(X1, X2)) we get C1(X1, X2) +H ( c5X 2 1 + c6X 2 2 + c7X1X2 ) 7−→ C1(X1, X2) +H ( c′7X1X2 ) +H2(c′8X1 + c′9X2) + c10H3 for some c′7, c ′ 8, c ′ 9, c ′ 10, hence we can assume that the F (3)(X1, X2) +HF (2)(X1, X2) = C1(X1, X2) + c7HX1X2 +O ( H2 ) using a matrix of the form A = 1 0 0 0 1 0 0 0 r  we can further assume that c7 ∈ {0, 1}. For the case of c7 = 0 we obtain the following proposition: Proposition 5.5. The stabilizer of the form F (3)(X1, X2) +HF (2)(X1, X2) +O ( H2 ) = C1(X1, X2) +O ( H2 ) is given by StabG ( C1(X1, X2) +O ( H2 )) = {( A 02 0 c ) \|A ∈ Ω(C1), 02 = 0 ∈ C2, c ∈ C∗ } . 20 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag Proof. It is easy to see that StabG ( C1(X1, X2) +O ( H2 )) ⊇ {( A 02 0 c ) \|A ∈ Ω(C1), 02 = 0 ∈ C2, c ∈ C∗ } . For inclusion in the other direction, let M ∈ StabG(C1(X1, X2)+O(H2)) then obviously M2 has to preserve C1(X1, X2), i.e., M2 ∈ Ω(C1). Hence the matrix ( M−1 ) 1,1 ( M−1 ) 1,2 0( M−1 ) 2,1 ( M−1 ) 2,2 0 0 0 1 M = 1 0 M1,3 0 1 M2,3 0 0 M3,3  as a product of two matrices in the stabilizer StabG(C1(X1, X2)+O(H2)) is also in the stabilizer. The result of the action of this matrix on C1(X1, X2) +O(H2) forces M1,3 = M2,3 = 0. � For the case of c7 = 1 we obtain the following proposition: Proposition 5.6. The stabilizer of the form F (3)(X1, X2) +HF (2)(X1, X2) +O ( H2 ) = C1(X1, X2) +HX1X2 +O ( H2 ) is given by StabG ( C1(X1, X2) +HX1X2 +O ( H2 )) =  1 0 0 0 1 0 0 0 1  . Proof. Following the same reasoning as in the previous proof we easily see that for M ∈ StabG(C1(X1, X2)+HX1X2+O(H2)) we must have M2 = ( 1 0 0 1 ) and then by direct calculation the rest of the proof follows. � 5.3.1 F(3)(X1, X2) = C1(X1, X2) and c7 = 0 Suppose that R2 = F (3)(X1, X2) +HF (2)(X1, X2) +H2F (1)(X1, X2) + c10H3 = C1(X1, X2) +H2(c8X1 + c9X2) + c10H3. Acting with A = α β 0 γ δ 0 0 0 c −1 ∈ StabG(C1(X1, X2) +O(H2)) on R2 we will have R2 = C1(X1, X2) +H2 (c8X1 + c9X2) + c10H3 7−→ C1(X1, X2) +H2(c′8X1 + c′9X2) + c′10H3, where c′8 = c2(αc8 + γc9), c ′ 9 = c2(βc8 + δc9), c ′ 10 = c3c10, and ( α β γ δ ) ∈ Ω(C1). Note that the size of the group Ω(C1) is 6. We now describe an algorithm for choosing a canonical form in this case. If c10 6= 0 then acting with 1 0 0 0 1 0 0 0 (c10) 1 3  we obtain c′10 = 1. Writing ( c′8 c′9 ) = ( reiθ ρeiφ ) with r, ρ ≥ 0 and Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 21 θ, φ ∈ [0, 2π) we choose as our canonical form the expression for c8 and c9 according to the following rules (note that the order is important) first make r is maximal, then θ minimal, then ρ minimal, and finally φ minimal. If c10 = 0 then again we act with A = α β 0 γ δ 0 0 0 1 −1 with( α β γ δ ) ∈ Ω(C1) and choose c8 and c9 as above and then we can act with a matrix of the form1 0 0 0 1 0 0 0 c  to normalize c8 to zero or one. 5.3.2 F(3)(X1, X2) = C1(X1, X2) and c7 = 1 Suppose that R2 = F (3)(X1, X2) +HF (2)(X1, X2) +H2F (1)(X1, X2) + c10H3 = C1(X1, X2) +HX1X2 +H2 (c8X1 + c9X2) + c10H3. (5.4) Since StabG ( C1(X1, X2) +HX1X2 +O ( H2 )) =  1 0 0 0 1 0 0 0 1  then for any c8, c9, c10 ∈ C equation (5.4) defines a canonical form. 5.4 Second case: a double root Suppose that F (3)(X1, X2) +HF (2)(X1, X3) = C2(X1, X2) +H ( c5X 2 1 + c6X 2 2 + c7X1X2 ) . Acting with A = 1 0 −1 2c7 0 1 −c5 0 0 1 −1 ∈ StabG(C2(X1, X2)) on R2 we have C2(X1, X2) +H ( c5X 2 1 + c6X 2 2 + c7X1X2 ) 7−→ C2(X1, X2) +H ( c′6X 2 2 ) +H2(c′8X1 + c′9X2) + c′10H3 for some c′6, c ′ 8, c ′ 9, c ′ 10. Hence we can assume that the F (3)(X1, X2) +HF (2)(X1, X3) = C2(X1, X2) + c6HX2 2 +O ( H2 ) using a matrix of the form A = 1 0 0 0 1 0 0 0 r  we can further assume that c6 ∈ {0, 1}. For the case of c6 = 0 we obtain the following proposition: 22 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag Proposition 5.7. The stabilizer of the form F (3)(X1, X2) +HF (2)(X1, X2) +O ( H2 ) = C2(X1, X2) +O ( H2 ) is given by StabG ( C2(X1, X2) +O ( H2 )) =  a 0 0 0 1 0 0 0 c  \| a, c ∈ C∗  . For the case of c6 = 1 we obtain the following proposition: Proposition 5.8. The stabilizer of the form F (3)(X1, X2) +HF (2)(X1, X2) +O ( H2 ) = C2(X1, X2) +HX2 2 +O ( H2 ) is given by StabG ( C2(X1, X2) +HX2 2 +O ( H2 )) =  r 0 0 0 1 0 0 0 r2  \| r ∈ C∗  . 5.4.1 F(3)(X1, X2) = C2(X1, X2) and c6 = 0 Suppose that R2 = F (3)(X1, X2) +HF (2)(X1, X2) +H2F (1)(X1, X2) + c10H3 = C2(X1, X2) +H2(c8X1 + c9X2) + c10H3. Acting with A = a 0 0 0 1 0 0 0 c −1 ∈ StabG(C2(X1, X2) +O(H2)) on R2 we have R2 = C2(X1, X2) +H2(c8X1 + c9X2) + c10H3 7−→ C2(X1, X2) +H2(c′8X1 + c′9X2) + c′10H3, where c′8 = c2a−1c8, c ′ 9 = c2a−2c9, c ′ 10 = c3a−2c10. For the canonical form, we normalize the first two non zero coefficients from c8, c9, c10 to be equal to 1. 5.4.2 F(3)(X1, X2) = C2(X1, X2) and c6 = 1 Suppose that R2 = C2(X1, X2) +HX2 2 +H2(c8X1 + c9X2) + c10H3. Acting with A = r 0 0 0 1 0 0 0 r2 −1 ∈ StabG(C2(X1, X2) +HX2 2 +O(H2)) on R2 we have R2 = C2(X1, X2) +HX2 2 + +H2(c8X1 + c9X2) + c10H3 7−→ C2(X1, X2) +HX2 2 +H2(c′8X1 + c′9X2) + c′10H3, where c′8 = r3c8, c ′ 9 = r2c9, c ′ 10 = r4c10. We define the canonical form to be with ck = 1, where k is the smallest integer among {8, 9, 10} such that ck 6= 0. Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 23 5.5 Third case: a triple root Suppose that F (3)(X1, X2) +HF (2)(X1, X2) = C3(X1, X2) +H ( c5X 2 1 + c6X 2 2 + c7X1X2 ) . Acting with A = 1 0 −1 3c5 0 1 0 0 0 1 −1 ∈ StabG(C3(X1, X2)) on R2 we have C3(X1, X2) +H ( c5X 2 1 + c6X 2 2 + c7X1X2 ) 7−→ C3(X1, X2) +H ( c′6X 2 2 + c′7X1X2 ) +H2(c′8X1 + c′9X2) + c′10H3 for some c′6, c ′ 7, c ′ 8, c ′ 9, c ′ 10. Hence we can assume that the F (3)(X1, X2) +HF (2)(X1, X3) = C3(X1, X2) + c6HX2 2 + c7HX1X2 using a matrix of the form A = d2 0 0 0 d 0 0 0 r  we can further assume that c6, c7 ∈ {0, 1}. For the case of c6 = c7 = 0 we obtain the following proposition: Proposition 5.9. The stabilizer of the form F (3)(X1, X2) +HF (2)(X1, X3) +O ( H2 ) = C3(X1, X2) +O ( H2 ) is given by StabG ( C3(X1, X2) +O ( H2 )) =  d2 0 0 γ d b 0 0 c  \| b, γ ∈ C, d, c ∈ C∗  . For the case of c6 = 0, c7 = 1 we obtain the following proposition: Proposition 5.10. The stabilizer of the form F (3)(X1, X2) +HF (2)(X1, X3) +O ( H2 ) = C3(X1, X2) +HX1X2 +O ( H2 ) is given by StabG(C3(X1, X2) +HX1X2 +O ( H2) ) =   d2 0 a −3a d d b 0 0 d3  \| a, b ∈ C, d ∈ C∗  . For the case of c6 = 1, c7 = 0 we obtain the following proposition: Proposition 5.11. The stabilizer of the form F (3)(X1, X2) +HF (2)(X1, X3) +O ( H2 ) = C3(X1, X2) +HX2 2 +O ( H2 ) is given by StabG ( C3(X1, X2) +HX2 2 +O ( H2 )) =  d2 0 0 0 d b 0 0 d4  \| b ∈ C, d ∈ C∗  . 24 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag For the case of c6 = 1, c7 = 1 we obtain the following proposition: Proposition 5.12. The stabilizer of the form F (3)(X1, X2) +HF (2)(X1, X3) = C3(X1, X2) +HX2 2 +HX1X2 +O ( H2 ) is given by StabG ( C3(X1, X2) +HX1X2 +HX2 2 +O ( H2 )) =   d2 0 d2 12(d2 − 1) 1 2d(1− d) d b 0 0 d4  \| b ∈ C, d ∈ C∗  . 5.5.1 F(3)(X1, X2) = C3(X1, X2), c6 = 0, and c7 = 0 Suppose that R2 = F (3)(X1, X2) +HF (2)(X1, X2) +H2F (1)(X1, X2) + c10H3 = C3(X1, X2) +H2 (c8X1 + c9X2) + c10H3. Acting with A = d2 0 0 γ d b 0 0 c −1 ∈ StabG(C3(X1, X2) +O(H2)) on R2 we have R2 = C3(X1, X2) +H2(c8X1 + c9X2) + c10H3 7−→ C3(X1, X2) +H2(c′8X1 + c′9X2) + c′10H3, where c′8 = c2(d−4c8 + d−6γc9), c ′ 9 = c2d−5c9, c ′ 10 = d−6(c2bc9 + c3c10). If c9 = 0 and c8 6= 0 we define the canonical form to be with c8 = 1 and c10 = reiθ with r ≥ 0 and θ ∈ [0, π). If c9 = 0 and c8 = 0 we define the canonical form to be with c10 ∈ {0, 1}. If c9 6= 0 then the canonical form is given by R2 = C3(X1, X2) +H2X2. 5.5.2 F(3)(X1, X2) = C3(X1, X2), c6 = 0, and c7 = 1 Suppose that R2 = F (3)(X1, X2) +HF (2)(X1, X2) +H2F (1)(X1, X2) + c10H3 = C3(X1, X2) +HX1X2 +H2(c8X1 + c9X2) + c10H3. Acting with A =  d2 0 a −3a d d b 0 0 d3 −1 ∈ StabG(C3(X1, X2) +HX1X2 +O(H2)) on R2 we have R2 = C3(X1, X2) +HX1X2 +H2(c8X1 + c9X2) + c10H3 7−→ C3(X1, X2) +HX1X2 +H2(c′8X1 + c′9X2) + c′10H3, where c′8 = b d + d2c8 − 3adc9, c ′ 9 = a d2 + c9d, c′10 = a3 d6 + ab d3 + ac8 + bc9 + d3c10. Hence we can always arrange that c8 = c9 = 0 and c10 ∈ {0, 1} and this will be the canonical form in this case. Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 25 5.5.3 F(3)(X1, X2) = C3(X1, X2), c6 = 1, and c7 = 0 Suppose that R2 = F (3)(X1, X2) +HF (2)(X1, X2) +H2F (1)(X1, X2) + c10H3 = C3(X1, X2) +HX2 2 +H2(c8X1 + c9X2) + c10H3. Acting with A = d2 0 0 0 d b 0 0 d4 −1 ∈ StabG(C3(X1, X2) +HX2 2 +O(H2)) on R2 we have R2 = C3(X1, X2) +HX2 2 +H2(c8X1 + c9X2) + c10H3 7−→ C3(X1, X2) +HX2 2 +H2(c′8X1 + c′9X2) + c′10H3, where c′8 = d4c8, c ′ 9 = 2 bd + c9d 3, c′10 = b2 d2 + d2bc9 + d6c10. Hence we can always arrange that c9 = 0 and either c8 = 0 and c10 ∈ {0, 1} or c8 = 1 and c10 = reiθ with r ≥ 0 and θ ∈ [0, π2 ). 5.5.4 F(3)(X1, X2) = C3(X1, X2), c6 = 1, and c7 = 1 Suppose that R2 = F (3)(X1, X2) +HF (2)(X1, X2) +H2F (1)(X1, X2) + c10H3 = C3(X1, X2) +HX1X2 +HX2 2 +H2(c8X1 + c9X2) + c10H3. Acting with A =  d2 0 d2 12(d2 − 1) 1 2d(1− d) d b 0 0 d4 −1 ∈ StabG(C3(X1, X2) + HX1X2 + HX2 2 + O(H2)) on R2 we have R2 = C3(X1, X2) +HX1X2 +HX2 2 +H2(c8X1 + c9X2) + c10H3 7−→ C3(X1, X2) +HX1X2 +HX2 2 +H2(c′8X1 + c′9X2) + c′10H3, where c′8 = b d − 1 48(d2 − 1)(d − 1)2 + d4c8 + 1 2d 3(1 − d)c9, c ′ 9 = 1 12d 3(d2 − 1) + 2 bd + c9d 3, c′10 = 1 123 (d2 − 1)3 + 1 12(d2 − 1)b + b2 d2 + 1 12d 4(d2 − 1)c8 + d2bc9 + d6c10. Hence we can assume that c9 = c8 = 0 and the canonical form is given by R2 = X3 1 +HX1X2 +HX2 2 + c10H3 with c10 ∈ C. 5.6 Fourth case: F̃ (3) = 0 A similar (but simpler) calculation to the one that was done in the previous section leads to the possibilities for the canonical forms for F ∈ C[3][x1, x2, x3] with a vanishing F (3). For example it easy to show the following lemma. Lemma 5.13. Given F ∈ C[3][x1, x2, x3] with a vanishing F (3) we can find an explicit matrix A ∈ G such that the F (2) part of A · F is equal to exactly one of the following three cases: X2 1 , X1X2, 0. 26 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag Table 1. List of canonical forms of R2 for the nondegenerate free quadratic algebras. Canonical forms of R2 for the nondegenerate free quadratic algebras R2 domain of parameters 1a X1X2(X1 +X2) + c8X1H2 + c9X2H2 +H3 c8, c9 ∈ C, see remark below 1b X1X2(X1 +X2) +X1H2 + c9X2H2 c9 ∈ C, see remark below 1c X1X2(X1 +X2) 1d X1X2(X1 +X2) +HX1X2 + c8X1H2 + c9X2H2 + c10H3 c8, c9, c10 ∈ C 2a X2 1X2 +X1H2 +X2H2 + c10H3 c10 ∈ C 2b X2 1X2 + c9X2H2 + c10H3 c9, c10 ∈ {0, 1} 2c X2 1X2 +HX2 2 +X1H2 + c9X2H2 + c10H3 c9, c10 ∈ C 2d X2 1X2 +HX2 2 +X2H2 + c10H3 c10 ∈ C 2e X2 1X2 +HX2 2 + c10H3 c10 ∈ {0, 1} 3a X3 1 +X1H2 + c10H3 c10 ∈ C 3b X3 1 +H3 3c X3 1 +X2H2 3d X3 1 +HX1X2 + c10H3 c10 ∈ {0, 1} 3e X3 1 +HX2 2 + c10H3 c10 ∈ {0, 1} 3f X3 1 +HX2 2 +X1H2 + reiθH3 r ≥ 0, θ ∈ [0, π 2 ) 3g X3 1 +HX1X2 +HX2 2 + c10H3 c10 ∈ C 4a HX2 1 +H2X2 4b HX2 1 +H2X1 + c10H3 c10 ∈ C 4c HX2 1 + c10H3 c10 ∈ {0, 1} 4d HX1X2 +H2(X1 +X2) + c10H3 c10 ∈ C 4e HX1X2 + c8H2X1 + c10H3 c8, c10 ∈ {0, 1} 4f H2X1 4g c10H3 c10 ∈ {0, 1} Remark 5.14. For each value of the parameter in the first two lines of Table 1 if c′8 = c2(αc8 + γc9), c′9 = c2(βc8 + δc9), c′10 = c3c10, for c ∈ C∗ and( α β γ δ ) ∈ Ω(C1) = {( 0 1 1 0 ) , ( 0 1 −1 −1 ) , ( 1 0 0 1 )} ∐{( −1 −1 0 −1 ) , ( 1 0 −1 −1 ) , ( −1 −1 1 0 )} then the system with parameters c8, c9, c10 isomorphic to the one with c′8, c ′ 9, c ′ 10. 5.7 Comparison of geometric and abstract nondegenerate quadratic algebras There is a close relationship between the canonical forms of abstract quadratic algebras and Stäckel equivalence classes of nondegenerate superintegrable systems. To demonstrate this we treat one example in detail. The superintegrable system S9, with nondegenerate potential, can be defined by R2 = L21L2 + L1L22 + L1L2(H− a4)− a2(H− a4)2 − 2a2L1(H− a4) − 2a2L2(H− a4)− (a3 + a2)L21 − (a3 + 3a2 + a1)L1L2 − (a2 + a1)L22 + ( 2a2a3 + 2a22 + 2a1a2 ) (H− a4) + 2 ( a22 + a2a3 + a1a2 ) L1 + 2 ( a22 + a2a3 + a1a2 ) L2 + 2a1a2a3 − 2a1a 2 2 − 2a22a3 − a2a23 − a2a21 − a32, Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 27 where the aj are the parameters in the potential. To perform a general Stäckel transform of this system with nonsingular transform matrix C = (cjk): 1) we set aj = 4∑ k=1 cjkbk, k = 1, . . . , 4 where the bk are the new parameters, 2) we make the replacements H → −b4, b4 → −H and 3) we then set all parameters bj = 0 to determine the free quadratic algebra. The result is R2 = c24 ( c214 + 2c14c24 − 2c14c34 + 2c14c44 + c224 + 2c24c34 + 2c24c44 + c234 + 2c34c44 + c244 ) H3 + (2c24(c14 + c24 + c34 + c44)L1 + 2c24(c14 + c24 + c34 + c44)L2)H2 + (c24 + c34)L21H+ (c14 + c24)L22H+ (c14 + 3c24 + c34 + c44)L2L1H + L21L2 + L1L22. We put this in canonical form by making the choices L1 = X1 + (c24 + c14)H, L2 = X2 + (c34 + c24)H. The final result is [1111] : R2 = X2 1X2 +X1X 2 2 +A1X1H2 +A2X2H2 +A3X1X2H+A4H3, where A1 = (c24 − c34)(c14 + c44), A2 = (c34 + c44)(c24 − c14), A3 = −c14 − c24 − c34 + c44, A4 = (c14 − c24 + c34 + c44)(c14c34 + c24c44). The possible canonical forms in Table 1 associated with the equivalence class [1111] depend on the possible choices of cij with detC 6= 0. The possible canonical forms are 1a, 1b, 1d all cases. The superintegrable system E1, with nondegenerate potential, can be defined by R2 = L1L2(H− a4) + L22L1 − a3(H− a4)2 − 2a3L2(H− a4) − (a3 + a2)L22 − a1L21 + 4a1a2a3. Going through the same procedure as above, we obtain the equivalence class [211] : R2 = −X2X 2 1 + ( 2c14c24 + 2c14c34 + 1 4c 2 44 ) X2H2 + c44(−c34 + c24)X1H2 + c14X 2 2H ( −2c14c24c34 + c14c 2 24 + c14c 2 34 + 1 2c 2 44c24 + 1 2c34c 2 44 ) H3. The canonical forms associated with this equivalence class are 2a, 2b, 2c, 2d, 2e, all cases. The superintegrable system E8, with nondegenerate potential, can be defined by R2 = L22L1 − a2(H− a4)L2 + 4a1a3L1 + a1(H− a4)2 − a3a22. The equivalence class is [22] : R2 = X2 1X2 − c24c44X1H2 + 4c14c34X2H2 + ( −c14c244 + c34c 2 24 ) H3. The canonical form associated with this equivalence class is 2a: all cases. The superintegrable system E2 can be defined by R2 = L31 + L1H2 − 2L21H+ (−2a4L1 − a2L2)H+ 2a4L21 + ( a2L2 + 4a1a3 + a24 ) L1 + 4a1L22 + a2a4L2 − 1 4a 2 2a3. The equivalence class is [31] : R2 = X3 1 + ( c14X1X2 − 4c34X 2 2 + c44X 2 1 ) H+ 4c34c24X1H2 + 1 4c24 ( c214 + 16c34c44 ) H3. 28 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag The canonical forms associated with this equivalence class are 3d: all cases, 3e: c10 = 0, 3f : all cases, 3g: c10 = 0. The superintegrable system E10 can be defined by R2 = L31 + 2a1L21 − a3L1L2 + a3(H− a4)2 + 2a2L1(H− a4) + 2a1a2(H− a4) + a21L1 + a22L2. The equivalence class contains [4] : R2 = X3 1 + c34X1X2H+ ( c224 + 2 3 c14c34 ) X2H2 + 1 27 ( 8c314c34 + 9c214c 2 24 + 54c14c24c34c44 + 54c324c44 − 27c234c 2 44 ) c34 H3, if c34 6= 0. If c34 = 0, c24 6= 0 it contains [4]′ : R2 = X3 1 − 2c214X 2 1H+ c224X2H2 + 2c14c24c44H3, and if c34 = c24 = 0 it contains [4]′′ : R2 = X3 1 + c214X1H2 − 2c14X 2 1H. The canonical form associated with [4] is 3d all cases. The canonical form associated with [4]′ is 3c: all cases, and the canonical form associated with [4]′′ is 3a: c10 6= 0. The superintegrable system E3′ can be defined by R2 = −4a1 ( L21 + L22 − L2H ) − 2a2a3L1 + ( a22 − a23 − 4a1a4 ) L2 − a23a4 + a23H. The canonical form is [0] : R2 = 4c14 ( X2 1 +X2 2 ) H− ( 4c14c44 − c224 − c234 )2 16c14 H3, if c14 6= 0; if c14 = 0 it is [0]′ : R2 = −2c24c34X1H2 + ( c224 − c234 ) X2H2 + c234c44H3. The canonical forms associated with [0] are 4d: all cases, 4e: all cases, and the canonical forms associated with [0]′ are 4f : all cases. Heisenberg systems. In addition there are systems that can be obtained from the geometric systems above by contractions from so(4,C) to e(3,C). These are not Bôcher contractions and the contracted systems are not superintegrable, because the Hamiltonians become singular. However, they do form quadratic algebras and many have the interpretation of time-dependent Schrödinger equations in 2D spacetime, so we also consider them geometrical. Some of these were classified in [23] where they were called Heisenberg systems since they appeared in quadratic algebras formed from 2nd order elements in the Heisenberg algebra with generators M1 = px, M2 = xpy, E = py, where E2 = H. The systems are all of type 4. We will devote a future paper to their study. The ones classified so far are 4a: all cases, 4c: c10 = 0, 4e: c10 = 0, 4f : all cases, 4g: all cases. All these results relating geometric systems to abstract systems are summarized in Table 2. Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 29 Table 2. Matching of geometric with abstract quadratic algebras. Class Canonical form 1 a: all cases b: all cases c: no 1 d: all cases 2 a: all cases b all cases c: all cases 2 d: all cases e: all cases 3 a: c10 6= 0 b: no c: all cases 3 d: all cases e: c10 = 0 f : all cases 3 g: c10 = 0 4 a: all cases b: no c: c10 = 0 4 d: all cases e: all cases f : all cases 4 4g: all cases 6 The quadratic algebras of the free 2D second order superintegrable systems In this section we list all canonical forms of the Casimirs of the quadratic algebras of free nondegenerate 2D superintegrable systems on a constant curvature space or a Darboux space. We list the canonical forms arising from superintegrable systems on a constant curvature spaces in Table 3 and those arising from superintegrable systems on a Darboux space in Table 4. In the next section we study contractions between these quadratic algebras. Table 3. Canonical forms of the Casimirs of quadratic algebras of free nondegenerate 2D superintegrable systems that lie inside U(so(3,C)) and U(e(2,C)). System Canonical forms of R2 Ẽ17 L2 1L2 Ẽ16 L2 1L2 +HL2 2 Ẽ1 L2 1L2 +H2L2 Ẽ8 L2 1L2 Ẽ′3 0 Ẽ2 L3 1 +H2L1 + 2i 3 √ 3 H3 Ẽ7 L2 1L2, ∀ a Ẽ9 L3 1 +H2L1 + 2i 3 √ 3 H3 Ẽ11 H2L1 Ẽ10 L3 1 Ẽ15 L3 1 Ẽ20 HL1L2 Ẽ19 L2 1L2 +H2L2 S̃9 L1L2(L1 + L2) +HL1L2 S̃4 L2 1L2 S̃7 L1L2(L1 + L2) +HL1L2 − 1 4 H2L1 − 1 4 H2L2 − 1 4 H3 S̃8 L1L2(L1 + L2) +HL1L2 S̃2 L2 1L2 S̃1 L3 1 30 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag Table 4. Canonical forms of the Casimirs of quadratic algebras of free nondegenerate 2D Darboux superintegrable systems. System Canonical forms of R2 D̃1A, b = 0 L3 1 +HL1L2 D̃1A, b 6= 0 L3 1 +HL1L2 +H3 D̃1B L3 1 +HL1L2 D̃1C H2L1 D̃2A L3 1 +H2L1 + 2i 3 √ 3 H3 D̃2B L2 1L2 +H2L1 +H2L2 + iH3 D̃2C L2 1L2 +HL2 2 +H2L2 D̃3A HL1L2 +H3 D̃3B L2 1L2 +HL2 2 +H2L2 D̃3C L2 1L2 +HL2 2 +H2L2 D̃3D L2 1L2 +HL2 1 +HL2 2 + i3 √ 2H3 D̃4A L2 1L2 D̃4(b)B, b 6= 0 L1L2(L1 + L2) +HL1L2 + b2−4 4b2 H2L1 D̃4(b)B, b = 0 L1L2(L1 + L2) +H2L1 D̃4(b)C, b 6= 0 L1L2(L1 + L2) +HL1L2 + 1 b2 H2L1 D̃4(b)C, b = 0 L1L2(L1 + L2) +H2L1 7 Abstract contractions of nondegenerate quadratic algebras arising from 2D second order superintegrable systems on constant curvature spaces and Darboux spaces We first recall the definition of contraction of quadratic algebras. Definition 7.1. Let A and A0 be quadratic algebras with generating sets {H,L1,L2} and {H0,L01,L02} respectively, satisfying the conditions of Definition 5.1. Let F(H,L1,L2) be the realization of the Casimir of A in the generating set {H,L1,L2} and similarly F0(H0,L01,L02) the Casimir of A0 in the generating set {H0,L01,L02}. We say that A0 is a contraction of A if there is a continuous curve (0, 1] −→ G, ε 7−→ A(ε) = A1,1(ε) A1,2(ε) A1,3(ε) A2,1(ε) A2,2(ε) A2,3(ε) 0 0 A3,3(ε)  such that lim ε−→0+ A(ε) · F (X1, X2, X3) = F 0(X1, X2, X3). Note that the action of G is defined in (5.3). Note that if A0 is a contraction of A then A0 is in the closure of the orbit of G that contains A. 7.1 Contractions of quadratic algebras In this section we study contractions between the quadratic algebras that arise from free nonde- generate 2D second order superintegrable system on a constant curvature space or a Darboux space. As we shall see below there are essentially 18 relevant quadratic algebras for classification purposes. For any two such quadratic algebras one can ask weather there is a contraction from Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 31 one to the other. In principal there are 324 = 182 cases to consider. We have studied most of these cases but our results do not give a complete classification. We discus our results in more details below. We shall give several contractions explicitly and write all those contractions that we were able to find in a diagram. At the end of this section we shall compare abstract contractions with Bôcher contractions. 7.1.1 The relevant quadratic algebras We first note that some quadratic algebras of different superintegrable systems coincide: 1) L1L2(L1 + L2) +HL1L2: S̃8, S̃9 , D̃4(b = ±2)C, 2) L1L2(L1 + L2) +H2L1: D̃4(b = 0)B, D̃4(b = 0)C, 3) L1L2(L1 + L2) +HL1L2 + γH2L1: D̃4(γ = b−2)B, D̃4(γ = b2−4 4b2 )C, 4) L21L2 +HL22 +H2L2: D̃2C, D̃3B, D̃3C, 5) L21L2: Ẽ17,Ẽ8, S̃2, S̃4, Ẽ7, D̃4A, 6) L21L2 +H2L2: Ẽ1, Ẽ19, 7) L31: Ẽ10, Ẽ15, S̃1, 8) L31 +H2L1 + i 2 3 √ 3 H3: Ẽ2, Ẽ9, D̃2A, 9) L31 +HL1L2: D̃1A(b = 0), D̃1B, 10) H2L1: Ẽ11, D̃1C. Hence it is enough to consider the eighteen quadratic algebras: Ẽ17, Ẽ16, Ẽ1, Ẽ′3, Ẽ2, Ẽ11, Ẽ10, Ẽ20, S̃9, S̃7, D̃4C (b 6= 0), D̃4C (b = 0), D̃2B, D̃2C, D̃1A (b 6= 0), D̃1A (b = 0), D̃3A, D̃3D. We divide the quadratic algebras into four sets according to the highest non-vanishing F (i) term in the decomposition R2 = F(H,L1,L2) = F (3)(L1,L2) +HF (2)(L1,L2) +H2F (1)(L1,L2) +H3F (0). Explicitly we define • subset A: F (3) 6= 0: Ẽ17, Ẽ16, Ẽ1, Ẽ2, Ẽ10, S̃9, S̃7, D̃4C (b 6= 0), D̃4C (b = 0), D̃2B, D̃2C, D̃1A (b 6= 0), D̃1A (b = 0), D̃3D, • subset B: F (3) = 0, F (2) 6= 0: Ẽ20, D̃3A, • subset C: F (3) = F (2) = 0, F (1) 6= 0: Ẽ11, • subset D: F (3) = F (2) = F (1) = 0: Ẽ′3. Since F (3) is a homogeneous polynomial of degree three in two variables, it has exactly three roots (zeros) on CP1 counting multiplicities. We divide subset A according to the number of different roots of F (3) as follows • three distinct roots, subset A1: S̃9, S̃7, D̃4C (b 6= 0), D̃4C (b = 0), • a repeated root, subset A2: Ẽ17, Ẽ16, Ẽ1, D̃2B, D̃2C, D̃3D, • a triple root, subset A3: Ẽ2, Ẽ10, D̃1A (b 6= 0), D̃1A (b = 0). 32 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag 7.1.2 Some general observations on contractions of quadratic algebras Note that the group G =  A1,1 A1,2 A1,3 A2,1 A2,2 A2,3 0 0 A3,3  ∈ GL(3,C)  is a complex algebraic group. The formula (A · F) (x1, x2, x3) = det(A2) 2F ( A−1(x1, x2, x3) ) defines an algebraic action of G on the complex algebraic variety C[3][x1, x2, x3], of homogeneous polynomials of degree three in three variables. It is well known (see, e.g., [2, Section 1.8]) that any orbit is an algebraic variety and the boundary of any orbit is also an algebraic variety of a smaller dimension. From this consideration it is clear that if O1 and O2 are two orbits such that O2 ⊂ O1 \O1 then O1 * O2. This imply that we have a partial order by inclusion of orbit closure. In our language this implies that if a quadratic algebra B is a contraction of a quadratic algebra A and A and B are not isomorphic then A is not a contraction of B. Hence for any contraction of quadratic algebras between non isomorphic ones we automatically get a proof of the nonexistence of a contraction in the opposite direction. Furthermore, under the action of G on C[3][x1, x2, x3] the sets A, A1, A2, A3, B, C, D are stable and hence consists of a union of orbits. It is easy to see that the hierarchy of the orbits allow us to consider contractions only in the following direction A1 −→ A2 −→ A3 −→ B −→ C −→ D. We further note that every quadratic algebra can be contracted to Ẽ′3 and Ẽ′3 can not be con- tracted further, hence we we shall ignore this system. In the rest of this section we realize many contraction of quadratic algebras and demonstrate how one can prove that some contractions do not exist. At the end of the section we summarize our results in a diagram. 7.2 Explicit contractions Using matrices of the form A(ε) = 1 0 0 0 1 0 0 0 ε −1 , A(ε) = 1 0 0 0 1 0 0 0 ε −1 , A(ε) =  1 0 0 ε−2 ε−1 0 0 0 ε−3 −1 , A(ε) = ε−2 ε−1/ √ 2 0 ε−2 −ε−1/ √ 2 0 0 0 1 −1 , A(ε) = ε−1 0 0 0 1 0 0 0 ε−3 −1 we can (respectively) realize contractions of the following forms: L21L2 +O(H) −→ L21L2 : D3D, D2C , D2B, E16, E1 −→ E17, L31 +O(H) −→ L31 : D1A, D1A, E2 −→ E10, L21L2 +O(H) −→ L31 : D3D, D2C , D2B, E16, E1, E17 −→ E10, L1L2(L1 + L2) +O(H) −→ L31 : S9, S7, D4C , D4C −→ E10, L1L2(L1 + L2) +O(H) −→ L21L2 : S9, S7, D4C , D4C −→ E17. Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 33 To get an idea of the type of contractions that exist, below we list realizations of all other abstract contractions of S9 that we have found. Contraction of S9 to E20: A(ε) = ε 0 0 0 ε 0 0 0 ε2 . Contraction of S9 to E1: A(ε) = ε−1 0 −iε−1 0 1 0 0 0 2iε−1 −1. Contraction of S9 to E11: A(ε) = ε−1 0 0 0 ε−1 −ε−3/2 0 0 1 −1. Contraction of S9 to E2 : A(ε) = 64ε2 64ε2 64ε2 + i128√ 3 ε i8ε −i8ε 0 0 0 −i128 √ 3ε . 7.3 Non-contractions Here we demonstrate how one can show that there are some quadratic algebras that can not be contracted to some others. Non-contraction of E10 to E11. Under a transformation of the formL1L2 H  = α(ε) β(ε) a(ε) γ(ε) δ(ε) b(ε) 0 0 c(ε) Lε1Lε2 Hε  = A Lε1Lε2 Hε  . We let (αδ − βγ) = \|A\| and we denote the coefficient of Li1L j 2Lk3 in the transformed expression for R2 by Ci,j,k. Then we see that C3,0,0 = α3 A2 −→ 0, C2,0,1 = 3a2α A2 −→ 1, C0,0,3 = a3 A2 −→ 0, which imply that α a −→ 0, a α −→ 0, which is a contradiction. All abstract contractions relating free constant curvature and Darboux quadratic algebras are listed in Diagram 1. There is an abstract contraction of Q(A) to Q(B) if and only if there is an arrow in the diagram pointing from A to B. 7.4 Comparison between abstract contractions and Bôcher contractions In this section we compare abstract contractions and Bôcher contractions. In previous sections we studied abstract contractions between the quadratic algebras of the free 2D nondegenerate second order superintegrable systems: Ẽ17, Ẽ16, Ẽ1, Ẽ ′ 3, Ẽ2, Ẽ11, Ẽ10, Ẽ20, S̃9, S̃7, D̃4C (b 6= 0), D̃4C (b = 0), D̃2B, D̃2C, D̃1A (b 6= 0), D̃1A (b = 0), D̃3A, D̃3D. By abuse of notation we denoted a superintegrable system and its corresponding free quadratic algebra by the same symbol (one of those 18 options above). It should be noted that different superintegrable systems may have the same free quadratic algebra, as was shown in Section 7.1.1. For this section we shall use the symbol S̃9 to denote the superintegrable system on the complex two sphere and use the symbol Q(S̃9) to denote the free quadratic algebra of S̃9. Similar conventions will be used for all other systems. For example, Q ( Ẽ17 ) = Q ( Ẽ8 ) = Q ( S̃2 ) = Q ( S̃4 ) = Q ( Ẽ7 ) = Q ( D̃4A ) . 34 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag S9 �� -- ## (( �� D4C(b 6= 0) �� D4C(b = 0) �� S7 rr {{ �� D3D (( ,, D2C ++ // "" )) �� E16 && �� ## �� D2B �� xx �� E1 ss }} �� xx �� E17 �� && D1A(b 6= 0) && ,, .. �� D1A(b = 0) && �� E2 xx �� E10 D3A // ,, E20 ++ E11 Diagram 1. Abstract contractions relating free nondegenerate 2D quadratic algebras. As we just observed superintegrable systems that share the same free quadratic algebra can still live on different manifolds. Note that in general superintegrable systems with identical free quadratic algebras are not even related by a Stäckel transform. In the above mentioned cases, Ẽ17, Ẽ8, and Ẽ7 belong to the same Stäckel equivalence class which is not the Stäckel equivalence class of the (Stäckel equivalent) systems S̃2, S̃4, and D̃4A. Since the classification of abstract contractions of abstract quadratic algebras is not complete we cannot simply compare Bôcher contractions and abstract contractions of quadratic algebras. Instead we are led to ask the following. Question. Let A and B be 2D second order nondegenerate superintegrable systems. Suppose that there is a contraction of free abstract quadratic algebras Q(A) −→ Q(B). Are there necessarily superintegrable systems A′ and B′ such that 1) Q(A) = Q(A′), Q(B) = Q(B′), 2) there is a Bôcher contraction from A′ to B′. The answer is no. Indeed the following 7 abstract contractions have no geometric counterpart as Bôcher contractions: 1) Q(S7)→ Q(E16), 2) Q(D4C) = Q(D4B)→ Q(E20), 3) Q(D2C) = Q(D3B) = Q(D3C)→ Q(E16), 4) Q(E16)→ Q(E20), 5) Q(E17) = Q(E8) = Q(S2) = Q(S4) = Q(E7) = Q(D4A)→ Q(E20), 6) Q(D1A)→ Q(D3A), 7) Q(D3A)→ Q(E20). These contractions are indicated in Diagram 1. In [27, Table 1] all Bôcher contractions of these systems are given. In these cases there is no chain of Bôcher contractions linking any of the Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 35 S9 E1 E1 E1 E8 E8 E3' Racah Dual HahnHahn Jacobi Jacobi Bessel Krawtchouk Meixner-Pollaczek Pseudo Jacobi Jacobi Jacobi Continuous Dual HahnContinuous Hahn Wilson Superintegrable system Finite dimensional Infinite dimensional Contraction description of the top half of the Askey Scheme This is the part of the scheme related to contractions of nondegenerate systems. The bottom half corresponds to restrictions of nondegenerate to degenerate systems, contractions of degenerate systems and contractions to Heisenberg (singular) systems. On the left side are the orthogonal polynomials that realize finite dimensional representations of the quadratic algebras and on the right those that realize infinite dimensional bounded below representations. Note that some of the contractions go from a superintegrable system to itself in a nontrivial Manner. We did not explicitly mention these in our classification since they are so numerous, but they are pointed out in references [4] and [16 ]. All of the contractions of the quadratic algebra representations are induced by geometric contractions of the corresponding superintegrale systems except for the 2 on the left and 2 on the right with the longest arrows, contractions of E1 to E3'. The limits of Hahn and dual Hahn polynomials to Krawtchouk polynomials and continuous Hahn and dual Hahn polynomials to Meixner-Pollaczek polynomials are abstract contractions of E1 to E3' not induced by geometric contractions. Figure 1. Contractions of nondegenerate systems and the top half of the Askey scheme. origin systems to the target system. However, there are ways that these abstract contractions can have practical significance. In the paper [32] Post shows that the structure equations for all of the quantum 2D quadratic algebras can be represented by either differential or difference operators depending on one complex variable. In some cases a model of one quadratic algebra contracts to a model of another quadratic algebra, even though there is no geometrical counterpart. An example of this can be found in [25] where the Askey scheme is described through contraction of a difference operator model of S9 to differential and difference operator models of other quadratic algebras, see Fig. 1. This is the part of the scheme related to contractions of nondegenerate systems, the top half. The bottom half corresponds to restrictions of nondegenerate to degenerate systems, contractions of degenerate systems and contractions to Heisenberg (singular) systems. On the left side are the orthogonal polynomials that realize finite-dimensional representations of the quadratic algebras and on the right those that realize infinite-dimensional bounded below representations. Note that some of the contractions go from a superintegrable system to itself in a nontrivial man- ner. We did not explicitly mention these in our classification since they are so numerous, but they are pointed out in references [27] and [23]. All of the contractions of the quadratic alge- bra representations are induced by geometric contractions of the corresponding superintegrable systems except for the 2 on the left and 2 on the right with the longest arrows, contractions of E1 to E3′. The limits of Hahn and dual Hahn polynomials to Krawtchouk polynomials and continuous Hahn and dual Hahn polynomials to Meixner–Pollaczek polynomials are ab- stract contractions of E1 to E3′ not induced by geometric contractions. This is an example of how abstract quadratic algebra contractions can be realized and shown to have practical significance. 7.5 Contractions between geometric quadratic algebras and abstract quadratic algebras In Section 5.7 we identified the canonical forms of the geometric quadratic algebras inside the space of all canonical forms of abstract quadratic algebras. In this section we give examples for contractions between geometric and abstract quadratic algebras. 36 M.A. Escobar-Ruiz, E.G. Kalnins, W. Miller Jr. and E. Subag 7.5.1 Contraction of an abstract quadratic algebra to a geometric one There are plenty of such contractions. The canonical forms of the geometric system Ẽ17 is given by L21L2. As noted in Section 5.7 (and following the labeling of Table 1), the case of 2a, that is, a canonical form that is given by L21L2 + L1H2 + L2H2 + c10H3 with c10 ∈ C is not arising from any free 2D, second order nondegenerate superintegrable sys- tem. The matrices A(ε) = diag(1, 1, ε−1) contract any of the systems above to the geometric system L21L2. Similarly, the same matrices realize contractions from the non-geometric quadratic algebras with canonical forms 3a with c10 = 0: L31 + L1H2, 3b: L31 +H3, and 3e with c10 = 1: L31 +HL22 +H3 to L31 that arises from the superintegrable system Ẽ10. 7.5.2 Contraction of a geometric quadratic algebra to a non-geometric one As noted in Section 5.7 the canonical form 1c, L1L2(L1 + L2) is not arising from any free 2D, second order nondegenerate superintegrable system. The matrices A(ε) = diag(1, 1, ε−1) realize contractions from the geometric quadratic algebras D̃4(b)B, D̃4(b)C (with any value of b), S̃7 and S̃9 to L1L2(L1 + L2). There are many other examples. 8 Conclusions and discussion In this paper we have solved the problem of classifying all 2D nondegenerate free abstract quadratic algebras, and have made major steps in determining which of these can be realized as the symmetry algebras of 2D 2nd order superintegrable systems with nondegenerate poten- tial. We have given a precise definition and classification of Bôcher contractions, which are the principle mechanisms for relating superintegrable systems via limit relations. We have made ma- jor steps toward a classification of contractions of abstract quadratic algebras and determining which of these can be realized as Bôcher contractions. In each case we have found some abstract algebras and contractions that cannot be realized geometrically as superintegrable systems or as Bôcher contractions. We know that some of these cases correspond to contractions of models irreducible representations of quadratic algebras belonging to superintegrable systems where the algebraic representations contract, but the geometrical systems do not. They already occur in the Askey scheme. However, other cases are as yet unclear. In his theory Bôcher introduces and some of the authors developed a limit procedure for obtaining so-called type 2 separable coordinate systems, see [26], which can be interpreted as limits where the null cone is preserved but the action is nonlinear. This may fill in gaps in our classification but has not been worked out. Up to now we have only classified abstract contractions of quadratic algebras that arise from superintegrable systems on constant curvature and Darboux spaces. We have not yet solved the problem of classifying contractions of abstract quadratic algebras that do not arise in this way, though the Bôcher contractions are known. One can see from the tables in [27] that in general there are often multiple distinct contrac- tions that link two geometric quadratic algebras, even multiple distinct contractions that take a quadratic algebra to itself. The abstract contractions classified here should be though of as providing existence proofs that a contraction between to abstract quadratic algebras does or does not exist, not giving information on the multiplicities of such contractions. In a paper under preparation we classify all abstract 2D 2nd order superintegrable systems with degenerate potential and, in this case, work out all possible abstract contractions and compare the results with those for Bôcher contractions of geometric superintegrable systems. Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras 37 All of the concepts introduced here are clearly also applicable for dimensions n ≥ 3 [3]. Already we have used the special Bôcher contractions for n = 3 to derive new families of super- integrable systems in 3 dimensions [6]. This paper can be considered as part of the preparation for these more complicated cases. 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Phys. 32 (1991), 2028–2033. https://doi.org/10.1016/j.aam.2009.11.014 http://arxiv.org/abs/0908.4316 https://doi.org/10.1063/1.1619580 https://doi.org/10.1063/1.1619580 http://arxiv.org/abs/math-ph/0307039 https://doi.org/10.1088/0305-4470/34/22/311 http://arxiv.org/abs/math-ph/0102006 https://doi.org/10.1142/S0219530514500377 https://doi.org/10.1142/S0219530514500377 http://arxiv.org/abs/1401.0830 https://doi.org/10.1088/1751-8113/43/3/035202 http://arxiv.org/abs/0908.4393 https://doi.org/10.3842/SIGMA.2013.057 http://arxiv.org/abs/1212.4766 https://doi.org/10.1098/rspa.1984.0075 https://doi.org/10.3842/SIGMA.2016.038 http://arxiv.org/abs/1512.09315 https://doi.org/10.1134/S1063778807030167 https://doi.org/10.1088/1751-8113/46/42/423001 https://doi.org/10.1088/1751-8113/46/42/423001 http://arxiv.org/abs/1309.2694 https://doi.org/10.1063/1.2400834 http://arxiv.org/abs/math-ph/0608018 https://doi.org/10.3842/SIGMA.2011.036 http://arxiv.org/abs/1104.0734 https://doi.org/10.1063/1.1386927 http://arxiv.org/abs/hep-th/0011209 https://doi.org/10.1088/1751-8113/49/26/26LT01 http://arxiv.org/abs/1603.02053 https://doi.org/10.1016/j.physrep.2016.06.002 http://arxiv.org/abs/1603.02992 https://doi.org/10.1063/1.529222 1 Introduction 1.1 The Helmholtz nondegenerate superintegrable systems 1.2 Contractions 2 2D conformal superintegrability of the 2nd order 2.1 The conformal Stäckel transform 3 Tetraspherical coordinates and Laplace systems 3.1 Relation to Cartesian coordinates (x,y) and coordinates on the 2-sphere (s1,s2,s3) 3.2 Relation to flat space and 2-sphere 1st order conformal constants of the motion 3.3 The 6 Laplace superintegrable systems with nondegenerate potentials 4 Definition and composition of Bôcher contractions 4.1 Composition of Bôcher contractions 4.2 Special Bôcher contractions 5 Classification of free abstract nondegenerate quadratic algebras. Identification of those from free nondegenerate 2nd order superintegrable systems 5.1 Free nondegenerate classical quadratic algebras 5.2 The algorithm for casting the Casimir to its the canonical form 5.2.1 The four cases for F(3) 5.3 First case: three distinct roots 5.3.1 F(3)(X1,X2)=C1(X1,X2) and c7=0 5.3.2 F(3)(X1,X2)=C1(X1,X2) and c7=1 5.4 Second case: a double root 5.4.1 F(3)(X1,X2)=C2(X1,X2) and c6=0 5.4.2 F(3)(X1,X2)=C2(X1,X2) and c6=1 5.5 Third case: a triple root 5.5.1 F(3)(X1,X2)=C3(X1,X2), c6=0, and c7=0 5.5.2 F(3)(X1,X2)=C3(X1,X2), c6=0, and c7=1 5.5.3 F(3)(X1,X2)=C3(X1,X2), c6=1, and c7=0 5.5.4 F(3)(X1,X2)=C3(X1,X2), c6=1, and c7=1 5.6 Fourth case: F"0365F(3)=0 5.7 Comparison of geometric and abstract nondegenerate quadratic algebras 6 The quadratic algebras of the free 2D second order superintegrable systems 7 Abstract contractions of nondegenerate quadratic algebras arising from 2D second order superintegrable systems on constant curvature spaces and Darboux spaces 7.1 Contractions of quadratic algebras 7.1.1 The relevant quadratic algebras 7.1.2 Some general observations on contractions of quadratic algebras 7.2 Explicit contractions 7.3 Non-contractions 7.4 Comparison between abstract contractions and Bôcher contractions 7.5 Contractions between geometric quadratic algebras and abstract quadratic algebras 7.5.1 Contraction of an abstract quadratic algebra to a geometric one 7.5.2 Contraction of a geometric quadratic algebra to a non-geometric one 8 Conclusions and discussion References

Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras

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