Contractions of Degenerate Quadratic Algebras, Abstract and Geometric

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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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2017
Hauptverfasser:	Escobar Ruiz, M.A., Subag, E., Miller Jr., W.
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Veröffentlicht:	Інститут математики НАН України 2017
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author	Escobar Ruiz, M.A. Subag, E. Miller Jr., W.
author_facet	Escobar Ruiz, M.A. Subag, E. Miller Jr., W.
citation_txt	Contractions of Degenerate Quadratic Algebras, Abstract and Geometric / M.A. Escobar Ruiz, Willard Miller Jr, E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 23 назв. — англ.
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
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 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between Bôcher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all Bôcher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric Bôcher contractions.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 099, 32 pages Contractions of Degenerate Quadratic Algebras, Abstract and Geometric Mauricio A. ESCOBAR RUIZ †, Willard MILLER Jr. ‡ and Eyal SUBAG § † Centre de Recherches Mathématiques, Université de Montreal, C.P. 6128, succ. Centre-Ville, Montréal, QC H3C 3J7, Canada E-mail: mauricio.escobar@nucleares.unam.mx ‡ School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA E-mail: miller@ima.umn.edu URL: https://www.ima.umn.edu/~miller/ § Department of Mathematics, Pennsylvania State University, State College, Pennsylvania, 16802 USA E-mail: eus25@psu.edu Received August 09, 2017, in final form December 26, 2017; Published online December 31, 2017 https://doi.org/10.3842/SIGMA.2017.099 Abstract. Quadratic algebras are generalizations of Lie algebras which include the symme- try 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 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free non- degenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on con- stant curvature spaces and Darboux spaces, and studied the relationship between Bôcher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all Bôcher contractions relating degenerate superintegrable systems and, sepa- rately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric Bôcher contractions. Key words: Bôcher contractions; quadratic algebras; superintegrable systems; conformal superintegrability; Poisson structures 2010 Mathematics Subject Classification: 22E70; 16G99; 37J35; 37K10; 33C45; 17B60; 81R05; 33C45 1 Introduction An abstract degenerate (quantum) quadratic algebra Q is a noncommutative multiparameter as- sociative algebra generated by linearly independent operators X, H, L1, L2, with parameters ai, such that H is in the center and the following commutation relations hold [17]: [X,Lj ] = ∑ 0≤e1+e2+e3+e4≤1 P (j) e1,e2,e3,e4L e1 1 L e2 2 H e3X2e4 , j = 1, 2, (1.1) mailto:mauricio.escobar@nucleares.unam.mx mailto:miller@ima.umn.edu https://www.ima.umn.edu/~miller/ mailto:eus25@psu.edu https://doi.org/10.3842/SIGMA.2017.099 2 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag [L1, L2] = ∑ 0≤e1+e2+e3+e4≤1 Te1,e2,e3,e4{L e1 1 , L e2 2 , X}H e3X2e4 . (1.2) Finally, there is the relation: G ≡ ∑ 0≤e1+e2+e3+e4≤2 Se1,e2,e3,e4 { Le11 , L e2 2 , X 2e4 } He3 + c1XL1X + c2XL2X = 0, (1.3) X0 = H0 = I, where {Le11 , L e2 2 , X 2e4} is the 6-term symmetrizer of three operators. The constants P (j) e1,e2,e3,e4 , Te1,e2,e3,e4 and Se1,e2,e3,e4 are polynomials in the parameters ai of degrees 1− e1 − e2 − e3 − e4, 1 − e1 − e2 − e3 − e4 and 2 − e1 − e2 − e3 − e4, respectively, while c1, c2 are of degree 0. If all parameters aj = 0 the algebra is free. For these quantum quadratic algebras there is a natural grading such that the operators H, Lj are 2nd order and X is 1st order. The field of scalars can be either R or C. An abstract degenerate (classical) quadratic algebra Q is a Poisson algebra with linearly inde- pendent generators X , H, L1, L2, and parameters ai, satisfying relations (1.1), (1.2), (1.3) with the commutator replaced by the Poisson bracket, H, Lj , X by H, Lj , X , and the symmetrizer {Le11 , L e2 2 , X e3} by the product Le11 L e2 2 X e3/3!. These structures arise naturally in the study of classical and quantum superintegrable systems in two dimensions, e.g., [23, 22], and, in the case of zero potential systems, they are examples of Poisson structures, on which there is a considerable literature [4, 6, 20]. A quantum 2D superintegrable system is an integrable Hamiltonian system on a 2-dimensional real or complex Riemannian manifold with potential: H = ∆2 + V , that admits 3 algebraically independent partial differential operators commuting with H, the maximum possible: [H,Lj ] = 0, L3 = H, j = 1, 2, 3. Here ∆2 is the Laplace operator on the manifold. (We call this a Helmholtz superintegrable system with eigenvalue equation HΨ = EΨ to distinguish it from a Laplace conformally superintegrable system, HΨ = (∆2 + V )Ψ = 0 [18].) A system is of order K if the maximum order of the symmetry operators Lj (other than H) is K; all such systems are known for K = 1, 2 [3, 11, 14, 15]. Superintegrability captures the properties of quantum Hamiltonian systems that allow the Schrödinger eigenvalue problem HΨ = EΨ to be solved exactly, analytically and algebraically. A classical 2D superintegrable system is an integrable Hamiltonian system on a real or complex 2-dimensional Riemannian manifold with potential: H = 2∑ j,k=1 gjk(x)pjpk + V (x) in local coordinates x1, x2, p1, p2 that admit 3 functionally independent phase space functions H, L1, L2 in involution with H, the maximum possible. {H,Lj} = 0, L3 = H, j = 1, 2, 3. A system is of order K if the maximum order of the constants of the motion Lj , j 6= 3, as polynomials in p1, p2 is K. Again all such systems are known for K = 1, 2, and, for them, there is a 1-1 relationship between classical and quantum 2nd order 2D superintegrable systems [13], i.e., the quantum system can be computed from the classical system, and vice versa. The possible superintegrable systems divide into six classes: 1. First order systems. These are the (zero-potential) Laplace–Beltrami eigenvalue equations on constant curvature spaces. The symmetry algebras close under commutation to form the Lie algebras e(2,R), e(1, 1), o(3,R) or o(2, 1). Such systems have been studied in detail, using group theory methods. Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 3 2. Free triplets. These are superintegrable systems with zero potential and all generators of 2nd order. The possible spaces for which these systems can occur were classified by Koenigs (1896). They are: constant curvature spaces, the four Darboux spaces, and eleven 4-parameter Koenigs spaces [19]. In most cases the symmetry operators will not generate a quadratic algebra, i.e., the algebra will not close. If the system generates a nondegenerate quadratic algebra we call it a free quadratic triplet. 3. Nondegenerate systems. These are superintegrable systems with a non-zero potential and the generating symmetries are all of 2nd order. The space of potentials is 4-dimensional: V (x) = a1V(1)(x) + a2V(2)(x) + a3V(3)(x) + a4. The symmetry operators generate a nondegenerate quadratic algebra with parameters aj . 4. Degenerate systems. There are 4 generators: one 1st order X and 3 second order H, L1, L2. Here, X2 is not contained in the span of H, L1, L2. The space of potentials is 2-dimensional: V (x) = a1V(1)(x) + a2. The symmetry operators generate a degenerate quadratic algebra with parameters aj . Relation (1.3) is an expression of the fact that 4 symmetry operators cannot be algebraically independent. The possible degenerate sys- tems, classified up to conjugacy with respect to the symmetry groups of their underlying spaces, are listed in Appendix A. 5. Exceptional system. E15: V = f(x− iy), f an arbitrary function. The exceptional case is characterized by the fact that the symmetry generators are func- tionally linearly dependent [10, 12, 13, 15]. This is the only 2nd order functionally linearly dependent 2D system but there are many such systems in 3D, including the Calogero 3-body system on the line. In 3D such systems have not yet been classified. Every degenerate superintegrable system occurs as a restriction of the 3-parameter potentials to 1-parameter ones, such that one of the symmetries becomes a perfect square: L = X2. Here X is a first order symmetry and a new 2nd order symmetry appears so that this restriction admits more symmetries than the original system, see Remark A.1. Basic results that relate these superintegrable systems are the closure theorems: Theorem 1.1. A free triplet, classical or quantum, extends to a superintegrable system with potential if and only if it generates a free quadratic algebra Q̃, degenerate or nondegenerate. Theorem 1.2. A superintegrable system, degenerate or nondegenerate, classical or quantum, with quadratic algebra Q, is uniquely determined by its free quadratic algebra Q̃. These theorems were proved for systems in [16]. The proofs are constructive: Given a free quadratic algebra Q̃ one can compute the potential V and the symmetries of the quadratic algebra Q. Thus as far as superintegrable systems are concerned, all information about the systems is contained in the free classical quadratic algebras. Remark 1.3. This paper is a companion to [5] where we studied nondegenerate quadratic algebras, and we assume that the reader has some familiarity with this prior work. In particular, Bôcher contractions, their properties and associated notation, are treated there and we use them in this paper without detailed comment. The layout of this paper is as follows: In Section 2 we show how degenerate Helmholtz superintegrable systems can be split into Stäckel equivalence classes of Laplace conformally superintegrable systems and we determine how each Helmholtz system can be characterized in its equivalence class. In Section 3 we determine all Bôcher, i.e., geometrical, contractions of the Laplace systems and obtain complete lists of the possible Helmholtz contractions. In Section 4 4 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag we classify all abstract free quadratic algebras and determine which of these can be realized as the quadratic algebra of a Helmholtz degenerate superintegrable system. In Section 5 we classify all abstract contractions of abstract free quadratic algebras and determine which of these can be realized as Bôcher and Heisenberg contractions of the quadratic algebras of Helmholtz degenerate superintegrable systems. In Fig. 2 we describe how restrictions of nondegenerate superintegrable systems to degenerate ones, and contractions of degenerate superintegrable systems account for the lower half of the Askey scheme. The upper half of the Askey scheme is described by contractions of nondegenerate systems [5]. In Section 6 we assess our results. A list of all Helmholtz degenerate superintegrable systems can be found in Appendix A. 2 Stäckel transforms and Laplace equations Distinct degenerate classical or quantum superintegrable systems can be mapped to one another by Stäckel transforms, invertible transforms that preserve the structure of the quadratic algebra. This divides the 15 systems into 6 Stäckel equivalence classes [22]. The most convenient way to understand the equivalence classes is in terms of Laplace-like equations [18]. Since every 2D space is conformally flat there always exist “Cartesian-like” coordinates x, y such that the Hamilton–Jacobi equation can be expressed in the form H = E where H = p2x+p 2 y λ(x,y) + αV and α is a parameter. This is equivalent to the Laplace-like equation p2x + p2y + a1V1 + a2V2 = 0 where V1 = λV , V2 = λ, a1 = α, a2 = −E, now with 2 parameters. Symmetries (constants of the mo- tion) for the Helmholtz equation correspond to conformal symmetries of the Laplace equation. The Hamilton–Jacobi equation is defined on one of a variety of conformally flat spaces but the Laplace equation is always defined on flat space with conformal symmetry algebra so(4,C) [18]. An important observation is that the Laplace equations are Stäckel equivalence classes: two Helmholtz systems are Stäckel equivalent if and only if they correspond to the same Laplace equation. Remark 2.1. Indeed, If the Laplace conformally superintegrable equation can be split in the form p2x + p2y + V0 − ẼW = 0, where Ẽ is an arbitrary parameter, W is a nonconstant function, and V0, W are independent of E, then W , by division, defines a conformal Stäckel transform to the superintegrable Helmholtz system H̃ = 1 W (p2x + p2y + V0) = Ẽ. If the Laplace system admits another splitting p2x + p2y + V ′0 − Ẽ′W ′ = 0, it determines another superintegrable Helmholtz system H̃ ′ = 1 W ′ (p2x + p2y + V ′0) = Ẽ′ and H̃ ′ can be obtained from H̃ by an invertible Stäckel transform W ′ W . Thus all Helmholtz systems that can be obtained from the Laplace equation by splitting the potential are Stäckel equivalent to one another. The Laplace equations for nondegenerate systems were derived in [18], see Table 1. The Laplace equations for degenerate systems are listed in Table 2. The notation ai in Table 2 describes how these systems can be obtained as restrictions of systems in Table 1, but with added symmetry. The Helmholtz systems corresponding to each Laplace system are: Stäckel equivalence classes: Here the notation refers to the Helmholtz superintegrable systems listed in the Appendix. 1. Class A (a3, a4): System S3 corresponds to (1, 0) and (0, 1). System S6 corresponds to (1, 1). System D4(b)D corresponds to (a3, a4) with a3a4(a3 − a4) 6= 0. 2. Class B (a1, a4): System S5 corresponds to (1, 0). System E6 corresponds to (0, 1). System D2D corresponds to (a1, a4) with a1a4 6= 0. 3. Class C (a3, a4): System E3 corresponds to (1, 0). System E18 corresponds to (0, 1). System D3E corresponds to (a3, a4) with a3a4 6= 0. Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 5 System Non-degenerate potentials V (x, y) [1, 1, 1, 1] a1 x2 + a2 y2 + 4a3 (x2+y2−1)2 − 4a4 (x2+y2+1)2 [2, 1, 1] a1 x2 + a2 y2 − a3(x2 + y2) + a4 [2, 2] a1 (x+iy)2 + a2(x−iy) (x+iy)3 + a3 − a4(x2 + y2) [3, 1] a1 − a2x+ a3(4x 2 + y2) + a4 y2 [4] a1 − a2(x+ iy) + a3(3(x+ iy)2 + 2(x− iy))− a4(4(x2 + y2) + 2(x+ iy)3) [0] a1 − (a2x+ a3y) + a4(x 2 + y2) (1) a1 (x+iy)2 + a2 − a3 (x+iy)3 + a4 (x+iy)4 (2) a1 + a2(x+ iy) + a3(x+ iy)2 + a4(x+ iy)3 Table 1. Four parameter Laplace systems. System Degenerate potentials V (x, y) A 4 a3 (x2+y2−1)2 − 4a4 (x2+y2+1)2 B a1 x2 + a4 C a3 − a4(x2 + y2) D a1 − a2x E a1 (x+iy)2 + a3 F a1 − a2(x+ iy) Table 2. Two-parameter Laplace systems. 4. Class D (a1, a2): System E5 corresponds to (1, 0). System D1D corresponds to (a1, a2) with a1a2 6= 0. 5. Class E (a1, a3): System E14 corresponds to (0, 1). System E12 corresponds to (a1, a3) with a1a3 6= 0. 6. Class F (a1, a2): System E13 corresponds to (a1, a2) with a2 6= 0. System E4 corresponds to (1, 0). Here, for example, system D3E belongs to class C and is obtained from the Laplace equation by dividing it by a3−a4(x2 + y2) where a3a4 6= 0, whereas E18 is obtained by the same division with a3 = 0, a4 = 1. The conformal symmetry of these Laplace equations is best exploited by using tetraspherical coordinates to linearize the action of the conformal symmetry group [1, 18]. These are projective coordinates x1, x2, x3, x4 on the null cone x21 + x22 + x23 + x24 = 0, related to flat space coordina- tes x, y by 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 , H ≡ p2x + p2y + V = (x3 + ix4) 2 ( 4∑ k=1 p2xk + VB ) = (1 + s3) 2  3∑ j=1 p2sj + VS  , V = (x3 + ix4) 2VB, (1 + s3) = −i(x3 + ix4) x4 , s1 = ix1 x4 , s2 = ix2 x4 , s3 = −ix3 x4 . 6 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag Thus the Laplace equation H ≡ p2x + p2y + V = 0 in Cartesian coordinates becomes 4∑ k=1 p2xk + VB = 0 in tetraspherical coordinates. Here, the sj refer to coordinates on the unit 2-sphere: s21 + s22 + s23 = 1, The possible limits of one superintegrable system to another can be derived and classified by using tetraspherical coordinates and special Bôcher contractions of so(4,C) to itself. The method is described in detail in [5, 18]. Here we just recall the basic definition of a Bôcher contraction, 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 `, 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, 2) x · x ≡ 4∑ j=1 xi(ε) 2 = y · y +O(ε). If, in addition, A ∈ O(4,C) for all ε 6= 0 the matrix A defines a special Böcher contraction. For a special Böcher contraction x ·x = y ·y, with no error term. (These contractions correspond to limit relations introduced by Bôcher to obtain all orthogonal separable coordinates for Laplace and wave equations as limits of cyclidic coordinates. There is an infinite family of such contrac- tions, but they can be generated by 4 basic contractions.) Related contraction methods that don’t make use of tetraspherical coordinates directly can be found in references such as [8, 9]. Bôcher contractions take a Laplace system to itself. The contraction process has already been described in [5, 18] and references therein, but we discuss, briefly, the main ideas. Suppose we have a degenerate Laplace superintegrable system with potential V (x,a) = a1V1(x) + a2V2(x) and generating conformal symmetriesX = X+W0, L1 = L1+W1, L2 = L2+W2, where X , L1, L2 are free 2nd order conformal symmetries and W0, W1, W2 are functions of the tetraspherical coordinates xi. Applying a Bôcher contraction A(ε) to the free symmetries we obtain X (ε) = εα0X ′ +O(ε), Lj(ε) = εαjL′j +O(ε), j = 1, 2, where X ′ ∈ so(4,C) and the L′j are quadratic in so(4,C). By a change of basis {L1, L2} if necessary, one can verify that X ′, L′j , j = 1, 2 generate a free conformal quadratic algebra, The action of the Bôcher contraction on the 2-dimensional potential space preserves its dimension and maps it smoothly as a function of ε, as follows from an examination of the Bertrand-Darboux equations. Thus we get a 2-dimensional potential space in the limit. To find an explicit basis for the contracted potential V ′(b,y) = b1V ′ 1(y)′+ b2V2(y)′ we put a1 = ∞∑ k=−∞ ckε k, a2 = ∞∑ k=−∞ dkε k, where only a finite number of the coefficients ck, dk can be nonzero. Then it is a linear algebra problem to determine the ck, dk such that lim ε→0 V (x(ε),a(ε)) = V ′(b,y) exists for independent potential functions V ′1(y)′, V ′2(y)′, where the nonzero ck, dk are linear in b1, b2. The limit is guaranteed to exist and is unique up to a change of basis {V ′1(y)′, V ′2(y)′} for the target potential. Only the last limit and the linear algebra problem need to be solved to identify the contraction. This work was carried out with the assistance of the symbol manipulation programs Maple and Mathematica. There is one additional complication; the results of the contraction Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 7 are not invariant under a permutation of the indices of the hyperspherical coordinates defining the contraction matrix Aij . Thus one Bôcher contraction applied to a source system can yield a multiplicity of target results, and all permutations need to be examined. The results are rather complicated. Fig. 1 provides a clearer idea of what is happening. There are 4 basic Böcher contractions of 2D Laplace systems and each one when applied to a Laplace system, and each permutation treated, yields another Laplace superintegrable system. A system in class K1 can be obtained from a system in class K2 via contraction provided there is a directed arrow path from K2 to K1. All systems follow from A for degenerate potentials, and A is a restriction of [1111] with increased symmetry. Fig. 1 describes only the existence or nonexistence of contractions, not the multiplicity of distinct contractions. Figure 1. Contraction relations for degenerate Laplace systems. Our basic interest, however, is in Helmholtz contractions, i.e., contractions of a Helmholtz superintegrable system to another such system. The key is to start with a Laplace system, take a conformal Stäckel transform to a Helmholtz system (which we initially interpret as another Laplace system) and then take a Bôcher contraction of the new system, which as described below gives a new Helmholtz system. The result is the contraction of one Helmholtz system to another This can be done in such a way the “diagrams commute”, i.e., a Helmholtz contraction is induced by a Bôcher contraction and a Stäckel transform [18]. For example, let H be the Hamiltonian for class A. In terms of tetraspherical coordinates a general conformal Stäckel transformed potential will take the form V = a3 x23 + a4 x24 b3 x23 + b4 x24 = VA F (x,b) , where F (x,b) = b3 x23 + b4 x24 , and the transformed Hamiltonian will be Ĥ = 1 F (x,b)H, where the transform is determined by the fixed vector (b3, b4). Now we apply the Bôcher contraction [1, 1, 1, 1] → [2, 1, 1] to this system. Depending on the permutation of the indices xj , in the limit as ε → 0 the potential VA → VB, or VA → VC , and H → H′, the B or C system. Now consider F (x(ε),b) = V ′(x′, b)εα + O ( εα+1 ) , where the integer exponent α depends upon our choice of b. From our theory, the system defined by Hamiltonian Ĥ′ = lim ε→0 εαĤ(ε) = 1 V ′(x′,b)H ′ is a superintegrable system that arises from the system A by a conformal Stäckel transform induced by the potential V ′(x′,b). Thus the Helmholtz superintegrable system with potential V = VA/F contracts to the Helmholtz superintegrable system with potential VS/V ′, where S = B or S = C. The contraction is induced by a generalized Inönü–Wigner Lie algebra contraction of the conformal algebra so(4,C). Always the V ′ can be identified with a specialization of the S potential. Thus a conformal Stäckel transform of A has been contracted to a conformal Stäckel transform of S. 8 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag 3 Degenerate Helmholtz contractions The superscript for each targeted Helmholtz system is the value of the exponent α associated with the contraction. In each table below, corresponding to a single Laplace equation equivalence class, the top line is a list of the Helmholtz systems in the class, and the lower lines are the target systems under the Bôcher contraction. A equivalence class contractions contraction S3 S6 D4D [1111] ↓ [211] E32 E184 E32 S50, E62 S50 S50 [1111] ↓ [22] E32 E184 E32 E142 E122 E122 [1111] ↓ [31] E52 E52 E52, D1D3 E62, S50 S50 S50 [1111] ↓ [4] E46 E46 E46, E138 B equivalence class contractions contraction S5 E6 D2D [1111] ↓ [211] E142 E140 E140 S50 E62 S50 [1111] ↓ [22] E142 E142 E122 [1111] ↓ [31] E52 E52 E52, D1D3 S50 E62 S50, E62 [1111] ↓ [4] E46 E46 E46, E138 C equivalence class contractions contraction E3 E18 D3E [1111] ↓ [211] E52 E52 E52,3 E32 E184 E32 [1111] ↓ [22] E32 E182,4 E32, D3E2 [1111] ↓ [31] E52 E52 E52, D1D3 [1111] ↓ [4] E46 E46 E46, E138 D equivalence class contractions contraction E5 D1D [1111] ↓ [211] E40 E13−1, E40 E52 E52, D1D3 [1111] ↓ [22] E42 E42, E132 [1111] ↓ [31] E52 E52, D1D2 [1111] ↓ [4] E46 E46, E138 E equivalence class contractions contraction E14 E12 [1111] ↓ [211] E42 E42, E133 E140 E140 [1111] ↓ [22] E42 E42, E134 E142 E122 [1111] ↓ [31] E42 E42, E133 [1111] ↓ [4] E46 E46, E138 Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 9 F equivalence class contractions contraction E4 E13 [1111] ↓ [211] E40,2 E13−1,3, E42 [1111] ↓ [22] E42 E132 [1111] ↓ [31] E42 E42 [1111] ↓ [4] E46 E46 4 Classif ication of free abstract degenerate classical quadratic algebras 4.1 Abstract quadratic algebras The special case of a 2D degenerate classical superintegrable system on a constant curvature space or a Darboux space with all parameters equal to zero (no potential) gives rise to a special kind of quadratic algebra which we call a free abstract quadratic algebra. Below is a precise definition. Definition 4.1. An abstract 2D free degenerate (classical) quadratic algebra is a complex Pois- son algebra possessing a linearly independent generating set {L1,L2,L3 = H,X} that satisfy: 1. The associative product is abelian. 2. A is graded: A = ⊕∞k=0Ak where each Ak is a complex vector space, the associative product takes Ak ×Al into Ak+l, and the Poisson brackets { , } goes from Ak ×Al into Ak+l−1. 3. A0 = C. 4. A1 = CX . 5. A2 = span{L1,L2,L3,X 2}. 6. The elements {L1,L2,L3,L4 := X 2} satisfy a relation given by a homogeneous polynomial of degree 2, G(L1,L2,L3,L4) = 0. 7. Any nonzero polynomial F of minimal degree such that F (L1,L2,L3,L4) = 0 is a multiple of G. 8. The center of A contains no elements of order 1 and any element of order two in the center is a multiple of L3 = H. 9. The polynomial G depends non-trivially on at least one of the non-central 2nd order generators L1, L2. We shall simply refer to such an algebra as a free abstract quadratic algebra. Remark 4.2. A free abstract quadratic algebra is a special case of a three-dimensional affine Poisson variety, see, e.g., [4, 20]. Description of some of the properties of such Poisson varieties, along with classification of such Poisson structures on the affine space C3 can be found in [20, Section 9.2]. Poisson structures on the manifold R3 were classified at [6]. It should be noted that the Poisson algebras considered in this paper, are not quadratic Poisson structures in the sense of [20, Section 8.2]. In addition our classification scheme is different from the above mentioned classifications. 10 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag Theorem 4.3. Keeping the same notations as of Definition 4.1, let A be a free abstract quadratic algebra. Then there exists a non zero K ∈ C such that: {L1,L2} = K ∂G ∂X , {X ,L1} = K ∂G ∂L2 , {X ,L2} = −K ∂G ∂L1 . (4.1) (We shall refer to equations (4.1) as the structure equations of A). Proof. Note that G(L1,L2,L3,L4) = 0 implies that 0 = {X , G} = ∂G ∂L1 {X ,L1}+ ∂G ∂L2 {X ,L2}. The above equation is a polynomial expression in χ2,L1, L2, H. Any one of the terms ∂G ∂L1 , {X ,L1}, ∂G ∂L2 , {X ,L2} is either zero or a polynomial of degree one in the variables χ2, L1, L2, H. Since at least one of the terms ∂G ∂L1 , ∂G ∂L2 is non-zero then we must have {X ,L1} = K ∂G ∂L2 , {X ,L2} = −K ∂G ∂L1 , for some K. If K = 0 then the first order element X must be in the center of A which is impossible. Hence K 6= 0. Similarly G(L1,L2,L3,L4) = 0 implies 0 = {L1, G} = ∂G ∂X {L1,X}+ ∂G ∂L2 {L1,L2}, 0 = {L2, G} = ∂G ∂X {L2,X}+ ∂G ∂L1 {L2,L1}, and along with our assumption that at least one of the terms ∂G ∂L1 , ∂G ∂L2 is non-zero we see that {L1,L2} = K ∂G ∂X . � Corollary 4.4. Keeping the same notations as of Definition 4.1, let A be a free abstract quadratic algebra. Then G(L1,L2,L3,L4) depends non-trivially in at least two of the three generators L1, L2, L4. Proof. It easily follows from the structure equations (4.1) that if G(L1,L2,L3,L4) depends on at most one of the generators L1, L2, L4 then the center of A contains a second order generator which is not a multiple of H. � Free abstract quadratic algebras appear as symmetry algebras of 2D degenerate free su- perintegrable systems. That is the main motivation for their study. For a given G there is a 1-parameter family of quadratic algebras, parametrized by K. However, these algebras are isomorphic. It follows that the particular choice of nonzero K in the classification theory to follow is immaterial. Note 4.5. If phase space generators L1, L2, H, X satisfy a 4th order (in the momentum variables) relation G = 0 but the closure relations for {X ,L1}, {X ,L2}, {L1,L2}, are not satisfied, then K is rational in the generators. For quantum degenerate systems, knowledge of the Casimir relation G(L1, L2, H,X, α) = 0 is sufficient to determine the quadratic algebra, subject to the same conditions as the classical case. Details can be found in [7]. Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 11 4.2 The symmetry group of a free abstract quadratic algebra In this section we determine the symmetry group of a free abstract quadratic algebra. Note that once we fixed a generating set {L1,L2,L3,X}, the Casimir G is given by a symmetric 4 × 4 matrix B(G) defined by G(L1,L2,L3,L4) = 4∑ i,j=1 B(G)ijLiLj , The pair (B(G),K) (consisting of the Casimir and the multiplicative constant in the structure equations) is defined only up to a constant. That is, the pair ( zB(G), z−1K ) for any nonzero z can also serve as a Casimir and a structure constant for the same quadratic algebra and the same basis. A free abstract quadratic algebra is completely determined by a generating set {L1,L2,L3,X} (satisfying the conditions of Definition 4.1) together with the pair (B(G),K). As noted previously, the constant K can always be normalized to 1. Given two sets of generators {L1,L2,L3,X}, { L̃1, L̃2, L̃3, X̃ } of the same algebra A such that: 1. 2 = deg(L1) = deg ( L̃1 ) = deg(L2) = deg(L̃2) = deg(L3) = deg ( L̃3 ) . 2. 1 = deg(X ) = deg ( X̃ ) . 3. L3 and L̃3 are in the center of A. Then there is a ’change of basis matrix’ A of the form A :=  A1,1 A1,2 A1,3 A1,4 0 A2,1 A2,2 A2,3 A2,4 0 0 0 A3,3 0 0 0 0 0 A4,4 0 0 0 0 0 A5,5  ∈ GL(5,C), (4.2) with A4,4 = A2 5,5 such that for L := (L1,L2,L3,L4,X )t, L̃ := ( L̃1, L̃2, L̃3, L̃4, X̃ )t L = AL̃. (4.3) For A as above we define Â =  A1,1 A1,2 A1,3 A1,4 A2,1 A2,2 A2,3 A2,4 0 0 A3,3 0 0 0 0 A4,4  . Besides linear change of basis we can also rescale the invariants (B(G),K). The full symmetry group of a free abstract quadratic algebra is given by the collection of pairs (A, z), A as in equation (4.2) and z ∈ C∗. We shall denote this group by Gdegn. The action of such (A, z) ∈ Gdegn on the generators is given by equation (4.3) and on (G,K) by B(G′) = zÂtB(G)Â, K ′ = z−1K(A1,1A2,2 −A1,2A2,1) −1A−15,5, where B(G) is the symmetric matrix representing G with respect to {L1,L2,L3,L4}, and B(G′) is the symmetric matrix representing G′ := (A, z) · G with respect to { L̃1, L̃2, L̃3, L̃4 } . We shall call the subgroup of Gdegn of all pairs of the form (A, 1) as the group of linear change of bases,we denote it by Glin degn. In the following we shall determine a canonical form for the pair 12 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag (B(G),K) of each abstract quadratic algebra. This means that we shall decide upon a unique representative from each orbit of Gdegn in the space of all possible pairs (B(G),K) that arise from an abstract quadratic algebra. Without loss of generality we can assume that K = 1 and determine representative from each orbit of the subgroup of Gdegn that fixes the value of K. Explicitly this group is given by GKdegn := { (A, z) ∈ Gdegn \| z = (A1,1A2,2 −A1,2A2,1) −1A−15,5 } . 4.3 The canonical form In this section after we identify some invariants of orbits of (B(G),K) under the action of GK degn we consider the reduction to a canonical form. That is, for each orbit we choose a representative which we call the canonical form (B(G),K). For a given Casimir (B(G),K) of a free abstract quadratic algebra A and a given (A, z) ∈ GK degn we introduce the following notations: Â =  A1,1 A1,2 A1,3 A1,4 A2,1 A2,2 A2,3 A2,4 0 0 A3,3 0 0 0 0 A4,4  = ( r s 0 t ) , B(G) =  b1,1 b1,2 b1,3 b1,4 b1,2 b2,2 b2,3 b2,4 b1,3 b2,3 b3,3 b3,4 b1,4 b2,4 b3,4 b4,4  = ( b c ct d ) . Direct calculation shows that ÂtB(G)Â = ( rtbr rt(bs+ ct) stbr + tctr st(bs+ ct) + t(cts+ dt) ) . Hence under the action GK degn the rank of B(G) and the rank of its upper left 2 by 2 block (which we denote by b) are preserved. We use the theory of symmetric bilinear forms and find (A, z) ∈ GK degn that cast B(G) into exactly one of the following forms depending on rank(b). 1. B(G) =  1 0 0 0 0 1 0 0 0 0 b3,3 b3,4 0 0 b3,4 b4,4  if rank(b) = 2, 2. B(G) =  1 0 0 0 0 0 b2,3 b2,4 0 b2,3 b3,3 b3,4 0 b2,4 b3,4 b4,4  if rank(b) = 1, 3. B(G) =  0 0 b1,3 b1,4 0 0 b2,3 b2,4 b1,3 b2,3 b3,3 b3,4 b1,4 b2,4 b3,4 b4,4  if rank(b) = 0. Below we analyze the different cases according to rank(b) we summarize the results of all cases in Table 3. Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 13 4.4 The rank 2 case If B(G) =  1 0 0 0 0 1 0 0 0 0 b3,3 b3,4 0 0 b4,3 b4,4  and (A, z) ·B(G) =  1 0 0 0 0 1 0 0 0 0 b̃3,3 b̃3,4 0 0 b̃4,3 b̃4,4  , for some (A, z) ∈ GK degn then Â =  A1,1 A1,2 0 0 A2,1 A2,2 0 0 0 0 A3,3 0 0 0 0 A4,4  , z = (A1,1A2,2 −A1,2A2,1) −1A−15,5, with √ z ( A1,1 A1,2 A2,1 A2,2 ) ∈ O2(C), A−15,5 = ±1, ( b̃3,3 b̃3,4 b̃3,4 b̃4,4 ) = ( zA2 3,3b3,3 zA3,3b3,4 zA3,3b3,4 zb4,4 ) . So if b3,3 6= 0 we define the canonical form of B(G) to be B(G)21(b3,4, b4,4) =  1 0 0 0 0 1 0 0 0 0 1 b3,4 0 0 b3,4 b4,4  , with b3,4 ∈ {0, 1}, b4,4 ∈ C. If b3,3 = 0 we define the canonical form of B(G) to be B(G)22(b3,4, b4,4) =  1 0 0 0 0 1 0 0 0 0 0 b3,4 0 0 b3,4 b4,4  , with b3,4, b4,4 ∈ {0, 1}. 4.5 The rank 1 case If B(G) =  1 0 0 0 0 0 b2,3 b2,4 0 b2,3 b3,3 b3,4 0 b2,4 b3,4 b4,4  and (A, z) ·B(G) =  1 0 0 0 0 0 b̃2,3 b̃2,4 0 b̃2,3 b̃3,3 b̃3,4 0 b̃2,4 b̃3,4 b̃4,4  , for some (A, z) ∈ GK degn then Â =  ±z−1/2 0 ∓z1/2b2,3A3,3A2,1 ∓z−1/2b2,4A4,4A2,1 A2,1 A2,2 A2,3 A2,4 0 0 A3,3 0 0 0 0 A4,4  , z = A−22,2A −1 4,4, 14 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag and b̃2,3 = b2,3A −1 2,2A3,3A −1 4,4, (4.4) b̃2,4 = b2,4A −1 2,2, (4.5) b̃3,3 = b22,3A 2 2,1A −4 2,2A 2 3,3A −2 4,4 + 2b2,3A2,3A −2 2,2A3,3A −1 4,4 + b3,3A −2 2,2A 2 3,3A −1 4,4, b̃3,4 = b2,3b2,4A 2 2,1A −4 2,2A3,3A −1 4,4 + b2,3A2,4A −2 2,2A3,3A −1 4,4 + b2,4A2,3A −2 2,2 + b3,4A −2 2,2A3,3, b̃4,4 = b22,4A 2 2,1A −4 2,2 + 2b2,4A2,4A −2 2,2 + b4,4A −2 2,2A4,4. From (4.4) we see that b̃2,3 = 0 if and only if b2,3 = 0, similarly from (4.5) we see that b̃2,4 = 0 if and only if b2,4 = 0. Hence we continue our analysis according to the vanishing of b2,3 and b2,4. 4.5.1 rank(b) = 1, b2,3 = 0 and b2,4 = 0 If B(G) =  1 0 0 0 0 0 0 0 0 0 b3,3 b3,4 0 0 b3,4 b4,4  and (A, z) ·B(G) =  1 0 0 0 0 0 0 0 0 0 b̃3,3 b̃3,4 0 0 b̃3,4 b̃4,4  , for some (A, z) ∈ GK degn then Â =  ±z−1/2 0 0 0 A2,1 A2,2 A2,3 A2,4 0 0 A3,3 0 0 0 0 A4,4  , z = A−22,2A −1 4,4, and ( b̃3,3 b̃3,4 b̃3,4 b̃4,4 ) = ( A−22,2A −1 4,4A 2 3,3b3,3 A−22,2A −1 4,4A3,3A4,4b3,4 A−22,2A −1 4,4A3,3A4,4b3,4 A−22,2A4,4b4,4 ) . We define the canonical form to be B(G)11(b3,3, b3,4, b4,4) =  1 0 0 0 0 0 0 0 0 0 b3,3 b3,4 0 0 b3,4 b4,4  , with b3,3, b3,4, b4,4 ∈ {0, 1}. 4.5.2 rank(b) = 1, b2,3 6= 0 and b2,4 6= 0 In this case we can act with (A, z) ∈ GK degn to obtain B(G)15(b3,4) =  1 0 0 0 0 0 1 1 0 1 0 b3,4 0 1 b3,4 0  , for a unique b3,4 ∈ {0, 1}. We define this form to be the canonical form in this case. Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 15 4.5.3 rank(b) = 1, b2,3 6= 0 and b2,4 = 0 In this case we can act with (A, z) ∈ GK degn to obtain B(G)16(b4,4) =  1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 b4,4  , with unique b4,4 ∈ {0, 1} which is defined to be the canonical form in this case. 4.5.4 rank(b) = 1, b2,3 = 0 and b2,4 6= 0 In this case we can act with (A, z) ∈ GK degn to obtain B(G)17(b3,3) =  1 0 0 0 0 0 0 1 0 0 b3,3 0 0 1 0 0  , with unique b3,3 ∈ {0, 1} which is defined to be the canonical form in this case. 4.6 The rank 0 case If B(G) =  0 0 b1,3 b1,4 0 0 b2,3 b2,4 b1,3 b2,3 b3,3 b3,4 b1,4 b2,4 b3,4 b4,4  and (A, z) ·B(G) =  0 0 b̃1,3 b̃1,4 0 0 b̃2,3 b̃2,4 b̃1,3 b̃2,3 b̃3,3 b̃3,4 b̃1,4 b̃2,4 b̃3,4 b̃4,4  , for some (A, z) ∈ GK degn then ( b̃1,3 b̃1,4 b̃2,4 b̃2,4 ) = z ( A1,1 A2,1 A1,2 A2,2 )( b1,3 b1,4 b2,3 b2,4 )( A3,3 0 0 A4,4 ) , and ( b̃3,3 b̃3,4 b̃4,3 b̃4,4 ) = z ( A1,3 A2,3 A1,4 A2,4 )( b1,3 b1,4 b2,3 b2,4 )( A3,3 0 0 A4,4 ) + z ( A3,3 0 0 A4,4 )( b1,3 b2,3 b1,4 b2,4 )( A1,3 A1,4 A2,3 A2,4 ) + z ( A3,3 0 0 A4,4 )( b3,3 b3,4 b3,4 b4,4 )( A3,3 0 0 A4,4 ) . Hence in the case of b = 0, the rank of c = ( b1,3 b1,4 b2,3 b2,4 ) is invariant under the action of GK degn. We continue our analysis according to the value of this rank. By Corollary 4.4 c 6= 0. 16 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag 4.6.1 rank(b) = 0, rank(c) = 1 In this case we have the following forms 0 0 0 0 0 0 1 1 0 1 b3,3 b3,4 0 1 b3,4 b4,4  ,  0 0 0 0 0 0 1 0 0 1 b3,3 b3,4 0 0 b3,4 b4,4  ,  0 0 0 0 0 0 0 1 0 0 b3,3 b3,4 0 1 b3,4 b4,4  . The first case can be reduced to the canonical form B(G)05 =  0 0 0 0 0 0 1 1 0 1 0 b3,4 0 1 b3,4 0  , for a unique b3,4 ∈ {0, 1}. The second case can be reduced to the canonical form B(G)06 =  0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 b4,4  , for a unique b4,4 ∈ {0, 1}. The case with b4,4 = 0 is not possible due to Corollary 4.4. The third case can be reduced to the following canonical form B(G)07(b3,3) =  0 0 0 0 0 0 0 1 0 0 b3,3 0 0 1 0 0  , for a unique b3,3 ∈ {0, 1}. 4.6.2 rank(b) = 0, rank(c) = 2 In this case we only have the form 0 0 1 0 0 0 0 1 1 0 b3,3 b3,4 0 1 b3,4 b4,4  , (4.6) which can be reduced to the canonical form B(G)08 =  0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0  . 4.7 Comparison of geometric and abstract free degenerate quadratic algebras We examine the entries in Table 3 to determine which can be realized as a classical 2D degenerate superintegrable system or a classical Lie algebra with linearly independent generators. There is a close relationship between the canonical forms of free abstract degenerate quadratic algebras Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 17 degenerate quadratic algebras # rank(b) invariant form canonical form 1 2 B(G)21(b34, b44) L21 + L22 + H2 + b44X 4 + 2b34HX 2, b34 ∈ {0, 1}, b44 ∈ C 2 2 B(G)22(b34, b44) L21 + L22 + b44X 4 + 2b34HX 2, b3,4, b44 ∈ {0, 1} 3 1 B(G)11(b33, b34, b44) L21+b33H2+b44X 4+2b34HX 2, b33, b34, b44 ∈ {0, 1}, b34 + b44 6= 0 4 1 B(G)15(b34) L21 + 2L2H+ 2L2X 2 + 2b34HX 2, b34 ∈ {0, 1} 5 1 B(G)16(b44) L21 + 2L2H+ b44X 4, b44 ∈ {0, 1} 6 1 B(G)17(b33) L21 + 2L2X 2 + b33H2, b33 ∈ {0, 1} 7 0 B(G)05(b34) 2L2H+ 2L2X 2 + 2b34HX 2, b34 ∈ {0, 1} 8 0 B(G)06 2L2H+ X 4 9 0 B(G)07(b33) 2L2X 2 + b33H2, b33 ∈ {0, 1} 10 0 B(G)08 2L1H+ 2L2X 2 Table 3. List of canonical forms of free abstract degenerate quadratic algebras. The canonical forms B(G)ab are given explicitly in Sections 4.4, 4.5, 4.6 above. and Stäckel equivalence classes of degenerate superintegrable systems. We demonstrate this by treating one example in detail. The superintegrable system S3, with degenerate potential, can be defined by G = L21 + L22 − L1H+ L1X 2 + a1X 2 + (a1 + a2)L1 = 0, 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 = 2∑ k=1 cjkbk, k = 1, 2 where the bk are the new parameters, 2) we make the replacements H → −b2, b2 → −H and 3) we then set all parameters bj = 0 to determine the free degenerate quadratic algebra. The result is [A] : G = L21 + L22 + L1X 2 − c12HX 2 − (c12 + c22)HL1 = 0, where \|c12\|+ \|c22\| > 0. The canonical forms in Table 3 associated with the equivalence class [A] are 1: b44 = 1 and 2: b44 = 1. The superintegrable system E6, with degenerate potential, can be defined by G = L21 − L2H+ L2X 2 + a1X 2 + a2L2 = 0. Going through the same procedure as above, we obtain the equivalence class [B] : G = L21 + L2X 2 − c12HX 2 − c22HL2 = 0, where \|c12\|+ \|c22\| > 0. The canonical form associated with this equivalence class is 4: all cases. The superintegrable system E3, with degenerate potential, can be defined by G = L21 + L22 − L1H+ a2 ( X 2 + a2 ) L1. The equivalence class is [C] : G = L21 + L22 − c12HX 2 − c22HL1 = 0, 18 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag rank canonical form 2 1: all cases 2: all cases 1 3: all cases except 4: all cases (b44, b34 = 1, b33 = 0, 1), (b44 = 0, b34, b33 = 1) 1 5: all cases 6: all cases 0 7: no 8: missing 2L2H+ X 4 = 0 0 9: missing L2X 2 = 0 10: all cases Table 4. Matching of geometric with abstract quadratic algebras. where \|c12\|+ \|c22\| > 0. The canonical forms associated with this equivalence class are 1: b44 = 0, and 2: b44 = 0. The superintegrable system E5 can be defined by G = L21 + X 4 −HX 2 + a1L2 + a2X 2 = 0. The equivalence class is [D] : G = L21 + X 4 − c22HX 2 − c12HL2 = 0, where \|c12\|+ \|c22\| > 0. The canonical forms associated with this equivalence class are 3: b44 = 1, b34 = 0 and 5: b44 = 1. The superintegrable system E14 can be defined by G = −L21 − L2X 2 + a1H− a1a2 = 0. The equivalence class is [E] : G2 = −L21 − L2X 2 − c12c22H2 = 0, where \|c12\|+ \|c22\| > 0. The canonical forms associated with [E] are 6: all cases. The superintegrable system E4 can be defined by G = H2 + X 4 + 2HX 2 − 4L2X 2 − 4ia1L1 − 2a2X 2 − 2a2H+ a22 = 0. The equivalence class is [F ] : G = X 4 − 4L2X 2 − 4ic12HL1 + 2c22HX 2 + c222H2 = 0, where \|c12\|+ \|c22\| > 0. The canonical forms associated with [F ] are 9: b33 = 1 and 10: all cases. Heisenberg systems: In addition there are systems that can be obtained from the degen- erate 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 be- come singular. However, they do form quadratic algebras and several have the interpretation of time-dependent Schrödinger equations in 2D spacetime, so we also consider them geometri- cal. Some of these were classified in [16] 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 possible canonical forms are 3: b33 = b44 = 0, b34 = 1 and 5: b44 = 0. All these results relating geometric systems to abstract systems are summarized in Table 4. We see that every abstract quadratic algebra is isomorphic to a quadratic algebra corresponding to a superintegrable system, with just 5 exceptions. Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 19 Theorem 4.6. Every free quadratic algebra realizable by functions on 4-dimensional phase space with grading the order of polynomials in the momenta is isomorphic to a free quadratic algebra of a superintegrable system. Proof: We show that the 5 exceptional free quadratic algebras cannot be realized in phase space. We assume in each case that the algebra is realizable in terms of functions on phase space and obtain a contradiction. 1. Case 3: L21 + ( H+X 2 )2 = 0. We can factor G as (H+X 2 + iL1)(H+X 2− iL1) = 0. This is possible only if one of the factors vanishes; hence the generators are linearly dependent. Impossible! 2. Case 3: L21 + 2HX 2 + X 4 = 0. Here, L21 = −X 2(2H + X 2) so L1 = XY where Y is a 1st order constant of the motion. Thus Y must be a multiple of X and the generators are linearly dependent. Impossible! 3. Case 3: L21 + H2 + 2HX 2 = 0. Here, L21 = −H(H + 2X 2). If L1 doesn’t factor then it must be a multiple of H and of H + 2X 2 simultaneously. Impossible! Suppose then that L1 = YZ, a product of two 1st order factors. If Z is a multiple of Y then Y is a constant of the motion, hence proportional to X . Impossible! Thus Y,Z must be distinct linear factors. If H is divisible by each factor then L1, H are linearly dependent. Impossible! So H must be divisible by the square of a single factor. By renormalizing Y and Z we can assume H = Y2. Thus Y is a 1st order constant of the motion, necessarily proportional to X . We conclude that H ∼ X 2. Impossible! 4. Case 8: 2L2H+ X 4 = 0. Since X is 1st order, and 2L2H = −X 4 both H and L2 must be divisible by X 2. Thus each is a perfect square of a 1st order symmetry, necessarily a scalar multiple of X . Hence the 2nd order generators are linearly dependent. Impossible! 5. Case 9: L2X = 0. Here at least one of the generators must vanish. Impossible.! Thus we have shown that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. The remaining 5 abstract systems must lie in different graded Poisson algebras. 5 Classif ication of contractions of free abstract quadratic degenerate 2D superintegrable systems on constant curvature spaces and Darboux spaces In this section we define contractions between free abstract quadratic algebras. Then we list the canonical forms of the Casimirs of free abstract quadratic algebras that arises as the symmetry algebras of degenerate 2D free superintegrable systems on constant curvature spaces or Darboux spaces. Finally using the canonical forms, we classify all possible contractions relating free abstract quadratic algebras of degenerate 2D superintegrable systems on constant curvature spaces or Darboux spaces. 5.1 Contraction of a free abstract quadratic algebra Definition 5.1. Let A, B be free abstract quadratic algebras with Casimirs (B(GA),KA) and (B(GB),KB), respectively. Suppose that there is a continuous map ε 7−→ (Aε, zε) from some punctured neighborhood of 0 in C∗, the nonzero complex numbers, into Gdegn and such that lim ε−→0 (Aε, zε) · (b(GA),KA) = (B(GB),KB). Then we say that B is a contraction of A. 20 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag The meaning of the definition is that in any generating set {L1,L2,L3,L4} of A (satisfying the same assumptions as before) the corresponding matrix B(GA) and KA satisfy lim ε−→0 zεÂε t B(GA)Âε = B(GB), (5.1) lim ε−→0 z−1ε KA ((Aε)1,1(Aε)2,2 − (Aε)1,2(Aε)2,1) −1 (Aε) −1 5,5 = KB, (5.2) where (B(GB),KB) is a realization of the Casimir of B in some generating set. In the clas- sification below we are using a more refined class of contractions. We shall call these contrac- tions algebraic. By definition an algebraic contraction is a contraction that can be realized via a map ε 7−→ (Aε, zε) from some punctured neighborhood of 0 in C∗ into Gdegn such that zε as well as the entries of Aε are rational functions in ε. Proposition 5.2. B is a contraction of A if and only if there is a continuous map ε 7−→ (Aε, zε) from some punctured neighborhood of 0 in C∗ into Gdegn such that lim ε−→0 zεÂε t B(GA)canÂε = B(GB)can, (5.3) lim ε−→0 z−1ε ((Aε)1,1(Aε)2,2 − (Aε)1,2(Aε)2,1, ) −1 (Aε) −1 5,5 = 1, (5.4) where (B(GA)can, 1) and (B(GB)can, 1) are the canonical forms of the Casimirs of A and B respectively. Proof. Obviously if equations (5.3)–(5.4) hold then B is a contraction of A. For the other direction assume that equations (5.1)–(5.2) hold. We can further assume that (B(GA),KA) is in its canonical form (B(GA)can, 1). Let (A, z) ∈ Gdegn such that (B(GB),KB) = (A, z) · (B(GB)can, 1), then the continuity of the action of Gdegn implies that lim ε−→0 ( (A, z)−1(Aε, zε) ) · (B(GA)can, 1) = (B(GB)can, 1). � Proposition 5.3. Let A, B be free abstract quadratic algebras and ε 7−→ (Aε, 1) a continuous map from C∗ into Gdegn such that lim ε−→0 Âε t B(GA)canÂε = B(GB)can. Then there is another continuous map ε 7−→ (Cε, zε) from C∗ into Gdegn such that lim ε−→0 (Cε, zε) · (B(GA)can, 1) = (B(GB)can, 1). Proof. Define zε := ((Aε)1,1(Aε)2,2 − (Aε)1,2(Aε)2,1) 4 (Aε) 4 5,5, Cε = ((Aε)1,1(Aε)2,2 − (Aε)1,2(Aε)2,1) −1 (Aε) −1 5,5Aε. � Note that Propositions 5.2 and 5.3 also hold for algebraic contractions. The conclusion from the last two proposition is that for the purpose of classifying contractions of free abstract quadratic algebras it is enough to take the Casimirs B(G)A and B(G)B in their canonical forms and to consider only the action of the group of invertible matrices of the form Â =  A1,1 A1,2 A1,3 A1,4 A2,1 A2,2 A2,3 A2,4 0 0 A3,3 0 0 0 0 A4,4  , (5.5) Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 21 on symmetric matrices B by B 7−→ ÂtBÂ. We shall denote the space of 4 by 4 complex sym- metric matrices by Sym(4,C) and the group of matrices of the form (5.5) by Ĝdegn. The group Ĝdegn is a complex algebraic group and the space Sym(4,C) is a complex algebraic variety on which Ĝdegn acts algebraically. As was explained in [5, Section 7.1.2] this implies that if B is a contraction of a quadratic algebra A then A is not a contraction of B (unless A and B are isomorphic). In addition if B is a contraction of A then the rank of any matrix that repre- sents B(G)B and the rank of its upper left 2 by 2 block can not exceed the corresponding ranks for any matrix that represents B(G)A. Hence there is certain hierarchy for contractions that is governed by the rank. By a rank of a free abstract quadratic algebra we mean (rank(B), rank(b)) of the corresponding matrices B and b in any bases. We shall make this more precise below. 5.2 Classif ication of abstract algebraic contractions of superintegrable systems Organized according to their rank, the canonical forms of the free abstract quadratic algebras of free triplets of 2D constant curvature spaces and Darboux spaces are given in Table 5 below. We shall classify all possible algebraic contractions between any two free abstract quadratic canonical forms for the Casimirs of free degenerate 2D 2nd order superintegrable systems on constant curvature and Darboux spaces # system rank(B) rank(b) canonical form 1 S6 4 2 B22(1, 1), L21 + L22 + 2HX 2 + X 4 2 E18 4 2 B22(0, 1), L21 + L22 + 2HX 2 3 D3E 4 2 B21(1, 0), L21 + L22 +H2 + 2HX 2 4 D4(b)D 4 2 B21(1,−2), L21 + L22 +H2 + 2HX 2 − 2X 4 5 S3 4 2 B21( √ 2ei 3π 4 ,−2i), L21 +L22 +H2 + 2 √ 2ei 3π 4 HX 2− 2iX 4 6 E3 3 2 B21(0, 0), L21 + L22 +H2 7 E12 4 1 B17(1), L21 + 2L2X 2 +H2 8 D1D 4 1 B16(1), L21 + 2L2H+H2 9 D2D 4 1 B15(1), L21 + 2L2H+ 2L2X 2 + 2HX 2 10 E6 3 1 B15(0), L21 + 2L2H+ 2L2X 2 11 E5 3 1 B11(0, 0, 1), L21 + 2HX 2 + X 4 12 E14 3 1 B17(0), L21 + 2L2X 2 13 S5 3 1 B17(0), L21 + 2L2X 2 14 E13 4 0 B08, 2L1H+ 2L2X 2 15 E4 3 0 B07(1), H2 + 2L2X 2 Table 5. List of canonical forms of free abstract degenerate quadratic algebras. algebras that appear in Table 5. More precisely for any two such algebras we determine if such a contraction is possible or not. If it does we give one realization of it, unless it is a contraction from a free abstract quadratic algebra to itself. We shall start with some general observa- tions. We note that the quadratic algebras of E14 and S5 coincide so we only keep E14 in our notation. We divide the free abstract quadratic algebras of the second order free degenerate superintegrable systems on 2D constant curvature spaces and 2D Darboux spaces according to their ranks: R4,2 := {S6, E18, D3E,D4(b)D,S3}, R3,2 := {E3}, R4,1 := {E12, D1D,D2D}, 22 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag S6 E18 D3E D4(b)D S3 E3 E12 D1D D2D E6 E5 E14 E13 E4 S6 + + – – – – + – – – + + + + E18 – + – – – – – – – – + – + + D3E – + + – – + – + – – + – + + D4(b)D – – – + – + + + – – + + + + S3 – – – – + + – + – + + + + + E3 – – – – – + – – – – + – – + E12 – – – – – – + – – – + + + + D1D – – – – – – – + – – + – + + D2D – – – – – – + + + + + + + + E6 – – – – – – – – – + + + – + E5 – – – – – – – – – – + – – + E14 – – – – – – – – – – + + – + E13 – – – – – – – – – – – – + + E4 – – – – – – – – – – – – – + Table 6. List of algebraic contractions between free abstract quadratic algebras of 2D free degenerate superintegrable systems on constant curvature spaces and Darboux spaces. A plus in the rubric placed in the i-th row and j-th column indicates that there is a contraction from the system listed in the i-th row of the first column to the system listed in the j-th column of the first row. A minus indicates that there is no such contraction. R3,1 := {E6, E5, E14}, R4,0 := {E13}, R3,0 := {E4}. By the discussion above, besides con- tractions between two algebras in the same class Ri,j we can potentially find a contraction only according to the following diagram: R4,2 ""\|\| R3,2 "" R4,1 ��\|\| R3,1 �� R4,0 \|\| R3,0. The classification is summarized in Table 6. Detailed analysis is presented below. Representa- tives for all possible contractions are given in Section 5.3.7. 5.2.1 Contractions between rank two free abstract quadratic algebras Note that for any of the Bcan matrices for the systems R4,2∪R3,2 = {S6, E18, D3E,D4(b)D,S3} ∪{E3} we have ÂtBÂ = ( rt 0 st t )( 1 0 0 d )( r s 0 t ) = ( rtr rts str sts+ tdt ) . (5.6) Consider A = A(ε) in equation (5.6) such that lim ε−→0 Â(ε) t BÂ(ε) = B0, (5.7) where B0 ∈ R4,2 ∪R3,2. Below we prove that without loss of generality we can assume that rtεrε is diagonal and sε is the zero two by two matrix. Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 23 Proposition 5.4. Suppose that {Bε}ε∈R+ is a family of 4× 4 symmetric matrices with entries in C[ε] and such that lim ε−→0+  (Bε)11 (Bε)12 (Bε)13 (Bε)14 (Bε)12 (Bε)22 (Bε)23 (Bε)24 (Bε)13 (Bε)23 (Bε)33 (Bε)34 (Bε)14 (Bε)24 (Bε)34 (Bε)44  =  1 0 0 0 0 1 0 0 0 0 l33 l34 0 0 l34 l44  . Then there exists a continuous function ε 7→ Aε from C∗ to Ĝdegn with each entry polynomial in ε such that Ât εBεÂε is of the form (B̃ε)11 0 0 0 0 (B̃ε)22 0 0 0 0 (B̃ε)33 (B̃ε)34 0 0 (B̃ε)34 (B̃ε)44  , and lim ε−→0 Ât εBεÂε = lim ε−→0 Bε. Proof. Note that for Ât ε = ( 1 0 0 0 −(Bε)12 (Bε)11 0 0 −(Bε)13 0 (Bε)11 0 −(Bε)14 0 0 (Bε)11 ) lim ε−→0 Ât ε is the identity matrix and Ât εBεÂε is a symmetric matrix with all entries in the first row and first column besides the (1, 1) entry equal to zero. Similar matrix take care of the second row and second column. � Now suppose that equation (5.7) holds. Using the last proposition this means that we can assume that Â(ε) = ( rε 0 0 tε ) with rε ∈ GL(2,C) and, as always, tε a diagonal invertible matrix. From this we easily see that the only possible contractions between two algebras in R4,2 ∪ R3,2 are S6 −→ E18, D3E −→ E18, D3E −→ E3, D4(b)D −→ E3, S3 −→ E3. 5.3 Contractions of rank two free abstract quadratic algebras to rank one algebras Proposition 5.5. Suppose that {Bε}ε∈R+ is a family of 4× 4 symmetric matrices with entries in C[ε] such that lim ε−→0+ Bε = L =  1 0 l13 l14 0 0 l23 l24 l13 l23 l33 l34 l14 l24 l34 l44  . Then there exists a continuous function ε 7→ Aε from C∗ to Ĝdegn with each entry polynomial in ε such that Ât εBεÂε is of the form (B̃ε)11 0 0 0 0 (B̃ε)22 (B̃ε)23 (B̃ε)24 0 (B̃ε)23 (B̃ε)33 (B̃ε)34 0 (B̃ε)24 (B̃ε)34 (B̃ε)44  , and lim ε−→0 Ât εBεÂε = lim ε−→0 Bε. If in addition l13 = 0 then (B̃ε)13 = 0 and similarly if l14 = 0 then (B̃ε)14 = 0. 24 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag The proof is similar to the proof of Proposition 5.4. Corollary 5.6. In any contraction from a rank two systems R4,2 ∪ R3,2 = {S6, E18, D3E, D4(b)D,S3, E3} to of one of the rank one systems R4,1∪R3,1 = {E12, D1D,D2D,E6, E5, E14} we can assume that Â(ε) t BÂ(ε) =  (Bε)11 0 0 0 0 (Bε)22 (Bε)23 (Bε)24 0 (Bε)23 (Bε)33 (Bε)34 0 (Bε)24 (Bε)34 (Bε)44  . (5.8) 5.3.1 Contractions of D3E to rank one algebras For A(ε) ∈ Ĝdegn the matrix Â(ε) t B(G)21(1, 0)Â(ε) is given by A2 11+A 2 21 A11A12+A21A22 A11A13+A21A23 A11A14+A21A24 A12A11+A22A21 A2 12+A 2 22 A12A13+A22A23 A12A14+A22A24 A11A13+A21A23 A12A13+A22A23 A2 13+A 2 23+A 2 33 A13A14+A23A24+A33A44 A11A14+A21A24 A12A14+A22A24 A13A14+A23A24+A33A44 A2 14+A 2 24  . Assuming that this matrix is in the form of (5.8) then this implies that exist β, γ, δ complex valued functions of ε define on C∗ such that (A12, A22) = β(A21,−A11), (A13, A23) = γ(A21,−A11), (A14, A24) = δ(A21,−A11). Hence Â(ε) t B(G)21(1, 0)Â(ε) takes the form A2 11 +A2 21 0 0 0 0 β2(A2 11 +A2 21) βγ(A2 11 +A2 21) βδ(A2 11 +A2 21) 0 βγ(A2 11 +A2 21) γ2(A2 11 +A2 21) +A2 33 γδ(A2 11 +A2 21) +A33A44 0 βδ(A2 11 +A2 21) γδ(A2 11 +A2 21) +A33A44 δ2(A2 11 +A2 21)  . We are going to consider the limit of ε goes to zero of such a matrix and check if it can be equal to one of the matrices of the canonical forms of the systems in R4,1 ∪ R3,1. This means that lim ε−→0+ (A2 11 +A2 21) = 1 and we can replace Â(ε) t B(G)21(1, 0)Â(ε) by 1 0 0 0 0 β2 βγ βδ 0 βγ γ2 +A2 33 γδ +A33A44 0 βδ γδ +A33A44 δ2  , with lim ε−→0+ β = 0. From this we can show that D3E can not be contracted to E6, E12, E14, D2D. 5.3.2 Contractions of E18 to rank one algebras Following the same steps as in the case of D3E we can replace Â(ε) t B(G)22(0, 1)Â(ε) by 1 0 0 0 0 β2 βγ βδ 0 βγ γ2 γδ +A33A44 0 βδ γδ +A33A44 δ2  , with lim ε−→0+ β = 0. From this we can show that E18 can not be contracted to E6, E12, D1D, D2D. Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 25 5.3.3 Contractions of S6 to rank one algebras Following the same steps as in the case of D3E we can replace Â(ε) t B(G)22(1, 1)Â(ε) by 1 0 0 0 0 β2 βγ βδ 0 βγ γ2 γδ +A33A44 0 βδ γδ +A33A44 δ2 +A2 44  , with lim ε−→0+ β = 0. From this we can show that S6 can not be contracted to E6, D1D, D2D. 5.3.4 Contractions of D4(b)D to rank one algebras Following the same steps as in the case of D3E we can replace Â(ε) t B(G)21(1,−2)Â(ε) by 1 0 0 0 0 β2 βγ βδ 0 βγ γ2 +A2 33 γδ +A33A44 0 βδ γδ +A33A44 δ2 − 2A2 44  , with lim ε−→0+ β = 0. From this we can show that D4(b)D can not be contracted to D2D, E6. 5.3.5 Contractions of S3 to rank one algebras Following the same steps as in the case of D3E we can replace Â(ε) t B(G)21( √ 2ei 3π 4 ,−2i)Â(ε) by  1 0 0 0 0 β2 βγ βδ 0 βγ γ2 +A2 33 γδ + √ 2ei 3π 4 A33A44 0 βδ γδ + √ 2ei 3π 4 A33A44 δ2 − 2iA2 44  , with lim ε−→0+ β = 0. From this we can show that S3 can not be contracted to D2D, E12. 5.3.6 Contractions of E3 to rank one algebras Following the same steps as in the case of D3E we can replace Â(ε) t B(G)21(0, 0)Â(ε) by 1 0 0 0 0 β2 βγ βδ 0 βγ γ2 γδ 0 βδ γδ δ2  , with lim ε−→0+ β = 0. From this we can show that E3 can not be contracted to E6 and E14. 5.3.7 Explicit contractions E13: E13 −→ E4 ε 0 1 2 0 0 1 0 0 0 0 1 0 0 0 0 1  . 26 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag E14: E14 −→ E4 E14 −→ E5 ε 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1  ,  1 0 0 0 0 ε 1 1 2 0 0 1 0 0 0 0 1  . E5: E5 −→ E4 ε2 ε 1 ε−1 0 1 0 0 0 0 i 0 0 0 0 iε−1  . D2D: D2D −→ E6 D2D −→ E12 D2D −→ D1D 1 0 1 0 0 ε−1 0 0 0 0 ε 0 0 0 0 ε  ,  1 0 0 0 0 ε − i√ 2 0 0 0 i√ 2 0 0 0 0 ε−1  ,  1 0 0 0 0 ε ε 2 0 0 0 ε−1 0 0 0 0 ε2  , D2D −→ E5 D2D −→ E14 D2D −→ E4 D2D −→ E13 1 0 0 0 0 ε 0 1 2ε 0 0 2ε 0 0 0 0 ε  ,  1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ε  ,  ε 0 1 1 0 ε 0 0 0 0 ε 0 0 0 0 ε  ,  0 ε 0 ε−1 1 0 0 0 0 0 1 0 0 0 0 iε−1  . E6: E6 −→ E14 E6 −→ E5 E6 −→ E4 1 0 0 0 0 1 0 0 0 0 ε 0 0 0 0 1  ,  1 0 0 0 0 ε ε ε−1 0 0 ε 0 0 0 0 1 2ε  ,  ε 0 1 0 0 1 0 0 0 0 ε 0 0 0 0 1  . E12: E12 −→ E14 E12 −→ E5 E12 −→ E4 E12 −→ E13 1 0 0 0 0 1 0 0 0 0 ε 0 0 0 0 1  ,  1 0 0 1 −1 ε 1 0 0 0 ε 0 0 0 0 1  ,  ε 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1  ,  ε 0 ε−1 0 0 1 0 0 0 0 iε−1 0 0 0 0 1  . The non-contraction of E12 to D1D, E6, D2D: Demanding that lim ε−→0 Â(ε) t B17(1)Â(ε) converge to the Canonical form of one of the systems D1D, E6, D2D, and observing the entries on places (1, 1), (2, 2), (3, 1), and (3, 2) in the matrix equation we obtain the equations: lim ε−→0 A2 1,1 = 1, lim ε−→0 A2 1,2 = 0, lim ε−→0 A1,3A1,1 = 0, lim ε−→0 A1,2A1,3 = 1, which obviously can not hold simultaneously. Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 27 D1D: D1D −→ E5 D1D −→ E13 D1D −→ E4 1 0 0 0 0 ε 0 1 0 0 1 0 0 0 0 1  ,  ε2 ε 0 ε−1 ε−1 ε 0 0 0 0 ε 0 0 0 0 iε−1  ,  ε2 ε 0 ε−1 0 ε 1/2 0 0 0 1 0 0 0 0 iε−1  . The non-contraction of D1D to E6, E12, S5: Demanding that lim ε−→0 Â(ε) t B16(1)Â(ε) converge to the canonical form of one of the systems E6, E12, S5, and observing the entries on places (1, 1), (1, 2), (4, 1), and (4, 2) in the matrix equation we obtain the equations: lim ε−→0 A1,1 = ±1, lim ε−→0 A1,2 = 0, lim ε−→0 A1,2A1,4 = 1, lim ε−→0 A1,1A1,4 = 0, which obviously can not hold simultaneously. E3: E3 −→ E5 E3 −→ E4 1 0 0 0 0 ε 1 1 0 0 i 0 0 0 0 1  ,  ε 0 1 −iε−1 iε ε i ε−1 0 0 1 0 0 0 0 1  . D4(b)D: D4(b)D −→ E4 D4(b)D −→ E14 D4(b)D −→ E5 ε 0 1 0 0 ε 0 ε−1 0 0 ε2 0 0 0 0 1√ 2ε  ,  1 0 0 0 0 ε 0 ε−1 0 0 ε2 0 0 0 0 1√ 2ε  ,  1 0 0 0 0 ε 1 1 0 0 i 0 0 0 0 ε  , D4(b)D −→ D1D D4(b)D −→ E12 D4(b)D −→ E13 1 0 0 0 0 −ε −ε−1 0 0 0 iε−1 0 0 0 0 ε2  ,  1 0 0 0 0 −ε 1√ 3 −ε−1 0 0 √ 2 3 0 0 0 0 1√ 2ε  ,  ε ε 1 2ε −1 1 2ε −1 iε −iε − i 2ε −1 i 2ε −1 0 0 ε 0 0 0 0 ε  , D4(b)D −→ E3 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ε  . S3: S3 −→ E5 S3 −→ E14 S3 −→ E4 1 0 0 0 0 ε i 0 0 0 1 0 0 0 0 1√ 2 e−i 3π 4  ,  1 0 ε −ε 0 ε ε3 ε−1 0 0 ε2 0 0 0 0 i ε √ 2 e−i 3π 4  ,  ε 0 1 0 0 ε 0 ε−1 0 0 ε2 0 0 0 0 i 1 ε √ 2 e−iπ/4  , S3 −→ D1D S3 −→ E13 S3 −→ E6 1 0 0 0 0 −ε −ε−1 0 0 0 iε−1 0 0 0 0 ε2  ,  ε ε 1 2ε −1 1 2ε −1 iε −iε − i 2ε −1 i 2ε −1 0 0 ε 0 0 0 0 ε  ,  1 0 0 0 0 −ε −ε−1 −ε−1 0 0 −iε−1 0 0 0 0 ei3π/4√ 2ε  , 28 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag S3 −→ E3 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ε  . S6: S6 −→ E12 S6 −→ E14 S6 −→ E5 1 0 0 0 0 ε 1 ε−1 0 0 i 0 0 0 0 iε−1  ,  1 0 0 0 0 ε 0 ε−1 0 0 ε2 0 0 0 0 iε−1  ,  1 0 0 0 0 ε 0 0 0 0 1 0 0 0 0 1  , S6 −→ E4 S6 −→ E13 S6 −→ E18 ε2 ε 1 ε−1 0 ε 0 0 0 0 i 0 0 0 0 iε−1  ,  ε −iε ε−1 0 0 ε iε−1 ε−1 0 0 −ε−1 0 0 0 0 iε−1  ,  1 0 0 0 0 1 0 0 0 0 ε−1 0 0 0 0 ε  . E18: E18 −→ E5 E18 −→ E13 E18 −→ E4 1 0 0 0 0 ε 0 1 0 0 1 0 0 0 0 1  ,  ε ε 1 2ε −1 1 2ε −1 ±iε ∓iε ∓ i 2ε −1 ± i 2ε −1 0 0 i√ 2 ε−1 0 0 0 0 i√ 2 ε−1  ,  ε2 ε 1 (1− i)(2ε)−1 0 ε 0 (1 + i)(2ε)−1 0 0 i 0 0 0 0 (1 + i)(2ε)−1  . D3E: D3E −→ E5 D3E −→ D1D D3E −→ E13 1 0 0 0 0 ε 1 1 0 0 i 0 0 0 0 ε  ,  1 0 0 0 0 −ε −ε−1 −1 0 0 iε−1 0 0 0 0 i  ,  ε−1 ε2 ε 1−i 1 ε2(1+i) iε−1 ε2 − ε 1−i 1 ε2(1−i) 0 0 − √ 2ε 1+i 0 0 0 0 1√ 2ε2  , D3E −→ E4 D3E −→ E3 D3E −→ E18 ε 0 1 0 0 ε 0 ε−1 0 0 ε2 −1+i√ 2ε 0 0 0 1√ 2ε  ,  1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ε  ,  1 0 0 0 0 1 0 0 0 0 ε 0 0 0 0 ε−1  . 5.4 Contractions of the degenerate quadratic algebras and the lower half of the Askey scheme The bottom half of the contraction Askey scheme relating orthogonal polynomials via contrac- tions to degenerate, singular and free superintegrable systems is presented in Fig. 2. The top half of the Scheme, relating to contractions of nondegenerate superintegrable systems can be found in [5] and the full scheme in [17]. On the left side are the orthogonal polynomials that realize finite-dimensional representations of the quadratic algebras via difference or differential operators and on the right those that realize infinite-dimensional bounded below representations. The arrows from the nondegenerate superintegrable system S9, E1, E8 and E3′ correspond to restriction/contractions to degenerate systems, i.e., the parameters in the 3-parameter poten- tials are restricted to the case of only 1 parameter and such that one of the symmetry operators Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 29 Figure 2. Contractions of degenerate systems and the bottom half of the Askey scheme. becomes a perfect square. This increases the symmetry algebra of the resulting degenerate system. Remark 5.7. For reference, the corresponding potentials are • E1: V = α(x2 + y2) + β x2 + γ y2 , • E3′: V = α(x2 + y2) + βx+ γy, • E8: V = α(x−iy) (x+iy)3 + β (x+iy)2 + γ ( x2 + y2 ) , on flat space and • S9: V = α s21 + β s22 + γ s23 , s21 + s22 + s23 = 1, on the 2-sphere. The arrows from one degenerate superintegrable system to another are the standard con- tractions studied above. The singular Laguerre and oscillator superintegrable systems have singular Hamiltonians, and for these systems knowledge of the free quadratic algebra does not necessarily determine the full superintegrable system. The singular Laguerre system is a restric- tion/contraction of E1. The resulting quadratic algebra is isomorphic to {H} ⊕ sl(2,R) [17], in the same sense that the quadratic algebra of the 2D Kepler system is said to be so(3,R). This is 30 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag true only if the system is restricted to an eigenspace of H. The Oscillator system is a contraction of E6 and its quadratic algebra is isomorphic to the 4-dimensional oscillator algebra [17]. The plane and sphere systems are restriction/contractions of degenerate superintegrable systems to free superintegrable systems on the plane and the 2-sphere, respectively. 6 Conclusions and discussion This paper is devoted to the study of the geometric quadratic algebras that correspond to 2D degenerate 2nd order superintegrable systems and general abstract degenerate quadratic alge- bras. Since the geometric quadratic algebras are uniquely determined by their free restrictions we studied only parameter-free geometric and abstract algebras. The geometric algebras were already known; in this paper we classified all abstract algebras in Table 3. We related the geo- metric and free quadratic algebras in Table 4. We showed that there were 5 abstract algebras with no geometric counterpart, but that it was impossible to represent them in phase space. In Section 3 we derived and classified all Bôcher contractions of 2D degenerate 2nd order superintegrable systems. In Fig. 2 we showed the relationship between our results and the bottom half of the Askey scheme. We derived and classified all abstract contractions of the geometric quadratic algebras, presenting the results in Table 6. Comparing the Bôcher and abstract contractions and taking into account the isomorphism of the E14 and S5 algebras, we see that there is a match except for 6 abstract contractions with no geometric realization: paren algebra abstract contracted algebra S6 D1D S3 D1 S3 E13 D3E E18 E12 E5 E14 E5 In two cases the failure of geometric realization is obvious: It is not possible to contract a constant curvature space to a Darboux space [7]. This paper is a partial warm-up for an analogous study of quadratics algebras for 3D superintegrable systems, e.g., [2] and for cubic algebras, e.g., [21]. A Summary of degenerate Laplace and Helmholtz systems The degenerate superintegrable systems can occur only on the 2-sphere, 2D flat space, or one of the 4 Darboux spaces. The notation for these systems is taken from [14, 15]. (We write the systems in classical form; the quantum analogs have the same potentials and the obvious replacements of classical momenta by quantum derivatives.) We assume all variables to be complex. Degenerate complex Euclidean systems H = p2x + p2y + αV (x, y): 1. E18: V = 1√ x2+y2 , Kepler potential, 2. E3: V = x2 + y2, harmonic oscillator, 3. E6: V = 1 x2 , radial potential, 4. E5: V = x, linear potential, 5. E12: V = x+iy√ (x+iy)2+c2 , 6. E14: V = 1 (x+iy)2 , Contractions of Degenerate Quadratic Algebras, Abstract and Geometric 31 7. E4: V = x+ iy, 8. E13: V = 1√ x+iy . The last 4 systems are real in Minkowski space. Degenerate systems on the complex sphere: We use the classical realization for o(3, C) with basis J1 = s2ps3 − s3ps2 , J2 = s3ps1 − s1ps3 , J3 = s1ps2 − s2ps1 , and Hamiltonian H = J2 1 + J2 2 + J2 3 + αV . Here s21 + s22 + s23 = 1. 1. S6: V = s3√ s21+s 2 2 , Kepler analog, 2. S3: V = 1 s23 , Higg’s oscillator, 3. S5: V = 1 (s1+is2)2 . The last system is real on the 2-sheet hyperboloid. Degenerate systems on Darboux spaces: 1. D1D: H = 1 4x(p2x + p2y) + α x , 2. D2D: H = x2 x2+1 (p2x + p2y) + α x2+1 , 3. D3E: H = 1 2 e2x ex+1(p2x + p2y) + α ex+1 , 4. D4(b)D: H = − sin2 2x 2 cos 2x+b(p 2 x + p2y) + α 2 cos 2x+b . Remark A.1. Every degenerate system occurs as a “restriction” of at least one nondegenerate system, although the symmetry algebra grows. For example the classical nondegenerate S9 system has the Hamiltonian H = J2 1 + J2 2 + J2 3 + a1 s21 + a2 s22 + a3 s23 and a basis of symmetries L1 = J2 3 + a1 s22 s21 + a2 s21 s22 , L2 = J2 1 + a2 s23 s22 + a3 s22 s23 , L3 = J2 2 + a3 s21 s23 + a1 s23 s21 , where H = L1 + L2 + L3 + a1 + a2 + a3. If we let a1 → 0, a2 → 0 we obtain the Hamiltonian for S3: H ′ = J2 1 + J2 2 + J2 3 + a3 s23 . However, now L1 → L′1 = J2 3 , L2 → L′2 = J2 1 + a3 s22 s23 , L3 → L′3 = J2 2 + a3 s21 s23 , and the restricted system now admits the 1st order symmetry J3 as well as a new 2nd order symmetry {J3, L′2}. These 4 symmetries are related by the Casimir. A table with all of the restrictions of nondegenerate systems on constant curvature spaces to degenerate systems can be found in [16] and a table with all restrictions of nondegenerate systems on Darboux spaces can be found in [7]. Acknowledgments 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). The author M.A. Escobar 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). We thank a referee for pointing out the relevance of references [4, 6, 20]. 32 M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag References [1] Bôcher M., Über die Riehenentwickelungen der Potentialtheory, B.G. Teubner, Leipzig, 1894. [2] Capel J.J., Kress J.M., Post S., Invariant classification and limits of maximally superintegrable systems in 3D, SIGMA 11 (2015), 038, 17 pages, arXiv:1501.06601. [3] Daskaloyannis C., Tanoudis Y., Quantum superintegrable systems with quadratic integrals on a two dimen- sional manifold, J. Math. Phys. 48 (2007), 072108, 22 pages, math-ph/0607058. 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publisher	Інститут математики НАН України
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spelling	Escobar Ruiz, M.A. Subag, E. Miller Jr., W. 2019-02-19T19:34:18Z 2019-02-19T19:34:18Z 2017 Contractions of Degenerate Quadratic Algebras, Abstract and Geometric / M.A. Escobar Ruiz, Willard Miller Jr, E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E70; 16G99; 37J35; 37K10; 33C45; 17B60; 81R05; 33C45 DOI:10.3842/SIGMA.2017.099 https://nasplib.isofts.kiev.ua/handle/123456789/149270 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 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between Bôcher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all Bôcher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric Bôcher contractions. 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). The author M.A. Escobar 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). We thank a referee for pointing out the relevance of references [4, 6, 20]. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Contractions of Degenerate Quadratic Algebras, Abstract and Geometric Article published earlier
spellingShingle	Contractions of Degenerate Quadratic Algebras, Abstract and Geometric Escobar Ruiz, M.A. Subag, E. Miller Jr., W.
title	Contractions of Degenerate Quadratic Algebras, Abstract and Geometric
title_full	Contractions of Degenerate Quadratic Algebras, Abstract and Geometric
title_fullStr	Contractions of Degenerate Quadratic Algebras, Abstract and Geometric
title_full_unstemmed	Contractions of Degenerate Quadratic Algebras, Abstract and Geometric
title_short	Contractions of Degenerate Quadratic Algebras, Abstract and Geometric
title_sort	contractions of degenerate quadratic algebras, abstract and geometric
url	https://nasplib.isofts.kiev.ua/handle/123456789/149270
work_keys_str_mv	AT escobarruizma contractionsofdegeneratequadraticalgebrasabstractandgeometric AT subage contractionsofdegeneratequadraticalgebrasabstractandgeometric AT millerjrw contractionsofdegeneratequadraticalgebrasabstractandgeometric

Contractions of Degenerate Quadratic Algebras, Abstract and Geometric

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