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 their quadratic algebras can be related by geometric contractions, induced by B\^ocher contractions of the conformal Lie algebra $\mathfrak{so}(4,\mathbb {C})$ 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 B\^ocher 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 B\^ocher 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 B\^ocher contractions.


Introduction
An abstract degenerate (quantum) quadratic algebra Q is a noncommutative multiparameter associative algebra generated by linearly independent operators X, H, L 1 , L 2 , with parameters a i , such that H is in the center and the following commutation relations hold [17]: Finally, there is the relation: G ≡ 0≤e 1 +e 2 +e 3 +e 4 ≤2 S e 1 ,e 2 ,e 3 ,e 4 L e 1 1 , L e 2 2 , X 2e 4 H e 3 + c 1 XL 1 X + c 2 XL 2 X = 0, (1.3) where {L e 1 1 , L e 2 2 , X 2e 4 } is the 6-term symmetrizer of three operators. The constants P (j) e 1 ,e 2 ,e 3 ,e 4 , T e 1 ,e 2 ,e 3 ,e 4 and S e 1 ,e 2 ,e 3 ,e 4 are polynomials in the parameters a i of degrees 1 − e 1 − e 2 − e 3 − e 4 , 1 − e 1 − e 2 − e 3 − e 4 and 2 − e 1 − e 2 − e 3 − e 4 , respectively, while c 1 , c 2 are of degree 0. If all parameters a j = 0 the algebra is free. For these quantum quadratic algebras there is a natural grading such that the operators H, L j 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 independent generators X , H, L 1 , L 2 , and parameters a i , satisfying relations (1.1), (1.2), (1.3) with the commutator replaced by the Poisson bracket, H, L j , X by H, L j , X , and the symmetrizer {L e 1 1 , L e 2 2 , X e 3 } by the product L e 1 1 L e 2 2 X e 3 /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, L j ] = 0, L 3 = 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 L j (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 g jk (x)p j p k + V (x) in local coordinates x 1 , x 2 , p 1 , p 2 that admit 3 functionally independent phase space functions H, L 1 , L 2 in involution with H, the maximum possible.
A system is of order K if the maximum order of the constants of the motion L j , j = 3, as polynomials in p 1 , p 2 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.
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: The symmetry operators generate a nondegenerate quadratic algebra with parameters a j . The exceptional case is characterized by the fact that the symmetry generators are functionally 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 = X 2 . 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: These theorems were proved for systems in [16]. The proofs are constructive: Given a free quadratic algebraQ 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 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.

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 = p 2 x +p 2 y λ(x,y) + αV and α is a parameter. This is equivalent to the Laplace-like equation x + p 2 y + V 0 −ẼW = 0, whereẼ is an arbitrary parameter, W is a nonconstant function, and V 0 , W are independent of E, then W , by division, defines a conformal Stäckel transform to the superintegrable Helmholtz systemH = 1 W (p 2 x + p 2 y + V 0 ) =Ẽ. If the Laplace system admits another splitting p 2 x + p 2 y + V 0 −Ẽ W = 0, it determines another superintegrable Helmholtz systemH = 1 W (p 2 x + p 2 y + V 0 ) =Ẽ andH can be obtained fromH 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 a i 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.
Here, for example, system D3E belongs to class C and is obtained from the Laplace equation by dividing it by a 3 − a 4 (x 2 + y 2 ) where a 3 a 4 = 0, whereas E18 is obtained by the same division with a 3 = 0, a 4 = 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 x 1 , x 2 , x 3 , x 4 on the null cone x 2 1 + x 2 2 + x 2 3 + x 2 4 = 0, related to flat space coordinates x, y by Thus the Laplace equation H ≡ p 2 x + p 2 y + V = 0 in Cartesian coordinates becomes 4 k=1 p 2 x k + V B = 0 in tetraspherical coordinates. Here, the s j refer to coordinates on the unit 2-sphere: s 2 1 + s 2 2 + s 2 3 = 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( )y, and x = (x 1 , . . . , x 4 ), y = (y 1 , . . . , y 4 ) be column vectors, and A = (A jk ( )) be a 4 × 4 matrix with matrix elements 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 If, in addition, A ∈ O(4, C) for all = 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 contractions, 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) = a 1 V 1 (x) + a 2 V 2 (x) and generating conformal symmetries X = X +W 0 , L 1 = L 1 +W 1 , L 2 = L 2 +W 2 , where X , L 1 , L 2 are free 2nd order conformal symmetries and W 0 , W 1 , W 2 are functions of the tetraspherical coordinates x i . Applying a Bôcher contraction A( ) to the free symmetries we obtain where X ∈ so(4, C) and the L j are quadratic in so(4, C). By a change of basis {L 1 , L 2 } 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 , 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 where only a finite number of the coefficients c k , d k can be nonzero. Then it is a linear algebra problem to determine the c k , d k such that lim →0 V (x( ), a( )) = V (b, y) exists for independent potential functions V 1 (y) , V 2 (y) , where the nonzero c k , d k are linear in b 1 , b 2 . 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 are not invariant under a permutation of the indices of the hyperspherical coordinates defining the contraction matrix A ij . 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 K 1 can be obtained from a system in class K 2 via contraction provided there is a directed arrow path from K 2 to K 1 . 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. 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 , and the transformed Hamiltonian will beĤ = 1 F (x,b) H, where the transform is determined by the fixed vector (b 3 , b 4 ). Now we apply the Bôcher contraction [1, 1, 1, 1] → [2, 1, 1] to this system. Depending on the permutation of the indices x j , in the limit as → 0 the potential where the integer exponent α depends upon our choice of b. From our theory, the system defined by HamiltonianĤ = lim 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 = V A /F contracts to the Helmholtz superintegrable system with potential V S /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.

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.
4 Classif ication of free abstract degenerate classical quadratic algebras

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 Poisson algebra possessing a linearly independent generating set {L 1 , L 2 , L 3 = H, X } that satisfy: 1. The associative product is abelian.
7. Any nonzero polynomial F of minimal degree such that F (L 1 , L 2 , L 3 , L 4 ) = 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 L 3 = H.
9. The polynomial G depends non-trivially on at least one of the non-central 2nd order generators L 1 , L 2 .
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 C 3 can be found in [20,Section 9.2]. Poisson structures on the manifold R 3 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.
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: (We shall refer to equations (4.1) as the structure equations of A).
The above equation is a polynomial expression in is either zero or a polynomial of degree one in the variables χ 2 , L 1 , L 2 , H. Since at least one of the terms ∂G ∂L 1 , ∂G ∂L 2 is non-zero then we must have for some K. If K = 0 then the first order element X must be in the center of A which is impossible.
and along with our assumption that at least one of the terms ∂G ∂L 1 , ∂G ∂L 2 is non-zero we see that Corollary 4.4. Keeping the same notations as of Definition 4.1, let A be a free abstract quadratic algebra. Then G(L 1 , L 2 , L 3 , L 4 ) depends non-trivially in at least two of the three generators L 1 , L 2 , L 4 .
Proof . It easily follows from the structure equations (4.1) that if G(L 1 , L 2 , L 3 , L 4 ) depends on at most one of the generators L 1 , L 2 , L 4 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 superintegrable 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. For quantum degenerate systems, knowledge of the Casimir relation G(L 1 , L 2 , 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].

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 {L 1 , L 2 , L 3 , X }, the Casimir G is given by a symmetric 4 × 4 matrix B(G) defined by 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 −1 K 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 {L 1 , L 2 , L 3 , 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 {L 1 , L 2 , L 3 , X }, L 1 , L 2 , L 3 , X of the same algebra A such that: Then there is a 'change of basis matrix' A of the form For A as above we define 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 G degn . The action of such (A, z) ∈ G degn on the generators is given by equation (4.3) and on (G, K) by where B(G) is the symmetric matrix representing G with respect to {L 1 , L 2 , L 3 , L 4 }, and B(G ) is the symmetric matrix representing G := (A, z) · G with respect to L 1 , L 2 , L 3 , L 4 . We shall call the subgroup of G degn of all pairs of the form (A, 1) as the group of linear change of bases,we denote it by G lin degn . In the following we shall determine a canonical form for the pair (B(G), K) of each abstract quadratic algebra. This means that we shall decide upon a unique representative from each orbit of G degn 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 G degn that fixes the value of K. Explicitly this group is given by

The canonical form
In this section after we identify some invariants of orbits of (B(G), K) under the action of G K 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) ∈ G K degn we introduce the following notations:

Direct calculation shows that
A t B(G) A = r t br r t (bs + ct) s t br + tc t r s t (bs + ct) + t(c t s + dt) .
Hence under the action G K 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) ∈ G K degn that cast B(G) into exactly one of the following forms depending on rank(b).
Below we analyze the different cases according to rank(b) we summarize the results of all cases in Table 3.

The rank 2 case
If

The rank 1 case
From If We define the canonical form to be

The rank 0 case
If Hence in the case of b = 0, the rank of c =  In this case we have the following forms The first case can be reduced to the canonical form

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 degenerate quadratic algebras # rank(b) invariant form canonical form 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 where the a j are the parameters in the potential. To perform a general Stäckel transform of this system with nonsingular transform matrix C = (c jk ): 1) we set a j =  Table 3 associated with the equivalence class [A] are 1: b 44 = 1 and 2: b 44 = 1.
The superintegrable system E6, with degenerate potential, can be defined by Going through the same procedure as above, we obtain the equivalence class where |c 12 | + |c 22 | > 0. The canonical form associated with this equivalence class is 4: all cases.
The superintegrable system E3, with degenerate potential, can be defined by The equivalence class is  where |c 12 | + |c 22 | > 0. The canonical forms associated with this equivalence class are 1: b 44 = 0, and 2: b 44 = 0.
The superintegrable system E14 can be defined by The equivalence class is where |c 12 | + |c 22 | > 0. The canonical forms associated with [E] are 6: all cases.
The superintegrable system E4 can be defined by The equivalence class is where |c 12 | + |c 22 | > 0. The canonical forms associated with [F ] are 9: b 33 = 1 and 10: all cases. Heisenberg systems: In addition there are systems that can be obtained from the degenerate 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 several have the interpretation of time-dependent Schrödinger equations in 2D spacetime, so we also consider them geometrical. 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 M 1 = p x , M 2 = xp y , E = p y , where E 2 = H. The systems are all of type 4. We will devote a future paper to their study. The possible canonical forms are 3: b 33 = b 44 = 0, b 34 = 1 and 5: b 44 = 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. 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: L 2 1 + H + X 2 2 = 0. We can factor G as (H + X 2 + iL 1 )(H + X 2 − iL 1 ) = 0. This is possible only if one of the factors vanishes; hence the generators are linearly dependent. Impossible! 2. Case 3: 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: L 2 1 + H 2 + 2HX 2 = 0. Here, L 2 1 = −H(H + 2X 2 ). If L 1 doesn't factor then it must be a multiple of H and of H + 2X 2 simultaneously. Impossible! Suppose then that L 1 = 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 L 1 , 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 = Y 2 . Thus Y is a 1st order constant of the motion, necessarily proportional to X . We conclude that H ∼ X 2 . Impossible! 4. Case 8: 2L 2 H + X 4 = 0. Since X is 1st order, and 2L 2 H = −X 4 both H and L 2 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: L 2 X = 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.

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.

Contraction of a free abstract quadratic algebra
. Then we say that B is a contraction of A.
The meaning of the definition is that in any generating set {L 1 , L 2 , L 3 , L 4 } of A (satisfying the same assumptions as before) the corresponding matrix B(G A ) and K A satisfy where (B(G B ), K B ) is a realization of the Casimir of B in some generating set. In the classification below we are using a more refined class of contractions. We shall call these contractions algebraic. By definition an algebraic contraction is a contraction that can be realized via a map −→ (A , z ) from some punctured neighborhood of 0 in C * into G degn such that z as well as the entries of A are rational functions in . Then there is another continuous map −→ (C , z ) from C * into G degn such that lim −→0 Proof . Define 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 on symmetric matrices B by B −→ A t B A. We shall denote the space of 4 by 4 complex symmetric matrices by Sym(4, C) and the group of matrices of the form (5.5) by G degn . The group G degn is a complex algebraic group and the space Sym(4, C) is a complex algebraic variety on which G degn 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 represents 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.

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 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 observations. 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: 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. By the discussion above, besides contractions between two algebras in the same class R i,j we can potentially find a contraction only according to the following diagram: The classification is summarized in Table 6. Detailed analysis is presented below. Representatives for all possible contractions are given in Section 5.3.7.

Contractions between rank two free abstract quadratic algebras
Note that for any of the B can matrices for the systems where B 0 ∈ R 4,2 ∪ R 3,2 . Below we prove that without loss of generality we can assume that r t r is diagonal and s is the zero two by two matrix. Then there exists a continuous function → A from C * to G degn with each entry polynomial in such that A t B A is of the form A t is the identity matrix and  Assuming that this matrix is in the form of (5.8) then this implies that exist β, γ, δ complex valued functions of define on C * such that  E5: D2D: E6: E12: The non-contraction of E12 to D1D, E6, D2D: Demanding that lim which obviously can not hold simultaneously.

D1D
: The non-contraction of D1D to E6, E12, S5: Demanding that lim 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(x 2 + y 2 ) + β x 2 + γ y 2 , • E3 : V = α(x 2 + y 2 ) + βx + γy, • E8: V = α(x−iy) (x+iy) 3 + β (x+iy) 2 + γ x 2 + y 2 , on flat space and The arrows from one degenerate superintegrable system to another are the standard contractions 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 restriction/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 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.

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 algebras. 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 geometric 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 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 = p 2 x + p 2 y + αV (x, y):