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Lowest dimensional example on non-universality of generalized In\"on\"u-Wigner contractions

We prove that there exists just one pair of complex four-dimensional Lie algebras such that a well-defined contraction among them is not equivalent to a generalized IW-contraction (or to a one-parametric subgroup degeneration in conventional algebraic terms). Over the field of real numbers, this pair of algebras is split into two pairs with the same contracted algebra. The example we constructed demonstrates that even in the dimension four generalized IW-contractions are not sufficient for realizing all possible contractions, and this is the lowest dimension in which generalized IW-contractions are not universal. Moreover, this is also the first example of nonexistence of generalized IW-contraction for the case when the contracted algebra is not characteristically nilpotent and, therefore, admits nontrivial diagonal derivations. The lower bound (equal to three) of nonnegative integer parameter exponents which are sufficient to realize all generalized IW-contractions of four-dimensional Lie algebras is also found.


Introduction
Limiting processes (contractions) of Lie algebras appear in different areas of physics and mathematics, e.g., in the study of representations, invariants and special functions. Perhaps the most important example of contraction of Lie algebras arising in physics is a singular transition from the Poincaré algebra to the Galilei one which corresponds to the limit transition from relativistic mechanics to classical mechanics when the velocity of light is assumed to go to infinity. Another important example is the transition from the Heisenberg algebra to the Abelian algebra. In physical terms the latter means taking the classical limit of quantum mechanics when the Planck constant goes to zero; the linear term in the expansion of the commutator in then yields the Poisson bracket. It is important to stress that contractions of Lie algebras provide only an initial symmetry background for limit transitions among physical theories. Careful analysis of such transitions necessarily includes, in particular, the study of contractions for representations of Lie algebras associated with these theories. For example, it was shown in [17] that Maxwell equations admit two possible nonrelativistic limits, accounting respectively for electric and magnetic effects. In terms of representations of Lie algebras this means that the representation of the Poincaré algebra corresponding to the Maxwell equations with currents and charges admits two inequivalent contractions corresponding to the contraction from the Poincaré algebra to the Galilei one. (See also discussion on applications of contractions in [21] and references therein.) The concept of the Lie algebra contraction was introduced by Segal [26] via limiting processes of bases. It became well known thanks to the papers by Inönü and Wigner [14,15] who invented the so-called Inönü-Wigner contractions (IW-contractions). A rigorous general definition of contraction, based on limiting processes of Lie brackets, was given by Saletan [25]. He also studied the entire class of one-parametric contractions whose matrices are first-order polynomials with respect to contraction parameters. IW-contractions form a special subclass in the class of Saletan contractions.
Another extension of the class of IW-contractions was introduced by Doebner and Melsheimer [9]. They used contraction matrices which become diagonal after choosing suitable bases in the initial and contracted algebras, with diagonal elements being real powers of the contraction parameters. (In fact, integer exponents are sufficient, see [24] for a simple geometric proof of this longstanding [28] conjecture.) In the modern physical literature, such contractions are usually called generalized Inönü-Wigner contractions, probably following [11], although a number of other names (p-contractions, Doebner-Melsheimer contractions and singular IWcontractions [18]) were previously used. In algebraic papers, similar contractions are called one-parametric subgroup degenerations (in a similar fashion, general contractions are extended to degenerations which are defined for Lie algebras over an arbitrary field in terms of the orbit closures with respect to the Zariski topology) [3,4,6,10]. Note that in fact a one-parametric subgroup degeneration is associated with a one-parametric matrix group only upon choosing special bases in the corresponding initial and contracted algebras. Unfortunately, this fact is often ignored.
For a long time it was not known whether any continuous one-parametric contraction can be represented by a generalized IW-contraction. As all continuous contractions arising in the physical literature enjoy this property, it was even claimed [28] that this is true for an arbitrary continuous one-parametric contraction but the proof presented in [28] is not correct [21].
The first crucial advance in tackling this problem was made in [3,4] where examples of contractions to characteristically nilpotent Lie algebras were constructed for all dimensions not less that seven. Since each proper generalized IW-contraction induces a proper grading for the contracted algebra and each characteristically nilpotent Lie algebra possesses only nilpotent derivations and hence has no proper gradings, any contraction to characteristically nilpotent Lie algebras is obviously inequivalent to a generalized IW-contraction. Unfortunately, these examples are not yet well-known to the physical community. This is why their detailed analysis and extension to other nilpotent algebras will be a subject of [5].
Contractions of low-dimensional Lie algebras were studied in a number of papers (see, e.g., [1,2,6,4,16,21,27] and the review of these results in [21]). Thus, it was shown in [21] that each contraction of complex three-dimensional Lie algebras is equivalent to a simple IWcontraction. Any contraction of real three-dimensional Lie algebras is realized by a generalized IW-contraction with nonnegative powers of the contraction parameter which are not greater than two. Moreover, only the contraction of so(3) to the Heisenberg algebra is inequivalent to a simple IW-contraction. The same result for continuous one-parametric contractions of real three-dimensional Lie algebras was also claimed in [27] but contractions within parameterized series of algebras were not discussed. All possible contractions of three-dimensional Lie algebras were realized by generalized IW-contractions much earlier (see, e.g., [8,13]). Therefore, the problem was to prove that there are no other contractions of three-dimensional Lie algebras. For the complex case, it made in a rigorous way in [6].
Almost all contractions of four-dimensional Lie algebras were realized in [21] via generalized IW-contractions. For the real case, the exceptions were the contractions where the above Lie algebras have the following nontrivial commutation relations:   Since the complexifications of the algebras 2A 2.1 and A 4.10 are isomorphic, there were only two exceptions in [21] for the complex case: 2g 2.1 → g 4.1 and 2g 2.1 → g 1 ⊕ g 3.2 . Here g ... denotes the complexification of the algebra A ... .
Recently Campoamor-Stursberg found that in fact both contractions to A 4.1 are equivalent to generalized IW-contractions [7]. As remarked by Nesterenko [20], the matrix proposed in [7] for the contraction 2A 2.1 → A 4.1 can be optimized via lowering the maximal parameter exponent. See Section 4 for the extended discussion of the contraction 2g 2.1 → g 4.1 .
In the present paper we first provide a detailed proof of the fact that the contraction 2g 2.1 → g 1 ⊕ g 3.2 is not equivalent to a generalized IW-contraction. As all other contractions relating complex four-dimensional Lie algebras were already realized as generalized IW-contractions in [7,21], we can state the main results of our paper. Theorem 1. There exists a unique contraction among complex four-dimensional Lie algebras (namely, 2g 2.1 → g 1 ⊕ g 3.2 ) which is not equivalent to a generalized Inönü-Wigner contraction. Corollary 1. There exist precisely two contractions among real four-dimensional Lie algebras (namely, 2A 2.1 → A 1 ⊕ A 3.2 and A 4.10 → A 1 ⊕ A 3.2 ) which cannot be realized as generalized Inönü-Wigner contractions.
Combining the results of [7], [21] with those from Section 4 of the present paper yields the following assertion.
Theorem 2. Any generalized Inönü-Wigner contraction among complex or real fourdimensional Lie algebras is equivalent to the one including only nonnegative integer parameter exponents which are not greater than three. This upper bound is exact, i.e., it cannot be totally decreased for all four-dimensional Lie algebras.

Contractions, generalized IW-contractions and gradings
Let g = (V, [·, ·]) be an n-dimensional Lie algebra with an underlying n-dimensional vector space V over R or C and a Lie bracket [·, ·], n < ∞. Usually a Lie algebra g is defined by the commutation relations in a fixed basis {e 1 , . . . , e n } of V . Namely, it is sufficient to write down the nonzero commutators [e i , e j ] = c k ij e k , i < j, where c k ij are components of the structure constant tensor of g. In what follows the indices i, j, k, i ′ , j ′ and k ′ run from 1 to n and the sum over the repeated indices is understood unless otherwise explicitly stated.
Using a continuous mapping U : (0, 1] → GL(V ) we construct a parameterized family of the Lie algebras g ε = (V, [·, ·] ε ), ε ∈ (0, 1], isomorphic to g. For each ε, the new Lie bracket [·, ·] ε on V is defined via the old one as follows: Definition 1. If for any x, y ∈ V there exists the limit then [·, ·] 0 is a well-defined Lie bracket. The Lie algebra g 0 = (V, [·, ·] 0 ) is called a one-parametric continuous contraction (or just a contraction) of the Lie algebra g. The procedure g → g 0 that yields the Lie algebra g 0 from the algebra g is also called a contraction.
If a basis of V is fixed, the operator U ε is defined by the corresponding matrix and Definition 1 can be restated in terms of structure constants. Let c k ij be the structure constants of the algebra g in the fixed basis {e 1 , . . . , e n }. Then Definition 1 is equivalent to requiring the limit to exist for all values of i ′ , j ′ and k ′ and, therefore, c k ′ 0,i ′ j ′ are components of the well-defined structure constant tensor of a Lie algebra g 0 . The parameter ε and the matrix-valued function U ε are called a contraction parameter and a contraction matrix, respectively.
The contraction g → g 0 is called trivial if g 0 is Abelian and improper if g 0 is isomorphic to g. Definition 2. The contractions g → g 0 andg →g 0 are called (weakly) equivalent if the algebras g andg 0 are isomorphic to g and g 0 , respectively.
Using the weak equivalence concentrates one's attention on existence and results of contractions and ignores differences in the ways contractions are performed. To take into account these different ways, we can introduce different notions of stronger equivalence. Let Aut(g) denote the group of automorphisms of the Lie algebra g. We identify automorphisms with the corresponding matrices in the canonical basis.
Definition 3. Two one-parametric contractions in the same pair of Lie algebras (g, g 0 ) with the contraction matrices U (ε) andŨ (ε) are called strongly equivalent if there exist δ ∈ (0, 1], mappingsÛ : (0, δ] → Aut(g) andǓ : (0, δ] → Aut(g 0 ) and a continuous monotonic function The concept of contraction is generalized to arbitrary fields in terms of orbit closures in the variety of Lie algebras [6,3,4,10,16]. Namely, let V be an n-dimensional vector space over a field F and L n = L n (F) denote the set of all possible Lie brackets on V . We identify µ ∈ L n with the corresponding Lie algebra g = (V, µ). L n is an algebraic subset of the variety V * ⊗ V * ⊗ V of bilinear maps from V × V to V . Indeed, upon fixing a basis {e 1 , . . . , e n } of V we have a bijection among L n and which is determined for any Lie bracket µ ∈ L n and any structure constant tuple (c k ij ) ∈ C n by the formula µ(e i , e j ) = c k ij e k . L n is called the variety of n-dimensional Lie algebras (over the field F) or, more precisely, the variety of possible Lie brackets on V . The group GL(V ) acts on L n in the following way: (It is a left action in contrast to the right action which is more usual for the 'physical' contraction tradition and defined by the formula (U · µ)(x, y) = U −1 µ(U x, U y) . Of course, this difference is not of fundamental significance. We use the right action throughout the rest of the paper.) Denote by O(µ) the orbit of µ ∈ L n under the action of GL(V ) and by O(µ) the closure of O(µ) with respect to the Zariski topology on L n .
For F = C the orbit closures with respect to the Zariski topology coincide with the orbit closures with respect to the Euclidean topology and Definition 4 is reduced to the usual definition of contractions.
In fact, the signature of a generalized IW-contraction C is defined up to a positive multiplier since the reparametrization ε =ε β , where β > 0, leads to a generalized IW-contraction strongly equivalent to C. Moreover, it is sufficient to consider signatures with integer components only. Theorem 3. Any generalized IW-contraction is equivalent to a generalized IW-contraction with an integer signature (and with the same associated constant matrices).
This result was believed to hold for a long time (see e.g. [28]) but a completely rigorous proof, which is surprisingly simple, was found only recently in [24].
Upon replacing the Lie algebras by isomorphic ones or, in other words, changing bases in the initial and contracted algebras, we can make the matrices A and P equal to the unit matrix. This is appropriate for some theoretical considerations but much less so for working with specific Lie algebras. If U ε = diag(ε α 1 , . . . , ε αn ) then the structure constants of the resulting algebra g 0 are given by the formula c k 0,ij = lim ε→+0 c k ij ε α i +α j −α k with no sums over the repeated indices. Therefore, the constraints α i + α j α k if c k ij = 0 are necessary and sufficient for existence of the well-defined generalized IW-contraction with the contraction matrix U ε , and c k 0,ij = c k ij if α i + α j = α k and c k 0,ij = 0 otherwise. This obviously implies that the conditions for existence of generalized IW-contractions and the structure of contracted algebras can be reformulated in the basis-independent fashion in terms of gradings of contracted algebras associated with filtrations on initial algebras [18,10]. (Probably, this was a motivation for introducing the purely algebraic notion of graded contractions [19].) In particular, the contracted algebra g 0 has to admit a derivation whose matrix is diagonalizable to diag(α 1 , . . . , α n ).
Certain amount of freedom in the matrices A and P is preserved even after fixing the canonical commutation relations. These matrices are defined up to the transformations where M and M 0 are matrices of automorphisms for algebras g and g 0 , respectively, and N is a matrix commuting with the diagonal part W ε . This means that the matrix N corresponds to an arbitrary change of basis within components of the grading of g 0 associated with W ε . The generalized IW-contractions with the matrices U ε andŨ ε =ÃW εP , whereW ε = diag(ε βα 1 , . . . , ε βαn ) for some β > 0 are obviously equivalent.
Let the canonical basis of g 0 be associated with a grading which is isomorphic to the one induced by the matrix W ε . Then the matrix P can be represented as a product P grad P aut , where P grad and P aut are matrices of a change of basis within the graded components and of an automorphism of g 0 , respectively. Therefore, in such a case we can get rid of the matrix P by setting it equal to the unit matrix up to the above equivalence. The guaranteed presence of nontrivial diagonal automorphisms in g 0 further enables us to set det A = 1 in order to simplify the form of the entries of A −1 . If U ε = AW ε , the structure constants of g 0 read , and there is no sum over i, j and k. Simple IW-contractions clearly form a subclass of generalized IW-contractions with signatures equivalent to tuples of zeros and units. They present limit processes of Lie algebras with contraction matrices of the simplest possible type. Most contractions of low-dimensional Lie algebras are equivalent to such contractions. Classifications of IW-contractions for three-and four-dimensional Lie algebras [8,12] can be easily derived from the classifications of subalgebras of such algebras obtained in [22].
3 Nonexistence of generalized IW-contraction from 2g 2.1 to g 1 ⊕ g 3.2 To prove Theorem 1, we use reductio ad absurdum. Namely, suppose that the contraction 2g 2.1 → g 1 ⊕ g 3.2 is realized as a generalized IW-contraction. First of all we should find out which gradings of the algebra g 1 ⊕ g 3.2 can be associated with this contraction.
The derivation algebra of g 1 ⊕g 3.2 consists of linear mappings whose matrices in the canonical basis have the form Therefore, the matrix of any diagonalizable derivation of g 1 ⊕ g 3.2 is reduced, by changing the basis, to the form diag(β, α, α, 0), i.e., each grading of g 1 ⊕ g 3.2 contains a nontrivial component of zero exponent. In view of this fact, the number of components for any grading associated with the contraction 2g 2.1 → g 1 ⊕ g 3.2 has to be greater than two because the contraction in question is not equivalent to a simple IW-contraction [12]. Hence the contraction 2g 2.1 → g 1 ⊕ g 3.2 can generate only gradings with three nonzero components L β , L α and L 0 , where 0 = α = β = 0, dim L β = dim L 0 = 1 and dim L α = 2. We prove that any such gradingG is equivalent, up to automorphisms of g 1 ⊕ g 3. General form of the matrices for the generalized IW-contractions from 2g 2.1 to g 1 ⊕ g 3.2 is U ε = AW ε P , where A and P are constant nonsingular matrices and W ε = diag(ε β , ε α , ε α , 1). Since P is a transition matrix among two graded bases with the same signature (β, α, α, 0), it admits the representation P = P grad P aut , where P grad and P aut are matrices of change of basis within the graded components and of an automorphism of g 1 ⊕ g 3.2 , respectively. The matrix P grad commutes with W ε and can be absorbed into the matrix A by passing from A tõ A = AP grad . The matrix P aut can be ignored as it does not affect the commutation relations of the contracted algebra. Therefore, it is sufficient to consider only contraction matrices of the form U ε = AW ε assuming that P is the unit matrix. Using the scaling automorphisms in 2g 2.1 we can further assume that det A = 1. This assumption significantly simplifies all computations by reducing the size of expressions for the entries of A −1 .
Each of the structure constants ( ij transformed using U ε includes a single power of the parameter ε. The set of possible values for the exponents is We treat two possible cases α > β and β > α separately. In each of these cases we further assume that α and β are positive. Moreover, in the second case we also assume that 2α > β. The systems of algebraic equations for the entries of the matrix A derived under the conditions α > β > 0 or (β > α > 0 and 2α > β) are minimal. Dropping the additional assumptions leads to the extension of the minimal systems with other algebraic equations. The parameters α and β affect only the limiting process ε → 0 and are not explicitly contained in the algebraic equations we have derived. For this reason their specific values are not essential. We can set α = 2 and β = 1 in the case α > β and α = 2 and β = 3 in the case α < β.
In what follows B = (b i j ) denotes the inverse A −1 of the matrix A. The indices p and q run from 1 to 2.
For the values α = 2 and β = 1, the conditions for the matrix of the generalized IWcontraction result in the equations .
Since assuming existence of generalized IW-contractions from 2g 2.1 to g 1 ⊕ g 3.2 leads to contradiction for all possible values of the parameter exponents, this assumption is not true. Taking into account the results of [7,21], we finally arrive at Theorem 1.
The ground field (complex or real) is not essential for the proof. Therefore, the statement on the contraction among the algebras 2g 2.1 and g 1 ⊕ g 3.2 can be directly reformulated for the contraction among their real counterparts 2A 2.1 and A 1 ⊕ A 3.2 . Moreover, if the contraction A 4.10 → A 1 ⊕ A 3.2 could be realized by a generalized IW-contraction over R then the same statement would be true over C for its complexification which is equivalent to the contraction 2g 2.1 → g 1 ⊕g 3.2 . This contradicts the proved nonexistence of generalized IW-contraction among 2g 2.1 and g 1 ⊕ g 3.2 . As a result, we obtain Corollary 1.
The two first quadruples are signatures of generalized IW-contractions from 2g 2.1 to g 4.1 . Considering them, we from the very beginning restrict ourselves to looking for the contraction matrices in the generalized Inönü-Wigner form (see Definition 5) with P equal to the unit matrix and det A = 1.
The quadruple (4, 3, 2, 1) leads to a system involving only three equations for entries of the matrix A: Recall that (b i j ) = A −1 . A particular solution of system (5) and (6) was implicitly found in [7] while constructing a contraction matrix with the signature (4, 3, 2, 1).
For the parameter exponents (3, 2, 1, 1) the system is extended with the single equation We obtain a solution of the whole system (5)- (7) under the constraint det A = 1. Hence the suggested matrix A will be admissible for generalized IW-contractions from 2g 2.1 to g 4.1 with both signatures (4, 3, 2, 1) and (3, 2, 1, 1). Since system (5)- (7) is underdetermined, we can choose simple values for the most of a i j without breaking compatibility of the equations that are not satisfied.

Discussion of technique applied
The proof of Theorem 1 has a number of special features which, when combined, form a technique applicable to a wide range of similar problems. For this reason we decided to list them below.
1. All necessary criteria for general contractions [4,6,21] do not work for the study of generalized IW-contractions since the contraction is known to exist and, therefore, the necessary criteria are definitely satisfied. The problem is to determine whether the contraction can be realized in a special way and this requires other tools.
2. There exists a simple criterion stating that a contraction is not equivalent to a generalized IW-contraction if the contracted algebra admits improper gradings only. In contrast with the contractions to characteristically nilpotent Lie algebras, this criterion is not applicable to the algebra g 1 ⊕ g 3.2 since the latter has nontrivial diagonal derivations and therefore possesses proper gradings.
3. In the canonical basis, the algebra g 1 ⊕ g 3.2 has a two-dimensional algebra of diagonal derivations. Therefore, we have to consider a number of different gradings for the contracted algebra. The study of the gradings aims at resolving a twofold challenge-to obtain possible values of parameter exponents and to understand the structure of constant components of contraction matrices. Thus, the structure of derivations of the algebra g 1 ⊕ g 3.2 implies that only signatures of the form (β, α, α, 0) are admissible.
4. Further restrictions on parameter exponents follow from the absence of simple IWcontractions from 2g 2.1 to g 1 ⊕ g 3.2 . Up to positive multipliers, any signature associated with a simple IW-contraction consists of zeros and units. Hence we have the condition 0 = α = β = 0.
5. The matrix P in the representation U ε = AW ε P of the contraction matrix U ε is determined up to changes of basis within graded components and up to automorphisms of the contracted algebra. Since in the case under consideration the matrix P provides an isomorphism among gradings, we can set P equal to the unit matrix. 6. A significant part of subcases for parameter exponents can be ignored as the associated systems of equations for entries of the matrix A are extensions of their counterparts for other subcases and hence the inconsistency of the former systems is immediate from that of the latter ones.
7. Using the scaling automorphisms of the contracted (or initial) algebra, we set det A = 1 to simplify the entries of the inverse matrix A −1 .
8. We consider each tuple of parameter exponents for which the corresponding system of algebraic equations for entries of the matrix A is minimal. This nonlinear system is represented in a specific form that allows us to apply methods of solving linear systems of algebraic equations. In particular, we try, wherever possible, to avoid writing out the entries of the inverse matrix B = A −1 in terms of entries of the matrix A.
Proving that a generalized IW-contraction 2g 2.1 → g 4.1 with nonnegative integer parameter exponents includes at least one exponent which is not less than three (see Section 4) is also based on the above technique.

Conclusion
The main result of the present paper is important from a number of different points of view. First of all, it gives the exact value of the lowest dimension for which some of well-defined contractions are not realized by generalized IW-contractions. This is the first example of such contractions in dimension less than seven. Moreover, this is also the first example of nonexistence of generalized IW-contraction for the case when the contracted algebra admits nontrivial diagonal derivations. The previous series of examples constructed by Burde [3,4] for dimensions greater than six involve characteristically nilpotent algebras possessing nilpotent derivations only.
Although the contractions considered in this paper do not yet have a direct physical interpretation, the very fact of ending the long-lived illusion of universality of generalized IWcontractions could be of interest for the physical community. In this connection it is important to stress that the Lie algebras involved are considerably less exotic than the characteristically nilpotent algebras and appear, for instance, in general relativity [23]. Thus, the algebra 2A 2.1 can be easily realized as the Lie algebra of the Lie group generated simultaneous scalings and translations in two directions.
Complete solution of the problem of characterizing generalized IW-contractions of fourdimensional complex (resp. real) Lie algebras leads to a number of other interesting open problems.
It is now known that all contractions of three-dimensional complex (resp. real) Lie algebras can be realized via generalized IW-contractions [21] and that this is not true for the dimension four (the present paper) and the dimensions greater than six ( [3,4]). Similar results for dimensions one and two are trivial. The problem of universality of generalized IW-contractions for five-and six-dimensional Lie algebras is still open. It is expected that for these dimensions the answer and the approach to this problem will be similar to those used in the dimension four.
Since generalized IW-contractions are not universal in the whole set of Lie algebras, the following question is natural and important: In which classes of Lie algebras closed under contractions any contraction is equivalent to a generalized IW-contraction? For example, the classes of four-and five-dimensional nilpotent algebras do have this property [5,10,21].
Although the total universality of generalized IW-contractions was disproved by counterexamples [3,4], it was conjectured in [7] after analyzing the classification of contractions of fourdimensional Lie algebras presented in [21] that any contraction of Lie algebras is a composition of generalized IW-contractions. Examples of [3,4] also provide counterexamples for the latter conjecture. There is a contraction among seven-dimensional characteristically nilpotent Lie algebras with orbit dimensions differing by 1. Therefore, this contraction is indecomposable and is not equivalent to a generalized IW-contraction. Representing general contractions of nilpotent algebras via generalized IW-contractions is studied in [5] at greater length. One can state a weaker conjecture that any contraction to a Lie algebra possessing nontrivial gradings is a composition of generalized IW-contractions. This conjecture does not contradict already known four-and seven-dimensional examples of contractions inequivalent to a generalized IW-contraction but it is expected that suitable counterexamples may be found.
The last but not least problem is to find criteria for existence of generalized IW-contractions which would be different from the simplest one, based on testing whether there are any gradings at all in contracted algebras, and would be powerful enough for the case when the contracted algebra possesses non-nilpotent derivations.