Application of the Gel'fand Matrix Method to the Missing Label Problem in Classical Kinematical Lie Algebras

We briefly review a matrix based method to compute the Casimir operators of Lie algebras, mainly certain type of contractions of simple Lie algebras. The versatility of the method is illustrated by constructing matrices whose characteristic polynomials provide the invariants of the kinematical algebras in (3+1)-dimensions. Moreover it is shown, also for kinematical algebras, how some reductions on these matrices are useful for determining the missing operators in the missing label problem (MLP).


Introduction
Casimir operators of Lie algebras were originally introduced in the frame of representation theory of semisimple algebras, and were soon recognized as a powerful tool for the structural analysis. They were determined by means of the universal enveloping algebra of a Lie algebra, and successive applications of Lie groups and algebras to other mathematical and physical problems led to alternative methods to compute these invariants, like the orbit method or the formulation in terms of differential equations, which allow to generalize the concept of invariant functions on Lie algebras beyond the pure algebraic frame.
In 1950 I.M. Gel'fand [1] showed how to use the generic matrix of standard representation of orthogonal groups to determine the Casimir operators of the algebra by means of characteristic polynomials. More specifically, for the pseudo-orthogonal Lie algebra so(p, q) (N = p + q) given by the 1 2 N (N − 1) operators E µν = −E νµ and having brackets: where g = diag (1, . . . , 1, −1, . . . , −1), the Casimir operators were obtained with the formula [1]: M p,q being the matrix 0 · · · −g jj e 1j · · · −g N N e 1N . . . . . . . . . e 1j · · · 0 · · · −g N N e jN . . . . . . . . . e 1N · · · g jj e jN · · · 0         (1) and Id N the identity matrix of order N . Once symmetrized, the functions C k provide the classical operators in the enveloping algebra of so(p, q). This direct matrix approach seems natural, taking into account that the eigenvalues of Casimir operators are essential for the labelling of representations of the Lie algebra. By application of the classical theory of semisimple Lie algebras, similar formulae can be developed for other semisimple Lie algebras [2]. It is natural to ask whether the essence of the Gel'fand method can be generalized to other non-semisimple Lie algebras, by means of extended matrices that possibly correspond to some representation of the algebra. The two main interesting cases are: 1. Generalized Inönü-Wigner contractions of (semi)simple Lie algebras. We have inhomogeneous algebras as a special case.
2. Semidirect products of simple and Heisenberg Lie algebras. The method usually employed to determine the invariants of a Lie algebra consists in solving of a system of partial differential equations (PDEs) related to a realization of the Lie algebra as differential operators [3,4]. If {X 1 , . . . , X n } is a basis of g and {C k ij } is the structure tensor, the realization of g in the space C ∞ (g * ) is given by: . , x n } is the corresponding dual basis, g * being the dual space to g. Invariants of the coadjoint representation are functions on the generators F (X 1 , . . . , X n ) of g such that [X i , F (X 1 , . . . , X n )] = 0, and are determined by solving the system of PDEs: Classical Casimir operators are recovered with replacing the variables x i by the corresponding generator X i (possibly after symmetrizing). The cardinal N (g) of a maximal set of independent solutions is described in terms of the following formula: where A (g) is the matrix representing the commutator table of g over a given basis, i.e., In this work we briefly review some recent work on matrix procedures that generalize the Gel'fand formula for various types of Lie algebras, specifically inhomogeneous Lie algebras. Applications to the computation of missing label operators are also given.

Contractions of simple Lie algebras
Let g be a simple Lie algebra of rank k such that the Casimir operators of g are given by the characteristic polynomial P (T ) of a matrix where Γ 1 is equivalent to the N -dimensional standard representation of g. Suppose further that the Lie algebra g ′ is a contraction of g and that N (g) = N (g ′ ). To compute the Casimir operators of g ′ using a matrix extension of A, the following procedure by steps has been proposed [5]: 2. Let Ψ ǫ be the automorphism giving the contraction 1 and consider the matrix 5. Define I j = C j as the polynomial coefficients of P (T ). 6. Check the functional independence of the functions I j .
7. If necessary, symmetrize the functions I j in order to recover the Casimir operators of g ′ .
The validity of the method is formally proved using the contraction explicitly. If Ψ ε is the automorphism of g defining the contraction, then the matrix expresses the Casimir operators of g over the transformed basis. Developing the corresponding characteristic polynomial P (T ) and taking into account the limit, after some algebraic manipulation we can obtain an expression of P (T ) that does not involve the contraction parameter ǫ anymore. By transitivity of contractions, the same algorithm can be applied formally to the contraction of reductive algebras.
For Lie algebras of the type ws = s − → ⊕ Γ h N , i.e., semidirect products of a simple and a Heisenberg Lie algebra, the argument remains valid for the contraction ws s − → ⊕ Γ (2N + 1)L 1 . The structure of representations of simple Lie algebras compatible with Heisenberg algebras was analyzed in [6]. Depending on this structure, we obtain by contraction either an inhomogeneous Lie algebra or a general affine algebra. The first step is to compute invariants of ws. This is done by constructing a copy of the Levi part s in the enveloping algebra of ws 2 , which can be used as the simple algebra to which the preceding procedure is applied. Remark 1. In general, the shape of P (T ) will depend essentially on the structure of the standard representation and the contraction considered, and possibly involves matrices depending on T . In particular, it will be different for the cases where this representation is by real or complex matrices, i.e., if it is of first or second genus [9]. This fact can also originate some dependence problems.

Inhomogeneous symplectic algebras
As an example of a general class to which the algorithm applies, we consider the inhomogeneous symplectic Lie algebras Isp(2N, R) and the matrix formula obtained for them in [5]. Over the basis 1 By this we mean that the limit [X, Y ] ∞ := lim exists for any X, Y ∈ g. This equation defines a Lie algebra g ′ called the contraction of g (by Ψǫ).
Then the Casimir operators C 2k of Isp (2N, R) are given by the coefficients of the polynomial and C 2N +1,2N +1 is the minor of C obtained deleting the last row and column. Moreover deg C 2k+1 = 2k + 1.

The kinematical algebras in (+ 1)-dimensions
As an interesting physical example, we apply the preceding procedure to the class of kinematical Lie algebras in (3 + 1)-dimensions. This choice is appropriate since it contains algebras of both types and because they are all related by contractions [10,11], thus by transitivity the procedure can be applied. Although their invariants have been obtained repeatedly in different contexts [12,13], it is worthy to be done on the basis of the above arguments, which, moreover, show that the matrix providing the invariants is not necessarily related to a faithful representation of the algebra.
Following the original notation of [10], kinematical Lie algebras are defined over the basis {J α , P α , K α , H} 1≤α≤3 , where J α are spatial rotations, P α spatial translations, K α the boosts and H the time translation, constrained to the condition of space isotropy as well as the assumption that time-reversal and parity are automorphisms of the group. Taking the compact notation [X, Y] = Z for [X α , Y β ] = ε αβγ Z γ , the brackets of the nonisomorphic kinematical Lie algebras are given in Table 1 3 . As shown in [10], any kinematical Lie algebra is obtained by contraction of the simple de Sitter Lie algebras so (3, 2) and so (4, 1). With the exception of the static algebra, all of the remaining ones possess two independent invariants.

De Sitter algebras
Since both de Sitter algebras are simple and pseudo-orthogonal, their Casimir operators follow at once from application of the Gel'fand formula. For obtaining the invariants over the kinematical basis above, the matrix (1) has to be slightly transformed.
1. Anti de Sitter algebra so (3,2) . The matrix related to the standard representations is: Computing the characteristic polynomial we have |D − T Id 5 | = T 5 + C 2 T 3 + C 4 T , where

De
Sitter algebra so (4, 1). The resulting matrix is similar to the previous one: For these two algebras, the result follows at once from the Gel'fand formula.

The nonrelativistic cosmological Lie algebras
The Newton algebras Ne + and Ne − are obtained as contractions of the de Sitter and Anti de Sitter algebras, respectively. Since the subalgebra generated by {K α , P α } is Abelian, it will follow that the invariants of these algebras will not depend on the rotation and time-translation generators.
1. The Newton algebra Ne − [oscillating universe]. It can be easily verified that this Lie algebra is indeed an extension by a derivation of the nine dimensional Lie algebra so (3) − → ⊕ 2adso(3) 6L 1 . By the comment above, the matrix giving the Casimir operators of Ne − will not be related to a faithful representation of the algebra. We have: Expanding the secular equation we arrive at |D − T Id 5 | = T 5 + C 2 T 3 + C 4 T , where 2. The Newton algebra Ne + [expanding universe]. This Lie algebra is also an extension by a derivation of so (3) − → ⊕ 2adso(3) 6L 1 . In this case the matrix to be used is:
1. The Poincaré Lie algebra Iso (3, 1). The matrix D is given by We define P (T ) = |D − T Id 5 | + T |D 55 − T Id 4 |, where D 55 is the minor of D obtained by deleting the fifth column and row. In particular, the matrix D 55 corresponds to that of the Lorentz algebra so (3, 1). Expanding the expression for P (T ), we get P (T ) = Moreover, the matrix D can be decomposed as where D 1 defines a faithful representation of Iso (3, 1).
2. The inhomogeneous algebra Iso (4). Here the polynomial is P (T ) = |D − T Id 5 |+T |D 55 − T Id 4 |, where D 55 is the minor of D obtained deleting the fifth column and row. The matrix D is given by Expanding P (T ), we get P (T ) = T 5 + C 2 T 3 + C 4 T , with In this case, D decomposes as and D 1 is the matrix related to a the faithful representation of Iso (4) by 5 × 5 matrices.

The Carroll Lie algebra
Among the classical kinematical Lie algebras, the Carroll Lie algebra is the only isomorphic to the semidirect product of a simple Lie algebra (the compact algebra so (3)) and a Heisenberg Lie algebra. Indeed the noncentral Casimir operator can be determined using the determinant procedure developed in [8]. However, the Casimir operators (the second is trivial, since the centre is nonzero) can also be obtained by the same method as before.
We define P (T ) = |D − T Id 5 |, where D is the matrix Observe that in this case, the matrix D is dependent on the variable T . Expanding, we get P (T ) = T 5 + C 2 T 3 + C 4 T , where The matrix D decomposes in this case as Again, the matrix D 1 gives rise to a faithful representation of the Carroll algebra.

The Galilei algebra G(2)
As happened for the Newton algebras, the Casimir operators of the Galilei algebra do not depend on the variables {j α , h}. Here we consider the polynomial P (T ) = |D − T Id 5 |, where Also in this case, the matrix D is dependent on the variable T . Developing the polynomial we obtain P (T ) = T 5 + C 2 T 3 + C 4 T , where By the remark above, the preceding matrix is not related to a faithful representation of the algebra.

The static Lie algebra
This algebra is nothing but the splittable affine Lie algebra so (3) − → ⊕ 2adso(3) 6L 1 ⊕ R. As commented, it has four invariants, all of the degree two, To obtain them in matrix form, we consider Simplifyng the coefficients we recover the basis of invariants above. Since the variables associated to the rotations J α do not appear in the invariants, the matrix D does not provide a representation of the static algebra.
For later use we consider the following functions: 4 Applications: The missing label problem As known, irreducible representations of a semisimple Lie algebra are labelled unambigously by the eigenvalues of Casimir operators. In a more general frame, irreducible representations of a Lie algebra g are labelled usingby means of the eigenvalues of its generalized Casimir invariants [14,15]. The number of internal labels needed is If we use a subalgebra h label the basis states of g, then we obtain 1 2 (dim h + N (h)) + l ′ labels from h, where l ′ is the number of invariants of g that depend only on variables of the subalgebra h [15]. In order to label irreducible representations of g uniquely, it is therefore necessary to find additional operators, which are usually called missing label operators. They are found by integrating the equations of system (2) corresponding to the subalgebra generators. The total number of available operators of this kind is easily shown to be m = 2n.
In this situation, it is plausible to think that whenever the Casimir operators of a Lie algebra g can be determined using determinants of (polynomial) matrices, the same procedure could hold for computing missing label operators according to some subalgebra h. In this section we analyze the missing label problem for the chain where g is a kinematical Lie algebra in (3 + 1)-dimensions and so (3) the compact subalgebra generated by the {J µν }. The missing label operators are among the solutions of the equations: Due to the space isotropy condition, the above system is valid for any kinematical Lie algebra. According to formula (6), there are (3))) + l ′ = 1 2 (6 − N (g)) + l ′ missing labels. In any case we have l ′ = 0. Moreover, for any kinematical algebra, with the exception of the static algebra, we obtain n = 2 and four available missing label operators, while for the static algebra we get n = 1 and m = 2. By using of the matrix notation, the system can be rewritten as: Since the matrix has rank three, there are seven independent solutions of the system. The number of solutions that do not depend on the variables {j α } of the subalgebra so (3) is given by [16]: It is straightforward to verify that a complete system of independent solutions is given by: {I 1 = h, I 2 = p α p α , I 3 = k α k α , I 4 = j α j α , I 5 = k α p α , I 6 = j α k α , I 7 = j α p α } .
In particular, the invariants I 1 , I 2 , I 3 , I 5 , which are the independent solutions not involving the variables j α , constitute a set of solutions for the static Lie algebra. The function M = ε αβγ j α p β k γ of (5) is functionally dependent on the previous functions, as shown by the relation M 2 = I 2 5 I 4 + I 2 7 I 3 − I 2 6 I 2 − I 2 I 3 I 4 − 2I 5 I 6 I 7 .
To see how the matrices used for the Casimir operators of kinematical algebras can also be applied to the MLP, we consider the Anti de Sitter algebra so(3, 2). In the notation of (8), the Table 2. Missing label operators for the chain so(3) ֒→ g.
Algebra g Casimir operators of g Missing label operators standard representation of the contracted algebra. Although it provides in many cases closed formulae of the invariants of contractions, the application of the Gel'fand formula is certainly only of interest when the contraction has the same number of invariants than the original algebra. Although in the algebras analyzed here no dependence problems have been encountered, one of the unsolved problems is to find sufficiency criteria to ensure that the contraction of independent invariants of an algebra provides also independent operators of the contraction. Work in this direction is in progress.
As concerns applications, we have seen that the MLP can be analyzed via the generalization of the Gel'fand method, by simply reducing the matrices by zeros corresponding to generators of the subalgebra considered. This point of view could also be interesting in combination with problems in symmetries of differential equations related to contractions of Lie algebras, such as the separation of variables [17].