ar X iv :m at h/ 02 08 19 5v 1 [ m at h. R A ] 2 6 A ug 2 00 2 Non-semisimple Lie algebras with Levi factor so(3), sl(2, R) and their invariants Rutwig Campoamor Departamento de Geometŕıa y Topoloǵıa Fac. CC. Matemáticas Universidad Complutense 28040 Madrid ( Spain ) rutwig@nfssrv.mat.ucm.es Abstract We analyze the number N of functionally independent generalized Casimir invariants for non-semisimple Lie algebras s −→ ⊕Rr with Levi factors isomorphic to so (3) and sl (2, R) in dependence of the pair (R, r) formed by a representation R of s and a solvable Lie algebra r. We show that for any dimension n ≥ 6 there exist Lie algebras s −→ ⊕Rr with non-trivial Levi decomposition such that N ( s −→ ⊕Rr ) = 0. 1 Introduction The important role played by invariant theory in Physics has been recognized long ago. Electroweak interactions and quantum numbers in the study of par- ticle states are based on the concept of symmetry, and their invariants provide fundamental information. Among the various types of symmetry, dynamical ones constitute one of the more important cases, as shown by Gell-Mann and Ne’eman in their hadron classification [1]. The analysis of the group SU (3) resulted in the prediction of new particles whose mass could be derived from the invariants of the group. The invariants of Lie algebras have also shown their efectiveness in the description of Hamiltonians [2], the labelling of irreducible representations or the study of coadjoint orbits [3, 4]. Other important appli- cations of invariants arise in their combination with the theories of Lie algebra contractions, deformations and rigidity [5, 6, 7, 8]. For example, all kinematical algebras are related by a contraction procedure, which has allowed a further analysis of these algebras [7, 9, 10]. The interest of invariants of rigid Lie al- gebras is fully justified by the fact that semisimple Lie algebras are rigid. The invariants of semisimple Lie algebras constitute a classical problem, and it is the only case which has been solved in a satisfactory manner. The invariants of solvable Lie algebras are only studied for specific classes, as they do not underly 1 http://arXiv.org/abs/math/0208195v1 to a structure theory like the classical algebras. What refers to the Lie alge- bras with non-trivial Levi decomposition, invariants are known for physically important algebras, like the special affine algebras sa(n, R), the kinematical Lie algebras and their subalgebras. A formula for the number N (g) of functionally independent invariants of the coadjoint representation of a Lie algebra g was given by Beltrametti and Blasi [11] and Pauri and Prosperi [12] in the mid 1960s. This fact reduces the computation of this number to the determination of the rank of a skew- symmetric matrix A (g) whose entries correspond to the Lie brackets of g. With some effort, this formula can be used to show that the number of invariants of semisimple Lie algebras coincides with its rank [13]. Moreover, it proves that for direct sums g1 ⊕ g2 of Lie algebras the number N (g1 ⊕ g2) is N (g1) + N (g2). One can ask whether for semi-direct sums g = s −→⊕Rr, s being the Levi factor of g, R a representation of s and r the maximal solvable ideal (called radical) of g some formula exists which allows to express N ( s −→⊕Rr ) in terms of N (s) ,N (r) and some quantity related to the representation R. The motivation of this problem lies in the study of the special affine algebras sa (n, R), which are a semidirect sum of the simple Lie algebra sl (n, R) and an n-dimensional abelian Lie algebra [14]. These algebras are known to have only one invariant (which turns out to be a Casimir operator), which shows that the representation plays a crucial role in the semidirect product, and that in principle the existence of a formula expressing the number of invariants in terms of the factors does not exist. The main reason for its nonexistence lies in the distinct possibilities of choice for radicals r for a fixed representation of s. The question that arises naturally in this context is if there exist Lie algebras s −→⊕Rr with non-trivial Levi decomposition (i.e. s 6= 0 and [s, r] 6= 0) such that N ( s −→⊕Rr ) = 0. In this work we show that such algebras exist for any dimension n ≥ 6. Moreover, by considering the simple algebras so (3) and sl (2, R), we analyze the number N ( s −→⊕Rr ) for various kinds of representations R and solvable Lie algebras r. Any Lie algebra g considered in this work is defined over the field R of real numbers. We convene that nonwritten brackets are either zero or obtained by antisymmetry. We also use the Einstein summation convention. Abelian Lie algebras of dimension n will be denoted by nL1. 2 Invariants of Lie algebras. The Beltrametti- Blasi formula The method to determine the invariants of a Lie algebra in terms of systems of partial differential equations (PDEs) has become standard in the physical literature [15, 16], and it is the one we will use here. Let {X1, .., Xn} be a basis of g and { Ck ij } be the structure constants over this basis. We consider the 2 representation of g in the space C∞ (g∗) given by: X̂i = −Ck ijxk∂xj , (1) where [Xi, Xj ] = Ck ijXk (1 ≤ i < j ≤ n, 1 ≤ k ≤ n). This representation is iso- morphic to ad (g), and therefore satisfies the brackets [ X̂i, X̂j ] = Ck ijX̂k. The invariants F (X1, .., Xn) of g: [Xi, F (X1, .., Xn)] , (2) are found by solving the system of linear first order partial differential equations: X̂iF (x1, .., xn) = −Ck ijxk∂xj F (x1, .., xn) = 0, 1 ≤ i ≤ n. (3) and then replacing the variables xi by the corresponding generator Xi (possibly after symmetrizing). In recent years new algorithms to solve system (3) have been developed, which simplify the calculation in some cases [16]. A maximal set of functionally independent solutions of (3) will be called a fundamental set of invariants. Polynomial solutions of system (3) are therefore polynomials in the generators which commute with g, thus correspond to the well known Casimir operators [15]. The system does not impose additional conditions which imply that the solutions are polynomials, so that a non-polynomial solution will be called, in some analogy with the classical case, a generalized Casimir invariant or simply an invariant of g. If F reduces to a constant we say that the invariant is trivial. In the case of semisimple Lie algebras, the solutions found are in fact Casimir operators, and the number of functionally independent invariants is given by the dimension of its Cartan subalgebra. However, for non-semisimple Lie algebras there is no reason to suppose that only the polynomial invariants are of physical interest. A classical example for a Hamiltonian being a nonlinear function of the Casimir operators was described by Pauli in [17]. Another important task is to find the maximal number N (g) of functionally independent solutions of (3). For the case of the classical groups this number depends only on the dimension of a Cartan subalgebra, while for solvable Lie algebras no such general formula exists [5]. However, for a fixed algebra, this number can be described in terms of the dimension and a certain matrix asso- ciated to the commutator table. More specifically, denote by A (g) the matrix whose (i, j)-entry is the bracket [Xi, Xj ]. Such a matrix has necessarily even rank. Then N (g) is given by N (g) = dim g − {rank A (g′) | g′ ≃ g} . (4) This formula was first described by Beltrametti and Blasi [11] and Pauri and Prosperi [12]. The number of polynomial solutions is generally lower than N (g) , up to certain special classes of Lie algebras (like semisimple and nilpotent) [15]. Invariants of Lie algebras have been determined for some classes of non- semisimple Lie algebras, like solvable Lie algebras in low dimensions [18, 19], the kinematical Lie algebras [7] or the special affine Lie algebras [16]. 3 We give an example to illustrate the general method of obtaining the invari- ants. Let s = so (3) and consider the representation R = ad so (3). Let us suppose that the radical of the six dimensional Lie algebra s −→⊕Rr is the three dimensional abelian algebra 3L1. The algebra s −→⊕Rr is of interest for multidimensional extensions of the Bianchi type-IX cosmology [20], and the corresponding vacuum Einstein field equations have been solved in [20]. Indeed this is the simplest embedding of a Bianchi type-IX algebra in an algebra with non-trivial Levi decomposition [21]. It can easily be verified that s −→⊕Rr satisfies N ( s −→⊕Rr ) = 2. The invariants are solution of the system: X̂1F = (−x3∂x2 + x2∂x3 − x6∂x5 + x5∂x6 )F = 0 X̂2F = (x3∂x1 − x1∂x3 + x6∂x4 − x4∂x6 )F = 0 X̂3F = (−x2∂x1 + x1∂x2 − x5∂x4 + x4∂x5 )F = 0 X̂4F = (−x6∂x2 + x5∂x3 ) F = 0 X̂5F = (x6∂x1 − x4∂x3 )F = 0 X̂6F = (−x5∂x1 + x4∂x2 ) F = 0    , (5) Since the equations { X̂iF = 0 } i=4,5,6 do not depend on ∂xi F for i = 4, 5, 6, we can extract the following system from (5) : X̂ ′ 1F = (−x6∂x5 + x5∂x6 )F = 0 X̂ ′ 2F = (x6∂x4 − x4∂x6 )F = 0 X̂ ′ 3F = (−x5∂x4 + x4∂x5 )F = 0    , (6) which has the solution I1 = x2 4 + x2 5 + x2 6. Now, as the rank of the coefficients matrix corresponding to this subsystem is two, the other solution of (5) will depend also on x1, x2, x3. This invariant can be chosen as I2 = x1x4 + x2x5 + x3x6. The important fact about this example is the solution found extracted from the subsystem (6). In a following section we will see that this is not casual, but a property that holds in general. 3 Semidirect sums of Lie algebras The classification of Lie algebras is simplified in some manner by the Levi de- composition theorem, which states that any Lie algebra is essentially formed from a semisimple Lie algebra s called the Levi factor of g and a maximal solv- able ideal r, called the radical [22]. Since the latter is an ideal, the Levi factor s acts on r, and there are two possibilities for this action: [s, r] = 0 [s, r] 6= 0 4 If the first holds, then g is a direct sum s ⊕ r, whereas the second possibility implies the existence of a representation R of s which describes the action, i.e., [x, y] = R (x) .y, ∀x ∈ s, y ∈ r (7) Unless there is no ambiguity, it is more convenient to write −→⊕R instead of −→⊕ , which is the common symbol for denoting semidirect products. Since (7) implies that the radical is a module over s, we have to expect severe restrictions on the structure of the radical, while for direct sums any solvable Lie algebra is suitable as radical. Proposition 1 Let s be a semisimple Lie algebra and R an irreducible repre- sentation. If s −→⊕Rr is the Levi decomposition of a Lie algebra, then r is an abelian algebra. The proof is immediate, since the Jacobi condition implies that the ideals r(0) := r, r(i) := [ r(i−1), r(i−1) ] for i ≥ 1 are invariant by the action of s. If R is irreducible, then either r(1) = 0 or r(1) = r, and since r is solvable, r(1) 6= r. Reasoning similarly, we can easily deduce that the radical r is mapped into its maximal nilpotent ideal n (usually called the nilradical of r), from which the following property follows: Proposition 2 Let s −→⊕Rr be a Levi decomposition. If the representation R does not posses a copy of the trivial representation, then the radical r is a nilpotent Lie algebra. This result is in some manner surprising, since it implies the existence of a copy of the trivial representation whenever the radical is not nilpotent. Of course it does not imply that a nilpotent Lie algebra cannot be the radical when the representation contains copies of the trivial representation. The Lie algebras having non-trivial Levi decomposition have been completely classified up to dimension 8 [20]. For dimensions 9 and 10 some partial results do also exist, mainly Levi factors isomorphic to rank one simple Lie algebras. Since the algebra so (3) is a real form of sl (2, C), the number of (real) representations of so (3) is lower than for for sl (2, R) [23], which implies the existence of much more Lie algebras having the latter as Levi factor. Lemma 1 Let g = s ⊕ r. Then N (g) = N (s) + N (r). This is an obvious consequence of the Beltrametti-Blasi formula. Since the sum is direct, we have that [s, r] = 0 and therefore the rank of the matrix A (g) is the sum of the ranks of A (s) and A (r). Now one can ask what happens whenever we have a non-trivial Levi decomposition. Here no apparent relation between the number of invariants of the Levi factor and the radical and the number of invariants of the semidirect sum seems to exist. If we consider the simple algebra sl (2, R) = {X1, X2, X3 | [X1, X2] = 2X2, [X1, X3] = −2X3, [X2, X3] = X1} and the representation R = D 1 2 ⊕ D0, D 1 2 being the irreducible representation of 5 highest weight λ = 1, there are two choices of r such that sl (2, R)−→⊕ r is a six dimensional Lie algebra with non-trivial Levi decomposition: either the 3- dimensional Heisenberg Lie algebra h1 = {X4, X5, X6 | [X4, X5] = X6} or the algebra A3,3 = {X4, X5, X6 | [Xi, X6] = Xi, i = 4, 5} (see [19] for this nota- tion). It is a straightforward verification that N (h1) = N (A3,3) = 1, thus the formula, if existing, should give the same value for both cases. Now the Lie algebra sl (2, R)−→⊕Rh1 admits two (polynomial) invariants I1 = x6 and I2 = 2x1x4x5 + 4x2x3x6 + 2x2x 2 5 − 2x3x 2 4 + x2 1x6, while the algebra sl (2, R) −→⊕A3,3 has no invariant. The conclusion is that the number of invariants will in general be not expressible in terms of its factors. This example points out another interesting fact: the existence of pairs (R, r) formed by representations R of a semisimple Lie algebra s and a solvable Lie algebra r with structure of s-module such that N (r) > 0 and N ( s −→⊕Rr ) = 0. (8) This also shows that it is not sufficient to determine the invariants of solvable Lie algebras to have an overview of invariants of Lie algebras, implicitly assumed in some early works. Thus the Levi decomposition theorem does not simplify the determination of Casimir operators of Lie algebras, up to the case where we obtain a direct sum. The next step is naturally to try the classification of pairs (R, r) for fixed Levi factor s such that (8) holds. This problem cannot be solved since the classification of solvable Lie algebras is probably not possible for dimensions n ≥ 7 (the classification of six dimensional real Lie algebras contains some errors and some omissions). We must restrict ourselves to certain special cases that are of interest, either for matematical or physical reasons. Table 1 shows the Lie algebras of dimension ≤ 8 with non-trivial Levi de- composition and having no invariants. Due to the low dimenions, the only Levi factors that appear are the simple Lie algebras so (3) and sl (2, R). These alge- bras are of own interest, since they play an important role in multidimensional cosmologies [24]. We convene that the term DJ denotes the real representation of sl (2, R) in its standard form, while R4 denotes the four dimensional real irreducible representation of so (3) and D0 denotes the trivial representation in both cases. Although the general classification of these algebras seems not realizable, since it is based on the possibility of classifying the solvable Lie algebras, once an example is known we can deduce the following generic result: Theorem 1 Let s be a semisimple Lie algebra and (R, r) be a pair formed by a representation of s and a solvable Lie algebra r such that N ( s −→⊕Rr ) = 0. Then, for any k ≥ 1 there exists a Lie algebra gk with Levi factor s and dimension n = dim ( s −→⊕Rr ) + 2k such that N (g) = 0. Proof. Consider the Lie algebra g = s −→⊕R′r′, where R′ = R ⊕ 2kD0 and the radical is r′ = r⊕kr 2, where r2 is the affine Lie algebra generated by Y, Z 6 Table 1: Lie algebras with non-trivial Levi factor and N = 0. Levi factor s dim Representation Nonzero structure constants sl (2, R) 6 D 1 2 ⊕ D0 C2 12 = 2, C3 13 = −2, C1 23 = 1, C4 14 = 1, C5 15 = −1 C4 25 = 1, C5 34 = 1, C4 46 = 1, C5 56 = 1 so (3) 8 R4 ⊕ D0 C3 12 = 1, C2 13 = −1, C1 23 = 1, C7 14 = 1 2 , C6 15 = 1 2 C5 16 = − 1 2 , C4 17 = − 1 2 , C5 24 = 1 2 , C4 25 = − 1 2 , C7 26 = 1 2 C6 27 = − 1 2 , C6 34 = 1 2 , C7 35 = − 1 2 , C4 36 = − 1 2 , C5 37 = 1 2 C4 48 = 1, C5 58 = 1, C6 68 = 1, C7 78 = 1 so (3) 8 R4 ⊕ D0 C3 12 = 1, C2 13 = −1, C1 23 = 1, C7 14 = 1 2 , C6 15 = 1 2 C5 16 = − 1 2 , C4 17 = − 1 2 , C5 24 = 1 2 , C4 25 = − 1 2 , C7 26 = 1 2 C6 27 = − 1 2 , C6 34 = 1 2 , C7 35 = − 1 2 , C4 36 = − 1 2 , C5 37 = 1 2 C4 48 = p, C6 48 = −1, C5 58 = p, C7 58 = −1, C4 68 = 1 C6 68 = p, C5 78 = 1, C7 78 = p sl (2, R) 8 2D 1 2 ⊕ D0 C2 12 = 2, C3 13 = −2, C1 23 = 1, C4 14 = 1, C5 15 = −1 C6 16 = 1, C7 17 = −, C4 25 = 1, C6 27 = 1, C5 34 = 1 C7 36 = 1, C4 48 = 1, C5 58 = 1, C4 68 = 1, C6 68 = 1 C5 78 = 1, C7 78 = 1 sl (2, R) 8 2D 1 2 ⊕ D0 C2 12 = 2, C3 13 = −2, C1 23 = 1, C4 14 = 1, C5 15 = −1 C6 16 = 1, C7 17 = −, C4 25 = 1, C6 27 = 1, C5 34 = 1 C7 36 = 1, C4 48 = 1, C5 58 = 1, C6 68 = p, C7 78 = p sl (2, R) 8 2D 1 2 ⊕ D0 C2 12 = 2, C3 13 = −2, C1 23 = 1, C4 14 = 1, C5 15 = −1 C6 16 = 1, C7 17 = −1, C4 25 = 1, C6 27 = 1, C5 34 = 1 C7 36 = 1, C4 48 = p, C6 48 = −1, C5 58 = p, C7 58 = −1 C4 68 = 1, C6 68 = p, C5 78 = 1, C7 78 = p sl (2, R) 8 D 1 2 ⊕ 3D0 C2 12 = 2, C3 13 = −2, C1 23 = 1, C4 14 = 1, C5 15 = −1 C4 25 = 1, C5 34 = 1, C4 46 = 1, C5 56 = 1, C8 78 = 1 sl (2, R) 8 D 3 2 ⊕ D0 C2 12 = 2, C3 13 = −2, C1 23 = 1, C4 14 = 3, C5 15 = 1 C6 16 = −1, C7 17 = −3, C4 25 = 3, C5 26 = 2, C6 27 = 1 C5 34 = 1, C6 35 = 2, C7 36 = 3, C4 48 = 1, C5 58 = 1 C6 68 = 1, C7 78 = 1 7 and brackets [Y, Z] = Z. The algebra r′ is obviously a s-module, and since N ( s −→⊕Rr ) = 0 and N (r2) = 0, the assertion follows from lemma 1. Corollary 1 For any dimension n ≥ 6 there exist Lie algebras g with non- trivial Levi decomposition such that N (g) = 0. This reduces the classification to the pairs (R, r) formed by a representation of s (this being fixed) and radicals r which are indecomposable, i.e., that do not decompose into a direct sum of ideals. Even for low dimensions like ten, it is far from being easy to find such pairs. As an example, consider the representation R = R4 ⊕ 3D0 of s0 (3) and the radical r defined by the brackets [Xi, X8] = Xi, 4 ≤ i ≤ 7 [X4, X9] = X6, [X5, X9] = X7, [X6, X9] = −X4 [X7, X9] = −X5, [X9, X10] = X10 over the basis {X4, .., X10}. This is the simplest non-decomposable solvable Lie algebra such that the semidirect sum so (3)−→⊕Rr has no non-trivial invariants (for the considered representation). In fact more is true, namely the nonexis- tence of solvable Lie algebras r such that the action of the generators X ∈ r−[r, r] over the nilradical [r, r] is diagonal. This will happen also for other representa- tions different from the one taken here. 4 Levi factors s = so (3) , sl (2, R) Theorem 1 is a general result which holds for any Lie algebra satisfying (8), and therefore non dependent on the particular Levi factor taken. Now an inspection of table 1 points out some interesting facts for the considered Levi factors s = s0 (3) and sl (2, R). In this section we analyze the semidirect sums s −→⊕Rr with these Levi subalgebras in more detail. Through this section, and unless otherwise stated, the notation s will refer either to sl (2, R) or to so (3). We saw in section 2 that in the computation of the invariants of the algebra so (3)−→⊕Rr with R = ad so (3) and r the three dimensional abelian algebra 3L1 there was an invariant depending only on the variables associated to 3L1. We claimed that the existence of this invariant, coming from a special subsystem of (5), was not casual. The next proposition shows that this property does not depend on the representation: Theorem 2 Let R be an irreducible representation of s. Then the semidirect sum s −→⊕Rr admits non-trivial invariants. Moreover, if dim(r) > dim (s), there exists a fundamental set of invariants formed by functions Fi depending only on variables associated to elements of r. Proof. We prove it for s = sl (2, R), the case of s0 (3) being similar. At first, we only need to prove the result for odd dimensional representations Dj , 8 since the remaining case follows at once from the odd dimensionality of the semidirect sum. By proposition 1, the radical r is abelian, and the maximal weight of R is λ = 2m−4 (m ≥ 3) . Let {X1, X2, X3, .., X2m} be a basis of s −→⊕Rr such that {X1, X2, X3} is a basis of sl (2, R) (with [X1, X2] = 2X2, [X1, X3] = −2X3, [X2, X3] = X1) and {X4, .., X2m} a basis of the abelian radical r. The system of PDEs giving the invariants of s −→⊕Rr is: X̂1F = ( −2x2∂x2 + 2x3∂x3 − ∑2m−4 i=0 (λ − 2i)x4+i∂x4+i ) F = 0 X̂2F = ( 2x2∂x1 − x1∂x3 − ∑2m−4 i=1 (λ − i + 1)x3+i∂x4+i ) F = 0 X̂3F = ( −2x3∂x1 + x1∂x2 − ∑2m−5 i=0 (i + 1)x5+i∂x4+i ) F = 0 X̂4+iF = ((λ − 2i)x4+i∂x1 − (i + 1)x5+i∂x2 + (λ − i + 1)x3+i∂x3 )F = 0, 0 ≤ i ≤ 2m − 4    , (9) Observe that since r is abelian, the equations { X̂4+iF = 0 } 0≤i≤2m−4 do not involve the partial derivatives ∂xi F for 4 ≤ i ≤ 2m. This allows us to extract the subsystem: X̂ ′ 1F = (∑2m−4 i=0 (λ − 2i)x4+i∂x4+i ) F = 0 X̂ ′ 2F = (∑2m−4 i=1 (λ − i + 1)x3+i∂x4+i ) F = 0 X̂ ′ 3F = (∑2m−5 i=0 (i + 1)x5+i∂x4+i ) F = 0    , (10) and any solution is obviously an invariant of s −→⊕Rr. The question reduces to show that the system (10) admits a non-trivial solution for any irreducible representation DJ . Observe that (10) can be written as   λx4 (λ − 2)x5 .. − (λ − 2)x2m−1 −λx2m 0 λx4 .. 2x2m−2 x2m−1 x5 2x6 .. λx2m 0     ∂x4 F . . ∂x2m F   = 0. (11) Now this matrix of coefficients has at most rank three (indeed three if m ≥ 4 and rank one if m = 3), so that (10) has always a solution, which shows that N ( s −→⊕Rr ) 6= 0. In particular the system (10) gives the following number of solutions: 1 if m = 3 2m− 6 if m ≥ 4 } , (12) Observe that for m = 3 the representation R is the adjoint representation, and in this case we can find another invariant which depends also on the variables x1, x2, x3. For m ≥ 4 is it not difficult to see that ∂xi F = 0 for i = 1, 2, 3, which shows that the (2m − 6) functionally independent solutions of (10) constitute a fundamental set of invariants for s −→⊕Rr. 9 Corollary 2 Let s = sl (2, R) , so (3). If the radical r is abelian then N ( s −→⊕Rr ) 6= 0. Proof. If the representation contains a copy of the trivial representation D0 or dim r is even , we automatically have solutions of the corresponding system (2). If R does not contain a copy of D0, we can again extract a subsystem from (2), since the radical is abelian and its equations do not contain the partial derivatives corresponding to elements of r. Now R is a sum of irreducible repre- sentations, of which at least one summand R0 must have even highest weight λ, in order to ensure the odd dimensionality of r. Moreover, the variables involved in R0 do not appear in the other summands of r, which ensures that we can apply the preceding theorem. This shows that there exists a nontrivial of the subsystem corresponding to R0, which, by the complete reducibility of R and the abelianity of r, is also an invariant of s −→⊕Rr. The following example illustrates the procedure used in this proof: Let s = sl (2, R) and consider the reducible representation D1 ⊕ D 1 2 . Suppose that the radical r is a five dimensional abelian Lie algebra. The invariants of s −→⊕Rr are the solutions of the system: (−2x2∂x2 + 2x3∂x3 − 2x4∂x4 + 2x6∂x6 − x7∂x7 + x8∂x8 )F = 0 (−2x2∂x1 + x1∂x3 + 2x4∂x5 + x5∂x6 + x7∂x8 )F = 0 (2x3∂x1 − x1∂x2 + x5∂x4 + 2x6∂x5 + x8∂x7 )F = 0 (−2x4∂x1 − x5∂x3 )F = 0 (−2x4∂x2 − 2x6∂x3 ) F = 0 (2x6∂x1 − x5∂x2 )F = 0 (−x7∂x1 − x8∂x3 ) F = 0 (x8∂x1 − x7∂x2 )F = 0    , (13) We extract a subsystem from the first three equations: (−2x4∂x4 + 2x6∂x6 − x7∂x7 + x8∂x8 )F = 0 (2x4∂x5 + x5∂x6 + x7∂x8 )F = 0 (x5∂x4 + 2x6∂x5 + x8∂x7 )F = 0    , (14) and any solution of this system is an invariant of the algebra. (14) can also be reduced to (−2x4∂x4 + 2x6∂x6 )F = 0 (2x4∂x5 + x5∂x6 )F = 0 (x5∂x4 + 2x6∂x5 )F = 0    , (15) which is the subsystem corresponding to the adjoint representation. Clearly the polynomial I1 = 4x4x6 − x2 5 is a solution of (14) and (15), and therefore an invariant of the algebra. Since the other summand of R is D 1 2 , the other invariant will depend on all the variables x4, .., x8. We find I2 = x4x 2 8−x5x7x8+ x6x 2 7. Thus I1, I2 form a fundamental set of invariants of s −→⊕Rr. 10 Table 2: Ten dimensional indecomposable Lie algebras with a compact subal- gebra of dimension n ≥ 7. Algebra Levi decomposition Representation R N L10,14 so (3)−→⊕R (7L1) R7 4 L10,15 so (3)−→⊕R (7L1) R4 ⊕ ad so (3) 4 L10,27 sl (2, R)−→⊕R (7L1) D3 4 L10,28 sl (2, R)−→⊕R (7L1) D2 ⊕ D 1 2 4 L10,29 sl (2, R)−→⊕R (7L1) D 3 2 ⊕ D1 4 L10,30 sl (2, R)−→⊕R (7L1) D1 ⊕ 2D 1 2 4 These two preceding results constitute an important restriction for a semidi- rect sum s −→⊕Rr to satisfy N ( s −→⊕Rr ) = 0. Any representation in such an algebra must be reducible and contain a copy of the trivial representation D0 (see table 1 and the examples in section 3). Proposition 3 Let s = so (3) , sl (2, R). If the radical r of s −→⊕Rr has a one dimensional centre, then the representation R describing the semidirect sum contains a copy of the trivial representation D0. In particular, N ( s −→⊕Rr ) 6= 0. Proof. Let z generate the centre Z (r) or r. For any X ∈ s and Y ∈ r we have [X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X ]] = 0, which shows that [s, Z(r)] ⊂ Z (r). Now [X2, [X3, Z]] + [Z, [X2, X3]] + [X3, [Z, X2]] = 0, which shows that [X1, Z] = 0. Similarly it is proven that [X2, Z] = [X3, Z] = 0, from which we deduce the existence of a copy of the trivial representation in the decomposition of R. Since the action of s over Z (r) is zero, we will obtain the monomial invariant I1 = z. The results obtained so far for the Levi factors so (3) and sl (2, R) have im- portant physical applications, like the classification of multidimensional space- times [21]. In this frame, all ten dimensional real Lie algebras having a (7 + d)- dimensional compact subalgebra have been determined. Of special interest are those which have non-trivial Levi decomposition, and which are the only candi- dates which could present the anomaly N ( s −→⊕Rr ) = 0. From the thirty classes found [20], only six are indecomposable, i.e., they do not decompose as a direct sum of lower dimensional Lie algebras. They have been listed in table 2, where the notation for the algebras is the same as in [24]: By theorem 2 and corollary 2 we see that, since the radical is always abelian, we will obtain non-trivial invariants. For these algebras, in contrast to the possible multidimensional cosmological models seen in section 3 and table 1, the existence of a compact subalgebra of dimension n ≥ 7 implies that the algebra has non-vanishing invariants. 11 5 Application to radicals with a codimension one abelian ideal In this section we analyze a special kind of radicals. We will suppose that r is a solvable non-nilpotent Lie algebra such that [r, r] is a codimension one abelian ideal. We will see that such radicals always imply the existence of invariants, up to the lower dimensional cases. In particular the radicals found in table 1 for the 8 dimensional algebras s −→⊕Rr will constitute the exception for radicals of this type. Theorem 3 Suppose that R = R′ ⊕ 2D0, where R′ is a representation of s. Then N ( s −→⊕Rr ) > 0. Proof. Since R contains at least two copies of the trivial representation, there exists an element Y ∈ [r, r] such that [s, Y ] = 0. Let T /∈ [r, r] and [T, Y ] = ∑ Yi∈[r,r] aiYi (ai ∈ R). The equation Ŷ F = 0 of system (2) has the form Ŷ F = −   ∑ Yi∈[r,r] aiyi   ∂T F = 0. (16) Now, if [T, Y ] = 0, the function F = y is an invariant of s −→⊕Rr. If the bracket [T, Y ] is nonzero, then (16) implies that ∂T F = 0 for any invariant F . The complete reducibility of the representation R (the ideal [r, r] has codimension one in r and is an s-module) implies that [s, T ] = 0. The number N ( s −→⊕Rr ) is given by the difference of the dimension of s −→⊕Rr and the rank of the matrix A ( s −→⊕Rr ) , which in this case has the form:   0 [X1, X2] [X1, X3] [X1, Z1] ... [X1, Zr] 0 0 [X2, X1] 0 [X2, X3] [X2, Z1] ... [X2, Zr] 0 0 [X3, X1] [X3, X2] 0 [X3, Z1] ... [X3, Zr] 0 0 [Z1, X1] [Z1, X2] [Z1, X3] 0 ... 0 0 [T, Z1] ... ... ... ... ... ... ... [Zr, X1] [Zr, X2] [Zr, X3] 0 ... 0 0 [T, Zr] 0 0 0 0 ... 0 0 [T, Y ] 0 0 0 [Z1, T ] ... [Zr, T ] [Y, T ] 0   , (17) where {Z1, .., Zr, Y, T} is a basis of r. Elementary methods show that the determinant of this matrix is the product of − [T, Y ] 2 and the following deter- 12 minant: det   0 [X1, X2] [X1, X3] [X1, Z1] ... [X1, Zr] [X2, X1] 0 [X2, X3] [X2, Z1] ... [X2, Zr] [X3, X1] [X3, X2] 0 [X3, Z1] ... [X3, Zr] [Z1, X1] [Z1, X2] [Z1, X3] 0 ... 0 ... ... ... ... ... [Zr, X1] [Zr, X2] [Zr, X3] 0 ... 0   , (18) which must be zero, since the rank of the matrix in (18) gives the number of invariants of the subalgebra s −→⊕R−2D0 (rL1), which is non-maximal in virtue of theorem 1. Therefore the rank of A ( s −→⊕Rr ) is less than its dimension, from which the existence of non-trivial invariants is ensured. It should be remarked that if R contains only one copy of D0 or the codi- mension of [r, r] is r is greater than one, then the conclusion is false, as can easily be extracted from table 1. We will finally see that radicals as considered in this section are only valid in low dimensions in order to obtain Lie algebras s −→⊕Rr such that N ( s −→⊕Rr ) = 0. Proposition 4 If dim (r) ≥ 7 then N ( s −→⊕Rr ) 6= 0. Proof. Like before, since [r, r] is a codimension one s-submodule of r, the action of s on a generator T ∈ r−[r, r] is zero. If dim (r) = 7, then dim ( s −→⊕Rr ) = 10 and the matrix A ( s −→⊕Rr ) has the form   0 [X1, X2] [X1, X3] [X1, Z1] ... [X1, Z6] 0 [X2, X1] 0 [X2, X3] [X2, Z1] ... [X2, Z6] 0 [X3, X1] [X3, X2] 0 [X3, Z1] ... [X3, Z6] 0 [Z1, X1] [Z1, X2] [Z1, X3] 0 ... 0 [T, Z1] ... ... ... ... ... ... [Z6, X1] [Z6, X2] [Z6, X3] 0 ... 0 [T, Z6] 0 0 0 [Z1, T ] ... [Z6, T ] 0   . (19) It is routine to verify that the determinant of (19) does not depend on the brackets, and that it is zero. Since for any radical r of the considered type such that dim (r) ≥ 7 the determinant of A ( s −→⊕Rr ) is a linear combination of matrices of type (19) and matrices like in (18), it follows that detA ( s −→⊕Rr ) = 0. Observe that this result explains, in terms of the representation theory of so(3), why the ten dimensional Galilei algebra has two (Casimir) invariants depending only on the translations Pi and the pure Galilean transformations Ki. 13 6 Conclusions We have seen that for any dimension n ≥ 6 there exist non-semisimple Lie algebras s −→⊕Rr with non-trivial Levi factor s and such that N ( s −→⊕Rr ) = 0. This constitutes a proof that the Levi decomposition theorem [22] does not reduce the number of generalized Casimir invariants of s −→⊕Rr to some combination of the numbers corresponding to the Levi factor s and the radical r, but depends essentially on the pair (R, r) formed by the representation R describing the semidirect sum and the radical. For the rank one simple Lie algebras so (3) and sl (2, R) the number of invari- ants of a semidirect sum s −→⊕Rr have been analized in some detail. In particular, the analysis undertaken has given a representation theoretic interpretation of the invariants obtained for the (3 + 1) kinematical algebras like the Galilei algebra. The interest of these Levi factors is therefore justified not only by kinematical problems, but also by the extensions of Bianchi type-IX cosmology [20, 21]. Specially interesting are those admissible extensions which have no invariants. Therefore invariant quantities for these algebras should be searched using distri- bution theory [19]. In particular, if the radical is abelian, we have proved that we will obtain solutions, some of them depending only on variables associated to elements of the radical. This confirms that the fact that the special affine algebras sa (n, R) have invariants (for being odd dimensional) is not an isolated case, but also the general pattern for those semidirect sums which are even di- mensional. From the computed examples, it seems reasonable to expect that, whenever the radical r is a nilpotent Lie algebra, the number of invariants of a semidirect sum s −→⊕Rr will be nonzero. However, for this case it is not sufficient to know which is the representation R that describes the semidirect sum. We need more precise information on the structure of n (not merely the value of very general invariants like the nilpotence index), which impedes to establish a general result as for the abelian case. The most important question that arises from our results is whether they can be extended to any semisimple Lie algebra of rank r ≥ 2. At least for direct sums of sl (2, R) and so (3) this seems to hold. An example which is worth to be analyzed is the Schrödinger algebra S in (3 + 1) dimensions [25]. Over the basis {Ji, Ki, Pi, P0, C, D}i=1,2,3 this algebra is given by the brackets: [Ji, Jj ] = εijkJk, [Ji, Kj] = εijkKk, [Ji, Pj ] = εijkPk, [Ki, P0] = Pi, [Pi, D] = Pi, [D, Kj] = Dj , [D, P0] = −2P0, [C, Pj ] = Kj, [C, P0] = −D, [C, D] = −2C. where P0 is the time translation, Pi the space translations, Ji the rotations and Ki the pure Galilean transformations. It can easily be verified that the subalgebra a generated by {Ki, Pi}i=1,2,3 is six dimensional and abelian, while {P0, C, D} generates a copy pf sl (2, R). Therefore we obtain the semisimple algebra so (3)⊕sl (2, R), and since a is an ideal, we have the Levi decomposition 14 of S (by abuse of notation we can denote the corresponding representation by D 1 2 ⊗ adso(3)). If we extract a subsystem of the corresponding system (2), as done in the proof of theorem 2, we obtain that S has a fourth order Casimir operator P4 depending only on the space translations and pure Galilei transfor- mations: P4 = K2 1 ( P 2 2 + P 2 3 ) + K2 2 ( P 2 1 + P 2 3 ) + K2 3 ( P 2 1 + P 2 2 ) −2 (P1P2K1K2 + P1P3K1K3 + P2P3K2K3) For other simple Lie algebras a direct calculation of the rank of matrices A ( s −→⊕Rr ) becomes a enormously difficult problem, and therefore the proofs of the generalization of the results obtained for rank one simple algebras, if they hold, must be approached by completely different means. Finally, these results are of interest for the study of non-semisimple (maxi- mal) regular subalgebras of simple Lie algebras. The example sl (2, R)−→⊕D 1 2 ⊕D0 3L1 of table 1 is a regular subalgebra of sl (3, R) and has no invariants. 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