Representations of Lie algebras, Casimir operators and their ...

Representations of Lie algebras, Casimiroperators and their applicationsLibor ŠnoblDecember 27, 2011AbstractWe intend to review some basic notions in the representation theoryof Lie algebras and focus in particular on the notion of Casimiroperator and its generalization.Outline:1. representations of Lie algebras, reducibility, Schur’s Lemma,2. universal enveloping algebra (UEA) of a given Lie algebra,3. Casimir operators as nontrivial central elements of UEA, theiressential role in labelling of irreducible representations,4. generalized Casimir invariants and when they can be expressedin terms of Casimir operators (time permitting),5. applications: irreducible representations of Lorentz group, hydrogenatom in quantum mechanics.1 Lie algebras and their representationsA Lie algebra g is a vector space over a field F equipped with a multiplication(also called a bracket), i.e. a bilinear map [ , ]: g × g → g,such that[y, x] = −[x, y] (antisymmetry) (1)0 = [ x, [y, z] ] + [ y, [z, x] ] + [ z, [x, y] ] (Jacobi identity)(2)for all elements x, y, z ∈ g. In what follows we shall consider the fieldsF = R, C and finite–dimensional Lie algebras only.1

The structure of the Lie algebra g can be represented in any chosenbasis (e j ) dim gj=1by the corresponding structure constants c jklin thebasis (e j ) dim gj=1[e j , e k ] =dim∑gl=1c jk l e l . (3)A fundamental theorem due to E. E. Levi [1, 2, 3] provides a generalscheme for the structure of Lie algebras.Theorem 1 (Theorem of Levi) Let g be a finite-dimensional Liealgebra over a field F and r = R(g) be its radical. Then there exists asemisimple subalgebra p of g such thatg = p ∔ r. (4)The subalgebra p is isomorphic to the factor algebra g/r and is uniqueup to automorphisms of g.Because r is a solvable ideal and p a semisimple subalgebra we have[p, p] = p, [p, r] ⊆ r, [r, r] r.A representation ρ of a given Lie algebra g on a vector space V isa linear map of g into the space gl(V ) of linear operators acting on Vρ: g → gl(V ): x → ρ(x)such that for any pair x, y of elements of gρ([x, y]) = ρ(x) ◦ ρ(y) − ρ(y) ◦ ρ(x) (5)holds. The field over which the vector space is defined must containF in order to have the representation well–defined, i.e. we may haverepresentations of real algebras on complex vector spaces but not viceversa. Dimension of the representation ρ is understood to be the sameas the dimension of the vector space V .A subspace W of V is called invariant ifρ(g)W = {ρ(x)w| x ∈ g, w ∈ W } ⊆ W.A representation ρ of g on V is reducible if a proper nonvanishinginvariant subspace W of V exists.A representation ρ of g on V is irreducible if no nontrivial invariantsubspace of V exists.2

A representation ρ of g on V is fully reducible when every invariantsubspace W of V has an invariant complement ˜W , i.e.V = W ⊕ ˜W , ρ(g) ˜W ⊆ ˜W . (6)In particular, any irreducible representation is also fully reducible.An important criterion for irreducibility of a given representationsisTheorem 2 (Schur’s Lemma) Let g be a complex Lie algebra andρ its representation on a finite–dimensional vector space V .1. Let ρ be irreducible. Then any operator A on V which commuteswith all ρ(x),[A, ρ(x)] = 0, ∀x ∈ g,has the form A = λ1 for some complex number λ.2. Let ρ be fully reducible and such that every operator A on Vwhich commutes with all ρ(x) has the form A = λ1 for somecomplex number λ. Then ρ is irreducible.The adjoint representation of a given Lie algebra g is a linear mapof g into the space gl(g) of linear operators acting on gad: g → gl(g): x → ad(x)defined for any pair x, y of elements of g viaad(x) y = [x, y]. (7)When convenient we may also use an alternative notation ad x = ad(x).The image of ad is denoted by ad(g).2 Universal enveloping algebras andCasimir operatorsUniversal enveloping algebra is an important object in the representationtheory of Lie algebras. It is defined as a certain factoralgebraof the tensor algebra of a given Lie algebra g.The tensor algebra (or free algebra) of the vector space V over thefield F is the vector spaceT (V ) = ⊕ ∞ k=0 V ⊗k = F ⊕ V ⊕ V ⊗ V ⊕ . . . ⊕ V ⊗k ⊕ . . .equipped with the associative multiplication generated by the multiplicationof decomposable elements(v 1 ⊗ v 2 ⊗ . . . ⊗ v k ) · (w 1 ⊗ . . . ⊗ w l ) = v 1 ⊗ v 2 ⊗ . . . ⊗ v k ⊗ w 1 ⊗ . . . ⊗ w l .3

When the vector space V is in addition a Lie algebra V = g,one may consider a two–sided ideal J in the associative algebra T (g)generated by the elements of the form x ⊗ y − y ⊗ x − [x, y], i.e.J = span {A ⊗ (x ⊗ y − y ⊗ x − [x, y]) ⊗ B | x, y ∈ g, A, B ∈ T (g)} .The factoralgebraU(g) = T (g)/J (8)is called the universal enveloping algebra of the Lie algebra g. It isobvious that universal enveloping algebras are associative algebras,i.e. the notion of a universal enveloping algebra allows us to constructan infinite dimensional associative algebra out of any Lie algebra in acanonical way.The main reason why universal enveloping algebras are useful isthe following observation: any representation ρ of a Lie algebra g ona (finite–dimensional, for simplicity) vector space V gives rise to arepresentation ˜ρ of the tensor algebra T (g) defined by˜ρ(x 1 ⊗ x 2 ⊗ . . . ⊗ x k ) = ρ(x 1 ) · ρ(x 2 ) . . . ρ(x k ).The definition of a representation ρ, equation (5), implies that ˜ρ(J ) =0. Consequently, ˜ρ defines also a representation ˆρ of the universalenveloping algebra U(g) on the vector space Vˆρ(a) = ˜ρ(A), a = A mod J ∈ U(g), A ∈ T (g).Casimir operators are elements of the center of the universal envelopingalgebra U(g) of the Lie algebra g [4, 5, 6], i.e. such c ∈ U(g)thatc · a = a · cholds for all a ∈ U(g). A necessary and sufficient condition for c to bea Casimir operator isc · x = x · c, ∀x ∈ g ≃ g ⊗1 /J .We shall consider nontrivial Casimir operators only, i.e. thosedifferent from elements of F/J ≃ F. In order to avoid writing mod Jat all times we adopt a convention that Casimir operators shall bewritten as totally symmetric expressions in the elements of g. Thiscan be always accomplished using the identityx ⊗ y mod J = 1 2 (x ⊗ y + y ⊗ x) + 1 [x, y] mod J2as many times as needed, starting from the highest order terms andproceeding order by order. Such a procedure also implies the uniquenessof such totally symmetric representative of the equivalence class4

mod J . We shall occasionally suppress the tensor product sign, i.e.xy ≡ x ⊗ y.The importance of Casimir operators for the representation theoryof complex Lie algebras comes from the Schur’s lemma, Theorem 2.In any representation ρ we have[ˆρ(c), ρ(x)] = 0, ∀x ∈ g.Consequently, if the representation ρ is irreducible, ˆρ(c) must be amultiple of the identity operator, λ1. The number λ depends on thechoice of the representation ρ and the Casimir operator c. If twoirreducible representations ρ 1 on V 1 and ρ 2 on V 2 are equivalent, i.e.if a linear transformation T : V 1 → V 2 exists such thatρ 2 (x) = T ◦ ρ 1 (x) ◦ T −1 , ∀x ∈ g,then necessarily we have λ 1 = λ 2 for the given Casimir invariant c.That means that the eigenvalues of ˆρ(c) can be used to distinguishinequivalent irreducible representations.If ρ is fully reducible but not irreducible then we may use theknowledge of Casimir operators of g in the decomposition of ρ into irreduciblecomponents. In particular, we construct common eigenspacesof all known Casimir operators and we know that each of them is aninvariant subspace (not necessarily irreducible for general g).The existence of nontrivial Casimir operators was established forcertain classes of Lie algebras only, e.g. for semisimple ones. AlsoLie algebras with nonvanishing center, including all nilpotent ones,do possess nontrivial Casimir operators; namely, the elements of thecenter themselves. On the other hand some Lie algebras are known tohave no nontrivial Casimir invariants.Let us consider a semisimple complex Lie algebra g and its Killingform K. Let us take any basis (e 1 , . . . , e dim g ) of g and find the dualbasis (ẽ 1 , . . . , ẽ dim g ) such thatK(e k , ẽ j ) = δ j k .Let us assume that c ij k are the structure constants (3) of the Liealgebra g in the basis (e 1 , . . . , e k ). By the invariance of the Killingform K we havedim∑gK(e k , [e a , ẽ j ]) = −K([e a , e k ], ẽ j ) = −c j ak = −K(e k , c amjẽ m )which by nondegeneracy of K implies that[e a , ẽ j ] =dim∑gm=15c majẽ m .m=1

Let us construct an element of the universal enveloping algebra U(g)of the formC =dim∑gk=1ẽ k ⊗ e k =dim∑gk=1e k ⊗ ẽ k (9)(its symmetry comes from the fact that the Killing form is symmetric).Suppressing the tensor product signs and computing mod J , wehave for the commutator between e a ∈ g and C ∈ U(g)[e a , C] ===dim∑gk=1dim∑gk=1dim gk=1)(e a ẽ k e k − ẽ k e k e a =()(e a ẽ k − ẽ k e a ) e k + ẽ k (e a e k − e k e a ) =dim g∑∑([e a , ẽ k ] e k + ẽ k [e a , e k ]) = (c lakẽ l e k + c aklẽ k e l ) = 0.k,l=1We conclude that C is a Casimir operator of g. It is called the quadraticCasimir operator [4]. For its application in the proof of Weyl’s theorem,see [5].We remark that the quadratic Casimir operator does not exhaustall independent Casimir operators of the semisimple Lie algebra gwhen we have rank g > 1. It is known that any semisimple Lie algebraof rank l has l independent Casimir operators which generate the wholecenter of the universal enveloping algebra U(g) through their productsand linear combinations. Their explicit form depends on the detailsof the structure of the considered algebra g.Casimir invariants are of primordial importance in physics. Theyoften represent such important quantities as angular momentum, elementaryparticle mass and spin, Hamiltonians of various physicalsystems etc.Example 1 Let us consider the angular momentum algebrawithso(3) = span{L 1 , L 2 , L 3 }[L j , L k ] =3∑ɛ jkl L l . (10)l=1The quadratic Casimir operator (9) isC = − 1 23∑L 2 l , (11)l=16

i.e. it coincides up to a numerical factor 1/2 with the square of angularmomentum, familiar from the construction of irreducible representationsof the angular momentum algebra in quantum mechanics.Notice that the sign of the Casimir operator (11) is in fact the sameas used in physics: in quantum mechanics the operators of angularmomentum ˆL j (measured in multiples of ) satisfy the commutationrelations[ˆL j , ˆL3∑k ] = iɛ jkl ˆLlwhich differ from the ones in equation (10) by an extra imaginary unit.This extra i factor can be traced to the requirement that observablesare described by Hermitean operators; the generators of unitary representationsof Lie groups are, on the contrary, anti–Hermitean. Anobvious remedy is to formally introduce a “physical” basis of a givenreal Lie algebraê j = ie j (12)l=1in which the original real structure constants[e j , e k ] = ∑ lf jk l e lbecome explicitly purely imaginary[ê j , ê k ] = ∑ lif jklê l .Example 2 Let us consider the Poincaré algebra iso(1, 3) (a.k.a. inhomogeneousLorentz algebra) spanned by M µν , P µ , µ, ν = 0, . . . , 3with the nonvanishing commutation relations[M µν , P ρ ] = η νρ P µ − η µρ P ν , (13)[M µν , M ρσ ] = η µσ M νρ + η νρ M µσ − η µρ M νσ − η νσ M µρ ,where η is the Minkowski metric η µν = η µν = diag(1, −1, −1, −1). Weshall use the metric η to move indices up and down, as is commonin the theory of relativity, and denote by ɛ µνρσ the covariant totallyantisymmetric tensor.The Poincaré algebra has a nontrivial Levi decomposition (4)iso(1, 3) = so(1, 3) ∔ rwith its semisimple factor being the Lorentz algebraso(1, 3) = span{M µν } µ,ν=0,1,2,37

and an Abelian radicalr = span{P µ } µ=0,1,2,3 .There are two independent Casimir operators of this Lie algebra,which are usually expressed asP 2 =3∑η µν P µ P ν and W 2 =µ=03∑η µν W µ W νµ=0where the quadruplet of quadratic elements of U(g)W µ = − 1 2 ɛ µνρσM νρ P σis called the Pauli–Lubanski vector. That means that in this case oneof the Casimir operators is of second order in generators whereas theother is of fourth order.These two Casimir operators are essential in the construction ofirreducible representations of the Poincaré algebra in relativistic quantumfield theory. Notice that in this case one constructs infinite–dimensional unitary representations.2.1 Energy spectrum of hydrogen atom in quantummechanicsIn order to further demonstrate the relevance of Casimir operatorsto physics, let us review another application, namely an algebraicdetermination of the hydrogen spectrum in quantum mechanics. Thiscomputation is originally due to Wolfgang Pauli [7].The Hamiltonian of an electron in hydrogen atom isĤ = 1 ∑ˆP j ˆPj − Q 2Mr , (14)jwhere ˆP j = −i ∂ ∂x jare operators of linear momenta in R 3 with the coordinatesx 1 , x 2 , x 3 , r = √ x 2 1 + x2 2 + x2 3 , M is the mass of the electronand Q =e24πɛ 0in SI units.The Hamiltonian (14) has three obvious integrals of motion, namelythe angular momentaˆL j = 1 ∑ɛ jkl ˆXk ˆPl ,k,l8

(chosen dimensionless for convenience) and three less obvious integralsof motion, namely the components of the Laplace–Runge–Lenz vectorˆK i = 12MQ∑ ∑kjɛ ikj ( ˆP k ˆLj + ˆL j ˆPk ) − 1 x i r . (15)The expression x irshould be interpreted as the operator of multiplicationby the given function of coordinates. For future reference, letus denote3∑3∑ˆL 2 = ˆL j ˆLj , ˆK2 = ˆK j ˆKj .j=1As it turns out, the knowledge of these integrals of motions andtheir algebraic structure is enough to determine the spectrum of boundstates in the hydrogen atom.The crucial ingredients are the commutators between various componentsˆL j and ˆK j . By a somewhat lengthy but straightforward calculationwe find[ˆL j , ˆL k ] = i[ˆL j , ˆK k ] = ij=13∑ɛ jkl ˆLl , (16)l=13∑ɛ jkl ˆKl , (17)l=1[ ˆK j , ˆK k ] = − 2iMQ 23∑ɛ jkl ˆLl Ĥ. (18)l=1Another important observation is the operator identity3∑j=1ˆK j ˆLj = 0. (19)The commutator (18) prevents the operators ˆL j , ˆK j from forming a Liealgebra. Nevertheless, this bothersome property can be circumventedif we consider a given energy level, i.e. a subspace H E of the Hilbertspace H consisting of all eigenvectors of Ĥ with the given energy E.Operators ˆL j , ˆK j can be all restricted to H E because they commutewith Ĥ. When such restriction is understood, the Ĥ in equation (18)can be replaced by a numerical factor E and the algebra of ˆL j , ˆK jcloses. In particular, when E < 0 it is isomorphic to the Lie algebraso(4) = so(3) ⊕ so(3). When E > 0 the difference in sign leads to adifferent real form of the same complex Lie algebra, namely to so(1, 3).We shall be interested in bound states here, i.e. we assume E < 0.9

Once the energy is fixed we may introduce the operators(ˆL (1)j = 1 √ )ˆL j + − MQ222E ˆK jandˆL (2)j = 1 2(ˆL j −√− MQ22E ˆK j)(notice that − MQ22Eis by assumption a positive number). The commutatorsof ˆL (1)j and ˆL (2)j now become[ˆL (1)j , ˆL (1)k ] = i[ˆL (2)j , ˆL (2)k ] = i[ˆL (1)j , ˆL (2)k ] = 0.3∑ɛ jkl ˆL(1)l ,l=13∑ɛ jkl ˆL(2)l ,That means that we have an explicit decomposition of our realizationof so(4) into the direct sum so(3)⊕so(3) and that the two independentCasimir operators of so(4) can be expressed asor equivalently asC 1 = 1 4C 1 =l=13∑ˆL 2 (1)j , C 2 =j=13∑ˆL 2 (2)j ,j=1( √ ) 2 3∑ˆL j + − MQ22E ˆK j , C 2 = 1 4j=1( √ ) 2 3∑ˆL j − − MQ22E ˆK j .(20)The sum of these two Casimir operators, i.e. C 1 + C 2 , gives thequadratic Casimir operator (9) of so(4).From the theory of angular momentum, i.e. of representations ofthe Lie algebra so(3), we know that in any irreducible representationof so(4) we havej=1C 1 = p(p + 1)1, C 2 = q(q + 1)1for some nonnegative integer or half–integer values of p and q. Theirreducible representation of so(4) = so(3)⊕so(3) determined by thesevalues of the Casimir operators has dimension equal to (2p+1)×(2q +1).10

When we expand the expressions for the Casimir operators (20)and subtract them, we find thatC 1 − C 2 =√− MQ22E3∑ˆL j ˆKjwhich vanishes in our representation, as we already know (cf. (19)).Therefore, only irreducible representations of so(4) with p = q arisein our problem.Let us now consider such a representation of so(4) with the givenvalues of E and p. The angular momentum ˆL j can be expressed asj=1ˆL j = ˆL (1)j + ˆL (2)j ,i.e. we can employ the standard result concerning the compositionof two independent angular momenta and conclude that ˆL 2 takes allinteger values between |p − p| = 0 and p + p = 2p. In particular, thes–state, i.e. the state with ˆL 2 = 0, exists in our representation and isof interest to us. Let ψ be any s–state, i.e. a vector ψ ∈ H such thatˆL j ψ = 0. Obviously, ψ is a function of the radial coordinate r only.We haveandˆL 2 ψ = 0ˆK 2 ψ = 2MQ 2 Ĥψ + 1 2 ψby inspection of both sides of the equation when expanded in termsof ˆXj , ˆP j etc.When ψ in addition belongs to our representation of so(4) determinedby the values of E and p, we have the following value for thequadratic Casimir operator (9) of so(4)(C 1 + C 2 )ψ = 2p(p + 1)ψ = 1 ∑)(ˆL 22jψ − MQ22EˆK j 2 ψj(= − MQ2 2E4E MQ 2 + 1 ) 2 ψ. (21)Thus we have arrived at the condition8p(p + 1) = −2 − MQ2 2 Ewhich is just a different formulation of the celebrated Rydberg formulaE = − MQ22 2 1(2p + 1) 2 (22)11

where the potentially half–integer valued parameter p is traditionallyreplaced by the integer n = 2p+1 > 0. Once we have established thatE is determined by the value of p by equation (22) we also see thatH E coincides with the representation space of the so(4) irreduciblerepresentation labelled by p and q = p. On H E we may also writeequation (21) in the form( MQ2C 1 + C 2 = −4 2Ĥ + 1 )(23)2since both C 1 + C 2 and Ĥ take a constant value on H E. While itmay be tempting to consider this to be an operator identity validon the whole Hilbert space H, we don’t consider such interpretationlegitimate. In particular, on scattering states (E > 0) we even havea different Lie algebra. Therefore, equation (23) should be consideredat most on the bound state sector of our Hilbert space H.To sum up, we have seen that the spectrum of hydrogen atom canbe derived using the theory of Lie algebras, without explicit constructionof eigenfunctions. More precisely, we have derived a necessarycondition (22) that any energy eigenvalue must satisfy. That this formulais physically relevant for all values of p ≥ 0 such that 2p ∈ Zis not a consequence of the computation just shown and shall be establishedby other means (e.g. by an explicit construction of s–statesintroduced above). Once the existence of at least one state with theenergy E n = − MQ2 1is shown, the degeneracy n 2 of the energy level2 2 n 2E n also follows directly from algebraic considerations.3 Generalized Casimir invariantsAs was shown by Kirillov in [8] and will be explained below, Casimiroperators are in one–to–one correspondence with polynomial invariantscharacterizing orbits of the coadjoint representation of g. Thesearch for invariants of the coadjoint representation is algorithmicand amounts to solving a system of linear first order partial differentialequations [9, 10, 11, 12, 13, 14, 15, 16, 17]. Alternatively, globalproperties of the coadjoint representation can be used [13, 18, 19, 20].In general, solutions are not necessarily polynomials and we shall callthe nonpolynomial solutions generalized Casimir invariants.For certain classes of Lie algebras, including semisimple Lie algebras,perfect Lie algebras, nilpotent Lie algebras, and more generallyalgebraic Lie algebras, all invariants of the coadjoint representationare functions of polynomial ones [9, 10].On the other hand, in the representation theory of solvable Liealgebras their invariants are not necessarily polynomials, i.e. they can12

e genuinely generalized Casimir invariants. In addition to their importancein representation theory, they may occur in physics. Indeed,Hamiltonians and integrals of motion of classical integrable Hamiltoniansystems are not necessarily polynomials in the momenta [21, 22],though typically they are invariants of some group action.In order to calculate the (generalized) Casimir invariants we considersome basis (e 1 , . . . , e n ) of g, in which the structure constants arec ij k . The coadjoint representation ad ∗ of g is the representation on g ∗obtained via transposition of the operators in the adjoint representation〈ad ∗ (x)φ, y〉 = −〈φ, ad(x)y〉, ∀x, y ∈ g, φ ∈ g ∗ .A basis for the coadjoint representation is given by the first orderdifferential operators acting on functions on g ∗ , i.e. vector fields,Ê k =n∑a,b=1be b c ∂ka , 1 ≤ k ≤ n. (24)∂e aIn equation (24) the quantities e a are commuting independent variables– the coordinates in the basis of the space g ∗ , dual to the algebrag. Using the relation (g ∗ ) ∗ ≃ g one can identify them with the basisvectors of g.The invariants of the coadjoint representation, i.e. the generalizedCasimir invariants, are solutions of the following system of partialdifferential equationsÊ k I(e 1 , . . . , e n ) = 0, k = 1, . . . , n. (25)The relation to Casimir operators, i.e. the 1–1 correspondencebetween polynomial solutions of equation (25) and the elements of thecenter of the enveloping algebra comes from the following observations.Firstly, it is obvious that both the operation on U(g) of taking thecommutator with a fixed element e k ∈ g and the application of thefirst order differential operator Êk satisfy Leibniz rule[e k , a 1 a 2 ] = [e k , a 1 ]a 2 + a 1 [e k , a 2 ], a 1 , a 2 ∈ U(g),Ê k (F 1 F 2 ) = Êk(F 1 )F 2 + F 1 Ê k (F 2 ), F 1 , F 2 ∈ C ∞ (g ∗ ).Further ingredient of the proof is the fact that [e k , ·] and Êk give thesame answer when applied to e l , namely[e k , e l ] =n∑c m kl e m , Ê k (e l ) =m=1n∑c m kl e m , (26)m=1where it is understood that e l ∈ g ⊂ U(g) in the first equality ande l ∈ (g ∗ ) ∗ in the second.13

Now, let us consider a polynomial function F on g ∗ . We expressit as a completely symmetric expression in the basis functionalse l ∈ (g ∗ ) ∗ – since as functions they commute that does not in factchange anything. Next, we associate to it an element ˜F of the universalenveloping algebra by simply changing the interpretation of thegenerators e k ∈ (g ∗ ) ∗ → e k ∈ g ⊂ U(g). Recalling that the totallysymmetric representative of a given element A ∈ U(g) is unique andobserving that [e k , ˜F ] is by construction again a totally symmetricexpression in the generators e l , we find that[e k , ˜F ] = 0 ⇔ Êk(F ) = 0by Leibniz rule and equation (26). Thus, polynomial invariants ofthe coadjoint representation can indeed be identified with Casimiroperators in a bijective way.Let us first determine the number of functionally independent solutionsof the system (25). We can rewrite this system asC · ∇I = 0 (27)where C is the antisymmetric matrix⎛0 c b 12 e b . . . c b 1n e b−c b 12 e b 0 . . . c b 2n e bC =⎜ ..⎝ −c b 1,n−1 e b . . . 0 c b n−1,n e b−c b 1n e b . . . −c b n−1,n e b 0⎞⎟⎠(28)in which summation over the repeated index b is to be understood ineach term and ∇ is the gradient operator ∇ = (∂ e1 , . . . , ∂ en ) t (wheret stands for transposition). The number of independent equations inthe system (25) is r(C), the generic rank of the matrix C. The numberof functionally independent solutions of the system (25) is hencen I = n − r(C). (29)Since C is antisymmetric, its rank is even. Hence n I has the same parityas n. Equation (29) gives the number of functionally independentgeneralized Casimir invariants.The individual equations in the system of partial differential equations(PDEs) (25) can be solved by the method of characteristics, or,equivalently by integration of the vector fields (24).14

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