Finding Lie-Algebra Invariants -- an Explicit Procedure?

[lpetrich]

I've found this paper: [physics/9712033] Closed Expressions for Lie Algebra Invariants and Finite Transformations and I've attempted to interpret it.

In some representation, a general Lie-algebra element A = a_iLⁱ for generators L (repeated dummy indices summed over, like i here).
det(I - x*A) = xⁱ*φ_i(A)

where the functions φ_i(A) are combinations of traces of powers of A: Tr(A^k). They can be expanded
φ_i(A) = a_j1a_j2...a_jiC^{D,j1,j2,...,ji}

giving a generalized Casimir invariant
C^D(i) = C^{D,j1,j2,...,ji}L_j1L_j2...L_ji

The D is for determinant. I don't get their jumping between lowered-index L's and raised-index L's. Something unstated somewhere, I think. I also don't get their jumping from φ's to C's.

But if one grants that, one can find an alternate set of invariants from the traces of products:
C^{T,j1,j2,...,ji} = Tr(L^j1L^j2...L^ji)

with
C^T(i) = C^{T,j1,j2,...,ji}L_j1L_j2...L_ji

with the C^M(i)'s in terms of the C^T(i)'s and vice versa. Each C^T(i) contains C^M(i) and vice versa.

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I've seen other forms of the invariants, like in terms of the commutator and the Killing form:
[L_i,L_j] = f_ij^kL_k, g_ij = f_ik^lf_jl^k, g^ij = inverse of g_ij

This makes it work only for semisimple Lie algebras, because the Killing form must be nonsingular here.

Construct matrix M_i^j = f_ik^jg^klL_l

The Casimir invariants are C^C(i) = Tr(Mⁱ) taking power i of the matrix M with its indices as defined above, then the trace with those indices.

So if that paper's results are valid, there must be some relationship between the C^C's and the C^T's and C^M's.

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One can make conjectures in some special cases as to the form of the Casimir invariants, notably for orthogonal and pseudo-orthogonal groups SO(n) and SO(n1,n2), n=n1+n2. Let the commutation go
[L_ijL_kl] = h_ikL_jl - h_ilL_jk - h_jkL_il + h_jlL_ik

where h is some real symmetric nonsingular n*n matrix. The Killing form is
g_i1i2,j1j2 = (something) * (h_i1,j2h_i2,j1 - h_i1,j1h_i2,j2)

Define a variant of the previous matrix M:
M_i^j = L_ikh^kj where h^ij is the inverse of h_ij.

Then C^O(i) = Tr(Mⁱ), trace of power i using this M.

One can show that the C^C's contain the C^O's, with C^C(i) containing C^O(i).

For even n, there is a possible one that uses the antisymmetric symbol:
C^X = ε^ijkl...L_ijL_kl...

It's easy to show that C^O(2n) contains the square of C^X.

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There are further questions about these operators. For semisimple Lie algebras, is there any halfway simple way of calculating them for generators in the Cartan-Weyl basis? That would give the operators' values in terms of the representations' highest weights. I've found A. M. Perelomov, V. S. Popov, “Casimir operators for semisimple Lie groups”, Izv. Akad. Nauk SSSR Ser. Mat., 32:6 (1968), 1368–1390, which does more-or-less that, but it's hard for me to see how Perelomov's and Popov's versions relate to the others.

I tried calculating C^D for some of the algebras that the papers' authors mentioned. It worked OK for the Lorentz group, SO(3,1), but not for the Poincaré group, SEuc(3,1) (Special Euclidean). For the Poincaré group, it got the same invariants as for the Lorentz group, though it got the right ones for the latter group. For the Galilean group, it only got the angular-momentum invariant.

I'll note a list of sizes of irreducible invariants. For rank n, an algebra has n of them.

• U(n): 1, 2, 3, ..., n
• A(n), SU(n+1): 2, 3, ..., n, n+1
• B(n), SO(2n+1): 2, 4, ..., 2n
• C(n), Sp(2n): 2, 4, ..., 2n
• D(n), SO(2n): 2, 4, ... 2n-2, n (instead of 2n)
• G2: 2, 6
• F4: 2, 6, 8, 12
• E6: 2, 5, 6, 8, 9, 12
• E7: 2, 6, 8, 10, 12, 14, 18
• E8: 2, 8, 12, 14, 18, 20, 24, 30

For a real or pseudoreal irrep, an odd-sized invariant will vanish. It's not surprising that irreducible odd-sized invariants only occur in the algebras that have complex irreps: U(n), SU(n) for n >= 3, SO(4n+2), E6.

 

[Greg Bernhardt]

I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?