Racah's Theorem on the Number of Casimir Operators
This appendix proves Theorem 18.21 of Lie Groups, Lie Algebras, and Fibre Bundles: for a finite-dimensional semisimple Lie algebra of rank \(\ell\) (Definition 18.20), the centre \(Z\left(U(\mathfrak{g})\right)\) of the universal enveloping algebra is a polynomial algebra on \(\ell\) algebraically independent Casimir elements, so that an irreducible representation carries exactly \(\ell\) invariant labels. The chapter needs the count in two places: to know that \(\mathfrak{su}(2)\) has one Casimir and \(\mathfrak{su}(3)\) two (Remark 18.19), and to convert the rank of \(\mathfrak{so}(p,q)\) computed in Proposition 18.22 into the number of Lorentz invariants.
The proof is a chain of three identifications:
where \(\mathfrak{h}\) is a Cartan subalgebra, \(W\) its Weyl group, and “invariant” means annihilated by every \(\ad_{X}\). The first link is proved here in full (The centre of the enveloping algebra and the invariant polynomials), and rests on the Poincaré–Birkhoff–Witt theorem of The Poincaré–Birkhoff–Witt Theorem. The second is the Chevalley restriction theorem and the third is Chevalley's theorem on finite reflection groups; both belong to the structure theory of semisimple Lie algebras, which this treatise does not build, and both are stated precisely as quoted inputs in The two quoted theorems and named again in Remark A32.8. The assembly and the count are The count, and the answer is checked against the chapter's own worked algebras in The count checked against the chapter's algebras.
Throughout, \(\mathfrak{g}\) is a finite-dimensional semisimple Lie algebra over a field \(\mathbb{K}\) of characteristic zero, with basis \(\set{T_{a}}\), Killing form \(\kappa\) — nondegenerate by Theorem A31.1 — and enveloping algebra \(U(\mathfrak{g})\) of Definition 18.5, filtered by \(U_{n}\) as in Definition A30.2. We write \(S(\mathfrak{g})\) for the symmetric algebra of Definition A30.7 and \(\omega\) for the symmetrization map Equation (A30.22).
Statement
Let \(\mathfrak{g}\) be a finite-dimensional semisimple Lie algebra of rank \(\ell\) over an algebraically closed field of characteristic zero. Then there are \(\ell\) Casimir elements \(C^{(1)},\ldots,C^{(\ell)} \in Z\left(U(\mathfrak{g})\right)\), algebraically independent, such that every element of the centre is a polynomial in them with scalar coefficients,
and no set of fewer than \(\ell\) elements of the centre generates it. By Corollary 18.18 each \(C^{(i)}\) acts on a finite-dimensional irreducible representation as a scalar, so such a representation carries \(\ell\) invariant labels and no more; that these \(\ell\) numbers also separate the finite-dimensional irreducible representations is the quoted Theorem A32.9. Rests on Definition 18.20, Definition 18.8 and Theorem A31.1.
The centre of the enveloping algebra and the invariant polynomials
For \(u \in U(\mathfrak{g})\) the following are equivalent: \(u\) is central; \(\comm{u}{X} = 0\) for every \(X \in \mathfrak{g}\); and \(u\) is invariant under the adjoint action \(\ad_{X}u = Xu - uX\) of \(\mathfrak{g}\) on \(U(\mathfrak{g})\). In particular the Casimir elements of Definition 18.8 are exactly the \(\ad\)-invariant elements of \(U(\mathfrak{g})\). Rests on Definitions 18.5 and 18.8.
Derives Lemma A32.2. The second and third conditions are the same statement written twice. A central element certainly commutes with every \(X \in \mathfrak{g}\). Conversely, the set of elements commuting with a fixed \(u\) is a subalgebra of \(U(\mathfrak{g})\) containing the unit — it is closed under sums and, by the Leibniz rule \(\comm{vw}{u} = v\comm{w}{u} + \comm{v}{u}w\), under products. If it contains \(\mathfrak{g}\) it therefore contains the algebra generated by \(\mathfrak{g}\) and the unit, which is \(U(\mathfrak{g})\) by Definition 18.5.
∎Let \(S(\mathfrak{g})^{\mathfrak{g}}\) denote the \(\ad\)-invariant elements of \(S(\mathfrak{g})\). Then:
-
\(\omega\) restricts to a linear isomorphism \(S(\mathfrak{g})^{\mathfrak{g}} \to Z\left(U(\mathfrak{g})\right)\);
-
\(S(\mathfrak{g})^{\mathfrak{g}}\) is a graded subalgebra of \(S(\mathfrak{g})\), and the symbol map Equation (A30.21) identifies the associated graded algebra of the centre with it,
\begin{equation}\tag{A32.3} \bigoplus_{n \ge 0} \frac{\left(Z\left(U(\mathfrak{g})\right) \cap U_{n}\right) + U_{n-1}}{U_{n-1}} \;\cong\; S(\mathfrak{g})^{\mathfrak{g}} \end{equation}as graded algebras.
Rests on Proposition A30.9, Definition A30.7 and Lemma A32.2.
Derives Proposition A32.3. (1) By Proposition A30.9 the map \(\omega\) is a linear isomorphism intertwining the two adjoint actions, so it carries the invariants of \(S(\mathfrak{g})\) bijectively onto the invariants of \(U(\mathfrak{g})\), and the latter are the centre by Lemma A32.2.
(2) The action of \(\ad_{X}\) on \(S(\mathfrak{g})\) is by a derivation that preserves degree, so an invariant decomposes into invariant homogeneous pieces and \(S(\mathfrak{g})^{\mathfrak{g}}\) is graded; it is a subalgebra because a derivation annihilating two elements annihilates their product.
For Equation (A32.3), note first that \(\ad_{X}\) maps \(U_{n}\) into \(U_{n}\) — by the Leibniz rule it replaces one letter of a word of length \(n\) by \(\comm{X}{\cdot}\) of it, which is again a letter — and that the induced map on \(U_{n}/U_{n-1} \cong S^{n}(\mathfrak{g})\) is exactly the derivation \(\ad_{X}\) of \(S^{n}(\mathfrak{g})\). Hence the symbol of a central element of \(U_{n}\) is an invariant of \(S^{n}(\mathfrak{g})\), which gives the inclusion “\(\subseteq\)” in Equation (A32.3). Conversely, if \(p \in S^{n}(\mathfrak{g})^{\mathfrak{g}}\) then \(\omega(p)\) lies in \(U_{n}\), is central by part (1), and has symbol \(p\) by Proposition A30.9(1); so the inclusion is an equality. The identification is multiplicative because the symbol map is (Definition A30.7).
∎Suppose \(S(\mathfrak{g})^{\mathfrak{g}}\) is a free polynomial algebra on homogeneous elements \(p_{1},\ldots,p_{\ell}\) of degrees \(d_{1},\ldots,d_{\ell}\). Put \(C^{(i)} = \omega(p_{i}) \in Z\left(U(\mathfrak{g})\right) \cap U_{d_{i}}\). Then the \(C^{(i)}\) are algebraically independent and generate \(Z\left(U(\mathfrak{g})\right)\) as an algebra, so Equation (A32.2) holds. Rests on Proposition A32.3 and Definition A30.7.
Derives Lemma A32.4. Give the monomial \(\left(C^{(1)}\right)^{\alpha_{1}}\cdots \left(C^{(\ell)}\right)^{\alpha_{\ell}}\) the weight \(\sum_{i}\alpha_{i}d_{i}\); by Equation (A30.3) a monomial of weight \(N\) lies in \(U_{N}\), and by Proposition A32.3 its symbol in \(S^{N}(\mathfrak{g})\) is the corresponding monomial \(p^{\alpha} = p_{1}^{\alpha_{1}}\cdots p_{\ell}^{\alpha_{\ell}}\).
Independence. Let \(F\) be a nonzero polynomial in \(\ell\) variables and let \(N\) be the largest weight carried by a monomial occurring in it. Then \(F\left(C^{(1)},\ldots,C^{(\ell)}\right) \in U_{N}\) and its symbol is \(\sum_{\text{weight }\alpha = N}\lambda_{\alpha}p^{\alpha}\), a nonzero element of \(S^{N}(\mathfrak{g})\) because the \(p_{i}\) are algebraically independent. A nonzero symbol means the element is not in \(U_{N-1}\); in particular it is not zero.
Generation. Let \(z\) be central and induct on the smallest \(n\) with \(z \in U_{n}\). Its symbol is a homogeneous invariant of degree \(n\), hence a weighted-homogeneous polynomial \(Q\) of weight \(n\) in the \(p_{i}\); then \(z - Q\left(C^{(1)},\ldots,C^{(\ell)}\right)\) is central and lies in \(U_{n-1}\), and the induction hypothesis applies. The base \(n = 0\) is \(z \in \mathbb{K}\).
∎The Killing form, being nondegenerate and invariant, is an isomorphism \(\mathfrak{g} \to \mathfrak{g}^{*}\) of \(\mathfrak{g}\)-modules, and hence induces an isomorphism of graded algebras
the right-hand side being the algebra of polynomial functions on \(\mathfrak{g}\) annihilated by the natural action of every \(\ad_{X}\). In components, an element of \(S^{r}(\mathfrak{g})^{\mathfrak{g}}\) is exactly a totally symmetric invariant tensor \(k^{a_{1}\cdots a_{r}}\) in the sense of Definition 18.14, and the corresponding Casimir element is Equation (18.19). Rests on Definition 18.10, Definition 18.14 and Theorem A31.1.
Derives Lemma A32.5. The map \(X \mapsto \kappa(X,\cdot)\) is injective because \(\kappa\) is nondegenerate (Theorem A31.1) and hence bijective by dimension; it intertwines the adjoint and coadjoint actions precisely because \(\kappa\) is invariant, Equation (18.16). An isomorphism of modules induces one of symmetric algebras, and it carries invariants to invariants. The component statement is Remark 18.15: writing an element of \(S^{r}(\mathfrak{g})\) as \(k^{a_{1}\cdots a_{r}}T_{a_{1}}\cdots T_{a_{r}}\) with \(k\) totally symmetric, the vanishing of \(\ad_{T_{c}}\) on it is Equation (18.18) term by term, and Corollary A30.10 identifies \(\omega\) of that element with the Casimir Equation (18.19).
∎So the first link of Equation (A32.1) is established, in the strong form of Lemma A32.4: the enveloping algebra contributes nothing of its own to the count. Counting Casimir operators is counting the algebraically independent invariant polynomials on \(\mathfrak{g}\), and that is a question about the adjoint action alone.
The two quoted theorems
Let \(\mathfrak{g}\) be a semisimple Lie algebra over an algebraically closed field of characteristic zero, \(\mathfrak{h} \subseteq \mathfrak{g}\) a Cartan subalgebra (Definition 18.20), and \(W\) the Weyl group of \(\mathfrak{g}\) acting on \(\mathfrak{h}\). Then restriction of functions from \(\mathfrak{g}\) to \(\mathfrak{h}\),
is an isomorphism of graded algebras. Rests on Definition 18.20 and Lemma A32.5.
Let \(\mathbb{V}\) be a vector space of dimension \(\ell\) over a field of characteristic zero and let \(W\) be a finite group of linear transformations of \(\mathbb{V}\) generated by reflections — elements fixing a hyperplane pointwise. Then the algebra of \(W\)-invariant polynomial functions on \(\mathbb{V}\) is a free polynomial algebra,
on exactly \(\ell\) algebraically independent homogeneous generators \(f_{1},\ldots,f_{\ell}\), whose degrees are determined by \(W\). Moreover, for \(\mathfrak{g}\) semisimple with Cartan subalgebra \(\mathfrak{h}\) of dimension \(\ell\), the Weyl group \(W\) is a finite group acting faithfully on \(\mathfrak{h}\) and generated by the reflections in the root hyperplanes, so Equation (A32.6) applies to it with \(\mathbb{V} = \mathfrak{h}\). Rests on Definition 18.20.
The free-polynomial statement is [Chevalley:1955]. That the Weyl group is a finite reflection group of rank \(\ell\) acting faithfully on \(\mathfrak{h}\) is standard structure theory and is quoted with it; no entry of this bibliography carries that half.
Two theorems are assumed and not proved, and it is worth being exact about what they are and why they are not proved here. Theorem A32.6 and the last sentence of Theorem A32.7 both belong to the root-space theory of semisimple Lie algebras: the decomposition \(\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha} \mathfrak{g}_{\alpha}\) into eigenspaces of \(\mathfrak{h}\), the root system it produces, the conjugacy of Cartan subalgebras, and the Weyl group generated by root reflections. That apparatus is the content of several chapters of a book on Lie algebras and this treatise does not build it: Lie Groups, Lie Algebras, and Fibre Bundles constructs the algebras it needs by hand and computes their ranks directly, as Proposition 18.22 does for \(\mathfrak{so}(p,q)\). Equation (A32.6) itself is a theorem of invariant theory, not of Lie theory, and is likewise quoted.
What is not quoted is the first link of Equation (A32.1), which is the part that could plausibly hide a circularity — it is proved in The centre of the enveloping algebra and the invariant polynomials from the Poincaré–Birkhoff–Witt theorem of The Poincaré–Birkhoff–Witt Theorem, itself proved from nothing but the Jacobi identity; and it is the part that fails without semisimplicity, which is why Remark 18.23 is careful to exclude the Poincaré algebra. Nor is the arithmetic quoted: the count in The count and the check in The count checked against the chapter's algebras are carried out here.
The count
Proof of Theorem A32.1. Derives Theorem A32.1. Chain the identifications. By Lemma A32.5, \(S(\mathfrak{g})^{\mathfrak{g}} \cong S\left(\mathfrak{g}^{*}\right)^{\mathfrak{g}}\) as graded algebras; by Theorem A32.6 the latter is \(S\left(\mathfrak{h}^{*}\right)^{W}\); and by Theorem A32.7 that is a free polynomial algebra on \(\ell = \dim\mathfrak{h}\) homogeneous generators \(f_{1},\ldots,f_{\ell}\), where \(\ell\) is the rank of \(\mathfrak{g}\) by Definition 18.20. Transporting the \(f_{i}\) back gives homogeneous \(p_{1},\ldots,p_{\ell} \in S(\mathfrak{g})^{\mathfrak{g}}\) that are algebraically independent and generate, and Lemma A32.4 converts them into Casimir elements \(C^{(i)} = \omega(p_{i})\) with the properties claimed in Equation (A32.2).
No fewer than \(\ell\). A commutative algebra generated by \(m\) elements has transcendence degree at most \(m\) over \(\mathbb{K}\), since every element is a polynomial in the generators and any \(m+1\) elements of such an algebra are algebraically dependent. By Equation (A32.2) the centre has transcendence degree exactly \(\ell\), the \(C^{(i)}\) being algebraically independent. Hence no \(m < \ell\) elements of the centre generate it, and no \(m < \ell\) Casimir eigenvalues can carry the information that all of them do.
∎Let \(\mathfrak{g}\) be semisimple over an algebraically closed field of characteristic zero. Two finite-dimensional irreducible representations of \(\mathfrak{g}\) on which every element of \(Z\left(U(\mathfrak{g})\right)\) takes the same scalar value are equivalent. Equivalently, the \(\ell\) numbers
determine the equivalence class of a finite-dimensional irreducible representation. Rests on Definition 18.8 and Corollary 18.18.
Theorem A32.1 says how many independent invariants there are; Theorem A32.9 says that their values are enough to tell two irreducible representations apart. The two are different assertions and the second does not follow from the first, which is why the chapter's phrase “a complete set of invariant labels” has been split here. Remark 18.19 makes the same point from the other side: one Casimir separates the irreducible representations of \(\mathfrak{su}(2)\) and does not separate those of \(\mathfrak{su}(3)\), and the reason is not a failure of Theorem A32.1 but the fact that \(\mathfrak{su}(3)\) has rank \(2\) — one label is simply not the whole set.
Theorem A32.1 is stated over an algebraically closed field, but the algebras of Lie Groups, Lie Algebras, and Fibre Bundles are real. The transition is harmless and worth stating once. For a real semisimple \(\mathfrak{g}\) one has \(U\left(\mathfrak{g}\right)\otimes_{\R}\C = U\left(\mathfrak{g}\otimes_{\R}\C\right)\), because both sides are generated by \(\mathfrak{g}\otimes\C\) with the same relations Equation (18.12), and taking the centre commutes with this extension: an element is central if and only if its real and imaginary parts are. So the number of independent Casimir elements of a real form equals that of its complexification, and the rank meant throughout is that of the complexification. This is what Proposition 18.22 in fact computes: its proof passes to \(\mathfrak{h}_{\C}\) at once and counts the eigenvalue pairs \(\pm\alpha_{i}\) there, which is the only sensible reading, since the adjoint action of \(J_{12}\) on the compact algebra \(\mathfrak{so}(3)\) has eigenvalues \(\pm\ii\) and is not diagonalizable over \(\R\) at all.
The count checked against the chapter's algebras
Any single generator spans a maximal abelian subalgebra of \(\mathfrak{su}(2) \cong \mathfrak{so}(3)\), whose complexification is \(\mathfrak{sl}(2,\C)\); the rank is \(\ell = 1\), in agreement with Equation (18.22) at \(D = 3\). Theorem A32.1 then predicts a single Casimir generator, and the invariant found in Definition 18.40 is \(\vect{J}^{2}\), of degree two — the free generator being \(f_{1}\) of degree \(2\). That it generates everything is visible in Theorem 18.43: the eigenvalue \(j(j+1)\) is injective in \(j \ge 0\), so by Theorem A32.9 nothing further is needed, and by Theorem A32.1 nothing further exists. Rests on Theorems 18.43 and A32.1.
The diagonal traceless matrices give a maximal abelian subalgebra of dimension \(2\) (Proposition 18.50), so \(\ell = 2\) and Theorem A32.1 predicts exactly two independent Casimir elements. They are found in Proposition 18.54: one quadratic, built from \(\kappa^{ab}\), and one cubic, built from the totally symmetric tensor \(d^{abc}\) of Equation (18.70) — degrees \(2\) and \(3\), which are the degrees Chevalley's theorem assigns to the invariants of the Weyl group of \(\mathfrak{su}(3)\). The count is not idle: by Proposition 18.55 the fundamental and antifundamental representations share the value of the quadratic invariant and are told apart only by the cubic one. One label would have failed; three would have been one too many. Rests on Theorem A32.1, Proposition 18.54 and Proposition 18.55.
By Corollary A31.15 the algebra \(\mathfrak{so}(3,1)\) is semisimple, and by Equation (18.22) its rank is \(\lfloor 4/2 \rfloor = 2\). Theorem A32.1 therefore gives exactly two independent Casimir operators for the Lorentz algebra of the observed spacetime, and they are the two familiar quadratics,
the second existing only because \(D = 4\) makes the Levi-Civita symbol carry exactly four indices — the same accident of dimension that Remark 18.67 records for the Pauli–Lubanski vector. Both are invariant by Lemma 18.62 and Corollary 18.12, and by Theorem A32.1 there is no third. Rests on Theorem A32.1, Corollary A31.15 and Equation (18.22).
Theorem 18.65 finds two invariants for the Poincaré algebra in \(3+1\) dimensions, \(P^{2}\) and \(W^{2}\), and it is tempting to read that as Theorem A32.1 applied to a rank-two algebra. It is not, and the temptation must be resisted for a reason that is now provable rather than asserted: the translations form an abelian ideal, so by Proposition A31.5 the Killing form of the Poincaré algebra is degenerate, the algebra is not semisimple, and every step of The centre of the enveloping algebra and the invariant polynomials — which inverts \(\kappa\) in Lemma A32.5 — fails at once. That the two counts agree at \(D = 4\) is the coincidence recorded in Remark 18.23, and Equation (18.107) shows the agreement breaking in odd dimension, where \(\lceil D/2 \rceil \neq \lfloor D/2 \rfloor\). The Galilei algebra of Example 18.80 makes the point sharper still: its invariants include the central mass, which is not an invariant of any semisimple algebra at all.
Where Theorem A32.1 is used for an inhomogeneous algebra, it is applied to a semisimple subalgebra and not to the whole: the count Equation (18.107) applies it to the little algebra \(\mathfrak{so}(D-1)\) of Lemma 18.68, which is semisimple for \(D-1 \ge 3\) by Corollary A31.15, and adds the mass by hand.
Racah's Theorem on the Number of Casimir Operators discharges the proof obligation of Theorem 18.21, modulo the two theorems of structure theory named in Remark A32.8 and the separation theorem Theorem A32.9. It is used at Proposition 18.22 to turn the rank \(\lfloor D/2 \rfloor\) into a number of Lorentz invariants, at Remark 18.67 to count the labels of a massive representation in general dimension, and throughout Section 18.2 whenever a representation is labelled by its Casimir eigenvalues. Its physical content is stated in Remark 18.24: the labels are pure numbers, and the measured quantity is the label multiplied by the power of \(\hbar\) that carries its SI dimension.