Classifying representations

The irreducible representations of the complex simple Lie algebras have been fully classified, and as it turns out, these representations apply directly to their compact real forms. Using similar reasoning to above, the irreducible representations of the compact connected Lie algebras and Lie groups have also been fully classified.

One can also classify the representations of finite groups, which are always completely reducible. In particular, Young tableaux are combinatorial diagrams used to enumerate the representations of the symmetric group \({S_{n}}\).

In one form of classification used in physics, the finite-dimensional complex irreps of a finite-dimensional Lie algebra \({\mathfrak{g}}\) may be characterized or labeled using the concept of a Casimir element. A Casimir element is an element of the universal enveloping algebra \({U\left(\mathfrak{g}\right)}\) of \({\mathfrak{g}}\), which as we recall is the associative algebra which includes \({\mathfrak{g}}\) as a subalgebra under the Lie commutator. A Casimir element \({E}\) is constructed from a basis of \({\mathfrak{g}}\) and a nondegenerate bilinear form \({B}\) (which \({E}\) is dependent upon); \({B}\) is also required to be “invariant” (AKA Ad-invariant, ad-invariant, associative), meaning that

\(\displaystyle B\left(\left[u,w\right],v\right)=B\left(u,\left[w,v\right]\right), \)

which if \({\mathfrak{g}}\) is the Lie algebra of a connected Lie group \({G}\) implies invariance under the adjoint rep, i.e. \({\forall g\in G}\)

\begin{aligned}B\left(g_{\mathrm{Ad}}\left(u\right),g_{\mathrm{Ad}}\left(v\right)\right) & =B\left(u,v\right).\end{aligned}

The key property of a Casimir element is that it can be shown to be an element of the center of \({U\left(\mathfrak{g}\right)}\) (commutes with all elements), which by Schur’s Lemma means that under any finite-dimensional complex irrep \({\rho}\) the Casimir operator \({\rho\left(E\right)}\) is a scalar multiple of the identity matrix \({C\left(\rho\right)I}\), where the scalar \({C\left(\rho\right)}\) (which is often described as the eigenvalue of the Casimir operator) is called the Casimir invariant, and may then be used to label the irrep.

Δ \({\rho\left(E\right)}\) denotes \({E}\) constructed from the basis vector irreps under \({\rho}\). Note that the Casimir invariant for a given \({B}\) is not necessarily different for each irrep.
Δ The terms Casimir element, Casimir operator, and Casimir invariant are often used interchangeably.

The above depicts the Casimir element, which is constructed from an invariant nondegenerate bilinear form \({B}\) on \({\mathfrak{g}}\), and is an element of the center of the universal enveloping algebra \({U\left(\mathfrak{g}\right)}\). It maps to the Casimir operator \({\rho\left(E\right)}\) under any finite-dimensional complex irrep \({\rho}\) of \({\mathfrak{g}}\), which by Schur’s Lemma is the identity matrix \({I}\) multiplied by the Casimir invariant \({C\left(\rho\right)}\).

Now, there exists an symmetric invariant bilinear form called the Killing form (AKA Cartan-Killing form) which may be defined on any Lie algebra; it is nondegenerate iff the Lie algebra is semisimple (a fact called the Cartan criterion), which makes it an invariant pseudo inner product \({\left\langle \right\rangle _{K}}\) on a real semisimple Lie algebra. For a real semisimple Lie algebra, we may therefore choose a basis \({\hat{e}_{j}}\) which is orthonormal under the Killing form, and which can be used to construct a specific Casimir element called the quadratic (AKA second order) Casimir element

\(\displaystyle e^{2}\equiv\sum_{j}\hat{e}_{j}\hat{e}_{j}. \)

Δ A Lie algebra including an invariant pseudo inner product is sometimes called a metric (AKA orthogonal, quadratic, self-dual) Lie algebra.

Furthermore, for a simple Lie algebra, it can be shown that any symmetric invariant bilinear form is proportional to the Killing form. Since the trace is a bilinear form on multiplied matrices, which can be verified to be symmetric and invariant due to its cyclic property, we may therefore define the Dynkin index (AKA index, second order Dynkin index) \({Y\left(\rho\right)}\) of a finite-dimensional complex rep \({\rho}\) by

\begin{aligned}\mathrm{tr}\left(\rho\left(v\right)\rho\left(w\right)\right) & =Y\left(\rho\right)\left\langle v,w\right\rangle _{K}\\
\Rightarrow\mathrm{tr}\left(\rho\left(\hat{e}_{j}\right)\rho\left(\hat{e}_{k}\right)\right) & =Y\left(\rho\right)\delta_{jk}.

Taking the trace of the quadratic Casimir operator under a finite-dimensional complex irrep then yields a relationship in terms of the vector space dimensions of \({\mathfrak{g}}\) and the space acted on by the rep \({\rho}\):

\begin{aligned}\mathrm{tr}\left(\sum_{j}\rho\left(\hat{e}_{j}\right)\rho\left(\hat{e}_{j}\right)\right) & =Y\left(\rho\right)\mathrm{dim}\left(\mathfrak{g}\right)\\
& =\mathrm{tr}\left(C\left(\rho\right)I\right)=C\left(\rho\right)\mathrm{dim}\left(\rho\right)\\
\Rightarrow C\left(\rho\right) & =\frac{Y\left(\rho\right)\mathrm{dim}\left(\mathfrak{g}\right)}{\mathrm{dim}\left(\rho\right)}

Δ Note that although for a simple Lie algebra any symmetric invariant bilinear form is proportional to the Killing form, there may be other nondegenerate invariant bilinear forms which can be used to construct other Casimir elements. In fact, it can be shown that the number of algebraically independent Casimir elements for a simple Lie algebra is equal to its rank; these elements algebraically generate the center of the algebra. It can also be shown that the negative of the Killing form is a (positive definite) inner product iff the real semisimple Lie algebra is a compact real form, and that the Killing form is actually invariant under any automorphism of \({\mathfrak{g}}\).

The main physical application of all this is to label simple algebra reps by the eigenvalues of the rep of \({e^{2}}\). In particular, in quantum physics, the state of a physical system is associated with a vector in a complex Hilbert space. The laws of physics are equations which are postulated to be invariant under various symmetry transformations, in particular those of \({SU(n)}\) and \({SO(r,s)}\), and hence their (simple) Lie algebras. The Hilbert space must therefore be acted on by (“carry”) a rep of each of these Lie algebras, under which the state vector is transformed in a way which leaves the laws invariant. Recalling Weyl’s theorem, these simple Lie algebras have completely reducible reps, and the subspace of state vectors acted on by a component irrep is assumed to correspond to a single elementary particle, which may then be labeled by its quadratic Casimir invariant.

Note that for a connected \({G}\) whose Lie algebra is \({\mathfrak{g}}\), \({\left\langle \right\rangle _{K}}\) is invariant under the adjoint rep, which for a matrix group is a similarity transformation, i.e.

\begin{aligned}\left\langle g_{\mathrm{Ad}}\left(u\right),g_{\mathrm{Ad}}\left(v\right)\right\rangle _{K} & =\left\langle u,v\right\rangle _{K}\\
& =\left\langle gug^{-1},gvg^{-1}\right\rangle _{K}.

Thus for the groups \({SU(n)}\) or \({SO^{e}(r,s)}\), which in gauge theories can be viewed as passive “rotations” (coordinate transformations preserving an inner product) on the vector space being acted upon, the Killing form (and therefore the trace) is a pseudo inner product on any \({\mathfrak{g}}\)-valued form which is independent of these coordinates, as one would naturally want to require. In fact, for these simple Lie algebras it is the unique invariant inner product up to a choice of units (since all others are proportional).

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