Why the Killing form?

You might find Thomas Hawkins' book "Emergence of the Theory of Lie Groups" an interesting place to look for first-principles explanations of Lie-theory facts. He explains how Killing, Cartan, and Weyl first came up with the structure theory for semi-simple Lie algebras. (See Section 6.2, in particular, for a detailed discussion of Cartan's contributions to Killing-style structure theory---including his introduction and use of the Killing form.)

According to Hawkins, one of Killing's insights in his structure theory for a Lie algebra $\mathfrak{g}$ was to consider the characteristic polynomial $$ {\rm det} (t I - {\rm ad}(X)) = t^n -\psi_1(X)t^{n-1} + \psi_2(X)t^{n-2} + \cdots + (-1)^n\psi_n(X) $$ as a function of $X$. (The start of the structure theory was to consider those $X$---regular elements---such that the eigenvalue $0$ has minimal multiplicity.) In general the coefficients $\psi_i(X)$ are polynomial functions on $\mathfrak{g}$ that are invariants for the adjoint action of $\mathfrak{g}$ on itself.

Consider, in particular, a simple Lie algebra $\mathfrak{g}$. Killing observed that the coefficient $\psi_1(X)$, which is a linear functional on $\mathfrak{g}$, must vanish identically, since its kernel is an ideal. Cartan considered the coefficient $\psi_2(X)$, which is a quadratic form on $X$. (The value $\psi_2(X)$ is essentially the sum of the squares of the eigenvalues---roots---of $X$, since $\psi_1(X)=0$.) The bilinear form associated to this quadratic form is the usual Killing form. The invariance of $\psi_2$ under the adjoint action translates to the "associativity" property of the Killing form. Cartan observed (in essence) that for $\mathfrak{g}$ simple, the kernel of the the associated bilinear form is either $0$ or $\mathfrak{g}$ (by invariance + simplicity), and he managed to prove that the kernel is always $0$, starting his repair of the faults in Killing's structure theory. One can look at Hawkins' book for the details of the story, stripped of modern efficiencies.

It is tempting to think that the simpler structure theory of finite-dimensional associative algebras (where non-degeneracy of the trace form also characterizes semi-simplicity) may have inspired Cartan. It seems (again according to Hawkins) that Molien introduced this form for associative algebras (as a bilinear form---as opposed to Cartan's quadratic form) independently in the same year (1893) that Cartan published his thesis.

Hi Ryan,

I presume given your description of the students that they know finite groups pretty well, and have seen the averaging idempotent $e=\frac{1}{|G|}\sum_{g\in G} g$, and how this can be used to construct an invariant inner product on any representation of a finite group. Perhaps you can convince them that compact groups admit the same sort of averaging idempotents via integral, and so perhaps you can construct the invariant inner product on finite dimensional representations of a compact group in more or less direct analogy with finite groups. Then you can derive the properties the Killing form should satisfy on the Lie algebra by setting g=e^tX, and taking derivatives of the axioms of the group's inner product?

This is the closest connection I can think of to finite group theory, which is hopefully well-understood by, or at least familiar to, your students.

What do you think? -david

A less algebraic answer, but one that really helped me to understand the role of the Killing form, is that it induces the unique G-invariant riemannian metric on symmetric spaces $G(\mathbb{R})/K$ (K maximal compact subgroup), another fact which was very dear to Cartan as well...