Killing Forms and Cartan's Criteria

Killing Form

Let be a finite dimensional Lie algebra over a field of characteristic . Then, the Killing form of is a bilinear, symmetric, ad-invariant form If the Lie algebra referred to is clear, we simply denote .

Cartan's Criteria for Solvability

If is a finite dimensional Lie algebra over a field of characteristic , then that is, for every .

Proof Sketch The proof for this is slightly complicated, but we show the (easier) forward direction: If is solvable, then we have that is solvable, and hence is nilpotent. So, is a nilpotent matrix, and hence $$\operatorname{tr}([\ad(\g), \ad(\g)] \ad(\g))=\kappa([\g, \g], \g) = 0.$$$\square$

Corollary

over a field of characteristic is solvable if and only if for all and .

Cartan's Criteria for Semisimplicity

If is a finite dimensional Lie algebra over a field of characteristic , then

e.g. We recall that is simple (and hence semisimple) when . Use the standard basis . Then, using the commutator relations we can deduce Using these matrices, we can compute the matrix : This has determinant , which is nonzero exactly when . So, by Cartan’s criterion, we know is semisimple if , which aligns with our previous findings.

Proof