Cartan's Criterion - Examples

Examples

Cartan's criteria fail in characteristic p>0; for example:

• the Lie algebra SL_p(k) is simple if k has characteristic not 2 and has vanishing Killing form, though it does have a nonzero invariant bilinear form given by (a,b) = Tr(ab).
• the Lie algebra with basis a_n for n∈Z/pZ and bracket = (i−j)a_i+j is simple for p>2 but has no nonzero invariant bilinear form.
• If k has characteristic 2 then the semidirect product gl₂(k).k2 is a solvable Lie algebra, but the Killing form is not identically zero on its derived algebra sl₂(k).k2.

If a finite dimensional Lie algebra is nilpotent, then the Killing form is identically zero (and more generally the Killing form vanishes on any nilpotent ideal). The converse is false: there are non-nilpotent Lie algebras whose Killing form vanishes. An example is given by the semidirect product of an abelian Lie algebra V with a 1-dimensional Lie algebra acting on V as an endomorphism b such that b is not nilpotent and Tr(b2)=0.

In characteristic 0, every reductive Lie algebra (one that is a sum of abelian and simple Lie algebras) has a non-degenerate invariant symmetric bilinear form. However the converse is false: a Lie algebra with a non-degenerate invariant symmetric bilinear form need not be a sum of simple and abelian Lie algebras. A typical counterexample is G = L/tnL where n>1, L is a simple complex Lie algebra with a bilinear form (,), and the bilinear form on G is given by taking the coefficient of tn−1 of the C-valued bilinear form on G induced by the form on L. The bilinear form is non-degenerate, but the Lie algebra is not a sum of simple and abelian Lie algebras.