Cartan's criterion

Cartan's criterion for semisimplicity

Cartan's criterion for semisimplicity states:

A finite-dimensional Lie algebra ${\displaystyle \mathfrak{g}}$ over a field of characteristic zero is semisimple if and only if the Killing form is non-degenerate.

Jean Dieudonné (1953) gave a very short proof that if a finite-dimensional Lie algebra (in any characteristic) has a non-degenerate invariant bilinear form and no non-zero abelian ideals, and in particular if its Killing form is non-degenerate, then it is a sum of simple Lie algebras.

Conversely, it follows easily from Cartan's criterion for solvability that a semisimple algebra (in characteristic 0) has a non-degenerate Killing form.

Examples

Cartan's criteria fail in characteristic ${\displaystyle p>0}$; for example:

• the Lie algebra ${\displaystyle {\mathfrak{s}\mathfrak{l}}_p(k)}$ is simple if k has characteristic not 2 and has vanishing Killing form, though it does have a nonzero invariant bilinear form given by ${\displaystyle (a,b)=\mathrm{tr}(ab)}$.
• the Lie algebra with basis ${\displaystyle a_n}$ for ${\displaystyle n\in \mathbb{Z}/p\mathbb{Z}}$ and bracket [a_i,a_j] = (i−j)a_i+j is simple for ${\displaystyle p>2}$ 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[t]/tⁿL[t] 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[t]-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.