Cartan's criterion

In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric [bilinear form] on defined by the formula
where tr denotes the trace of a linear operator. The criterion was introduced by.

Cartan's criterion for solvability

Cartan's criterion for solvability states:
The fact that in the solvable case follows from Lie's theorem that puts in the upper triangular form over the algebraic closure of the ground field. The converse can be deduced from the nilpotency criterion based on the Jordan–Chevalley decomposition, as explained there.
Applying Cartan's criterion to the adjoint representation gives:

Cartan's criterion for semisimplicity

Cartan's criterion for semisimplicity states:
gave a very short proof that if a finite-dimensional Lie algebra 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 has a non-degenerate Killing form.

Examples

Cartan's criteria fail in characteristic ; for example:

the Lie algebra is simple if k has characteristic not 2 and has vanishing Killing form, though it does have a nonzero invariant bilinear form given by.
the Lie algebra with basis for and bracket = a_i+''j is simple for but has no nonzero invariant bilinear form.
If k'' has characteristic 2 then the semidirect product gl₂.k² is a solvable Lie algebra, but the Killing form is not identically zero on its derived algebra sl₂.k².

If a finite-dimensional Lie algebra is nilpotent, then the Killing form is identically zero. 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=0.
In characteristic 0, every reductive Lie algebra 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/''tn''L 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.