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 states: A Lie subalgebra of endomorphisms of a finite-dimensional vector space over a field of characteristic zero is solvable if and only if whenever 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 trace can be computed after extending the ground field). The converse can be deduced from the nilpotency criterion based on the Jordan–Chevalley decomposition (for the proof, follow the link). Applying Cartan's criterion to the adjoint representation gives: A finite-dimensional Lie algebra over a field of characteristic zero is solvable if and only if (where K is the Killing form). Cartan's criterion for semisimplicity states: A finite-dimensional Lie algebra over a field of characteristic zero is semisimple if and only if the Killing form is non-degenerate. 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. 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 [ai,aj] = (i−j)ai+j is simple for but has no nonzero invariant bilinear form. If k has characteristic 2 then the semidirect product gl2(k).