# Cartan's criterion

Cartan's criterion

Cartan's criterion is an important mathematical theorem in the foundations of Lie algebra theory that gives conditions for a Lie agebra to be nilpotent, solvable, or semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on $mathfrak\left\{g\right\}$ defined by the formula: $K\left(u,v\right)=operatorname\left\{tr\right\}\left(operatorname\left\{ad\right\}\left(u\right)operatorname\left\{ad\right\}\left(v\right)\right),$where tr denotes the trace of a linear operator. The criterion is named after Élie Cartan.

Formulation

Cartan's criterion states: : "A finite-dimensional Lie algebra $mathfrak\left\{g\right\}$ over a field of characteristic zero is semisimple if and only if the Killing form is nondegenerate. A Lie algebra $mathfrak\left\{g\right\}$ is solvable if and only if $K\left(mathfrak\left\{g\right\}, \left[mathfrak\left\{g\right\},mathfrak\left\{g\right\}\right] \right)=0.$"

More generally, a finite-dimensional Lie algebra $mathfrak\left\{g\right\}$ is reductive if and only if it admits a nondegenerate invariant bilinear form.

References

* Jean-Pierre Serre, "Lie algebras and Lie groups." 1964 lectures given at Harvard University. Second edition. Lecture Notes in Mathematics, 1500. Springer-Verlag, Berlin, 1992. viii+168 pp. ISBN 3-540-55008-9

* Modular Lie algebra

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