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 $mathfrak\{g\}$ defined by the formula: $K(u,v)=operatorname\{tr\}(operatorname\{ad\}(u)operatorname\{ad\}(v)),$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 $mathfrak\{g\}$ over a field of characteristic zero is semisimple if and only if the Killing form is nondegenerate. A Lie algebra $mathfrak\{g\}$ is solvable if and only if $K(mathfrak\{g\},\; [mathfrak\{g\},mathfrak\{g\}]\; )=0.$"

More generally, a finite-dimensional Lie algebra $mathfrak\{g\}$ 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

See also

* Modular Lie algebra