Cartan's criterion

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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.

[edit] Formulation

Cartan's criterion states:

A finite-dimensional Lie algebra \mathfrak{g} over a field of characteristic zero is nilpotent if and only if the Killing form is identically zero, and 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.

[edit] 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

[edit] See also