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 $\mathfrak{g}$ 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 was introduced by Élie Cartan (1894).^[1]

Cartan's criterion for solvability

Cartan's criterion for solvability states:

A Lie subalgebra $\mathfrak{g}$ of endomorphisms of a finite-dimensional vector space over a field of characteristic zero is solvable if and only if $Tr(ab)=0$ whenever $a\in\mathfrak{g},b\in[\mathfrak{g},\mathfrak{g}].$

The fact that $Tr(ab)=0$ in the solvable case follows immediately from Lie's theorem that solvable Lie algebras in characteristic 0 can be put in upper triangular form.

Applying Cartan's criterion to the adjoint representation gives:

A finite-dimensional Lie algebra $\mathfrak{g}$ over a field of characteristic zero is solvable if and only if $K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0$ (where K is the Killing form).

Cartan's criterion for semisimplicity

Cartan's criterion for semisimplicity states:

A finite-dimensional Lie algebra $\mathfrak{g}$ over a field of characteristic zero is semisimple if and only if the Killing form is non-degenerate.

Dieudonné (1953) 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.

Examples

Cartan's criteria fail in characteristic p>0; for example:

the Lie algebra SL_p(k) is simple if k has characteristic not 2 and has vanishing Killing form, though it does have a nonzero invariant bilinear form given by (a,b) = Tr(ab).
the Lie algebra with basis a_n for n∈Z/pZ and bracket [a_i,a_j] = (i−j)a_i+j is simple for p>2 but has no nonzero invariant bilinear form.
If k has characteristic 2 then the semidirect product gl₂(k).k² is a solvable Lie algebra, but the Killing form is not identically zero on its derived algebra sl₂(k).k².

If a finite-dimensional Lie algebra is nilpotent, then the Killing form is identically zero (and more generally the Killing form vanishes on any nilpotent ideal). 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(b²)=0.

In characteristic 0, every reductive Lie algebra (one that is a sum of abelian and simple Lie algebras) 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[t]/tⁿL[t] 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[t]-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.

Notes

↑ Cartan, Chapitre IV, Théorème 1

References

Cartan, Élie (1894), Sur la structure des groupes de transformations finis et continus, Thesis, Nony
Dieudonné, Jean (1953), "On semi-simple Lie algebras", Proceedings of the American Mathematical Society 4: 931–932, doi:10.2307/2031832, ISSN 0002-9939, JSTOR 2031832, MR 0059262
Serre, Jean-Pierre (2006) [1964], Lie algebras and Lie groups, Lecture Notes in Mathematics 1500, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-70634-2, ISBN 978-3-540-55008-2, MR 2179691