# 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 Élie Cartan (1894).^{[1]}

## Contents

## Cartan's criterion for solvability[edit]

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 immediately from Lie's theorem that solvable Lie algebras in characteristic 0 can be put in upper triangular form.^{[clarification needed]}

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[edit]

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.*

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[edit]

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**/*p***Z**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 gl_{2}(*k*).*k*^{2}is a solvable Lie algebra, but the Killing form is not identically zero on its derived algebra sl_{2}(*k*).*k*^{2}.

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(*b*^{2})=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*]/*t*^{n}*L*[*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 *t*^{n−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[edit]

**^**Cartan, Chapitre IV, Théorème 1

## References[edit]

- 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