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Cartan's criterion - Wikipedia

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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 𝑔 defined by the formula where tr denotes the trace of a linear operator. The criterion was introduced by Élie Cartan (1894).[1] Cartan's criterion for solvability states: The fact that tr ⁡ ( 𝑎 𝑏 ) = 0 in the solvable case follows from Lie's theorem that puts 𝑔 in the upper triangular form over the algebraic closure of the ground field (the trace can be computed after extending the ground field). The converse can be deduced from the nilpotency criterion based on the Jordan–Chevalley decomposition, as explained there. Applying Cartan's criterion to the adjoint representation gives: Cartan's criterion for semisimplicity states: Jean Dieudonné (1953) gave a very short proof that if a finite-dimensional Lie algebra (in any charac

Cartan's criterion - Wikipedia Jump to content From Wikipedia, the free encyclopedia 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 g {\displaystyle {\mathfrak {g}}} defined by the formula κ ( u , v ) = tr ⁡ ( ad ⁡ ( u ) ad ⁡ ( v ) ) , {\displaystyle \kappa (u,v)=\operatorname {tr} (\operatorname {ad} (u)\operatorname {ad} (v)),} where tr denotes the trace of a linear operator . The cri

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