Isomorphism of Simple Lie Algebras

Abstract

Let $\mathcal {L}$ and $\mathcal {L}’$ be finite dimensional simple Lie algebras over a field of characteristic zero. A necessary and sufficient condition is given for $\mathcal {L}$ and $\mathcal {L}’$ to be isomorphic. The anisotropic kernel of $\mathcal {L}$ is also studied. In particular, a result about this kernel in the rank one reduced case is proved. This result is then used to prove a conjugacy theorem for the simple summands of the anisotropic kernel in the general reduced case. The results and methods of this paper are rational in the sense that they involve no extension of the base field.