Simple Lie algebras of small characteristic. II: Exceptional roots
Part I, cf. J. Algebra 189, 419-480 (1997; Zbl 0878.17019). This article represents another step in extending to characteristic \(p > 3\) the classification of finite-dimensional simple Lie algebras, which has already been accomplished when the base field \(F\) is algebraically closed of characteristic \(p > 7\). The authors investigate finite-dimensional simple Lie algebras of absolute toral rank 2 where \(F\) has characteristic \(p > 3\) and show that, for such an algebra \(L\) with \(T\) a 2-dimensional torus in the semisimple \(p\)-envelope of \(L\), all roots of \(\Gamma (L,T)\) are nonexceptional unless \(L\) is isomorphic to a Melikyan algebra. It appears that this generalization of a step in the classification by \textit{R. E. Block} and \textit{R. L. Wilson} of the restricted simple Lie algebras for \(p > 7\) [J. Algebra 114, 115-259 (1988; Zbl 0644.17008)] overcomes a major obstacle in achieving the classification in characteristics 5 and 7. To do this the authors prove that for every root \(\alpha \in \Gamma (L,T)\) the subalgebra \(K' (\alpha)\) generated by \(\sum_{i\in {\mathbb F}_p^\ast} K_{i\alpha}\) (where \(K_{i\alpha} = \{ x \in L_{i\alpha} \mid \alpha ([x,L_{-\alpha L} ]) = 0\}\)) acts triangulably on \(L\). Some additional results which may be used in other steps of the classification proof are also obtained.
simple Lie algebras of absolute toral rank 2
scientific article; zbMATH DE number 1309257
scientific article