On the structure of the graded Lie algebra associated with a noncontractible filtration
The theory of graded Lie algebras is the main tool of investigating nonclassical simple Lie algebras over an algebraically closed field of characteristic \(p>0\). This is the reason why it is of particular interest to study graded Lie algebras \(G=\oplus_{i\geq -q}G_i\) not having nontrivial ideals in \(G_0+G_+\) and \(G_-\), and in which \(G_{-1}\) is an irreducible \(G_0\)-module. Let \(\widehat G\) be the minimal ideal of \(G\). In contrast to the case of characteristic zero, the subalgebra \(G_0\) of \(G\) may have a noncentral radical. The main result of this paper is the description of the solvable radical \(\text{solv } G_0\) of \(G_0\). It is proved that when \(p>3\) either \(\widehat {G}_0\) or \(\text{solv}G_0\) is abelian. Using this result the author proves that for \(p>3\) a simple Lie algebra with solvable maximal subalgebra is isomorphic either to \(sl(2)\) or to the Zassenhaus algebra \(W(1:m)\). Many speculations are true for infinite-dimensional Lie algebras. In particular, in the paper the structure of simple linear compact Lie algebras with solvable maximal subalgebra of finite codimension is obtained. It should be pointed out that many technical results on graded Lie algebras are of independent interest.
scientific article; zbMATH DE number 1097705
scientific article