The classification of the simple modular Lie algebras. IV: Determining the associated graded algebra (Q690119)

This is the fourth in a series of papers presenting steps in the classification of the simple modular Lie algebras over an algebraically closed field of characteristic \(p>7\) [the previous papers are: I: Ann. Math., II. Ser. 130, 643-677 (1989; Zbl 0699.17016); II: J. Algebra 151, 425-475 (1992; Zbl 0776.17018); III: Ann. Math., II. Ser. 133, 577-604 (1991; Zbl 0736.17021)]. The author verifies the Kostrikin-Shafarevich conjecture in the case where there exists a nonclassical root and no 2-section has a core \(H(2;\mathbf{1};\Phi(\tau))^{(1)}\) by showing that all such algebras are of Cartan type. This follows from results he obtains from an investigation of the filtration (and an associated graded algebra) defined by \(Q(L,T)\) for the Lie algebra \(L\) and an optimal torus \(T\) in a \(p\)-envelope of \(L\), where \(Q(L,T)\) is a maximal subalgebra constructed from the compositionally classical subalgebras of the 1-sections and studied in [the author, \textit{G. Benkart} and \textit{J. M. Osborne}, Trans. Am. Math. Soc. 341, 227-252 (1994)].