Maximal subalgebras of simple modular Lie algebras (Q1770460)

Let \(L\) be a restricted simple Lie algebra of Cartan type over an algebraically closed field of characteristic \(p>3\). The aim of the paper under review is to classify the maximal graded subalgebras \(M=M_{-2}\oplus M_{-1}\oplus M_0\oplus M_\oplus\cdots \oplus M_r\) of \(L\) where \(L\) has the standard grading \(L=L_{-2}\oplus L_{-1}\oplus L_0\oplus L_1\oplus\cdots\oplus L_r\) (and \(L_{-2}=0\) unless \(L\) is a contact algebra). A classification of those maximal graded subalgebras \(M\) of \(L\) (with \(L_{-2}=0\)) that contain \(L_{-1}\oplus L_0\) has previously been established by \textit{A. I. Kostrikin} and \textit{I. R. Shafarevich} [Izv. Akad. Nauk SSSR, Ser. Mat. 33, 251--322 (1969; Zbl 0211.05304)]. In the paper under review the author considers the remaining cases. He obtains an explicit construction unless \(M=L_{-2}\oplus L_{-1}\oplus M_0\oplus M_1\oplus\cdots\oplus M_r\) and \(M_0\) is a maximal subalgebra of \(L_0\) that acts irreducibly on \(L_{-1}\). In this case the problem is reduced to the classification of the irreducible maximal subalgebras of the simple classical Lie algebras which still is not complete.