A. I. Kostrikin and I. R. Shafarevich [Izv. Akad. Nauk SSSR, Ser. Mat. 33, 251–322 (1969; Zbl 0211.05304)] have given a uniform construction of some families of simple algebras in characteristic \(p>0\) and have pointed out a possible way of classification of such algebras. The paper under review is a step in the direction pointed by the above-mentioned authors. It considers graded Lie algebras \(G=\sum G_i\) such that (1) \([x, G_{l,\text{signum}(i)}]\not = 0\) \(\forall x\in G_i\). (2) \(G_{-1}\) is irreducible under \(G_0\), (3) \(G_0\) is a restricted Lie algebra and a direct sum of classical simple Lie algebras and the center, (4) the representation of \(G_0\) on \(G_{-1}\) is restricted.
The paper is analogous as to results obtained and methods used to the paper of the author [Math. USSR, Izv. 2, 1271–1311 (1968); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1323–1367 (1968; Zbl 0222.17007)] where he treats algebras of characteristic zero. The result is that graded algebras of the type under consideration are either classical or of the type considered in the paper of Kostrikin and Shafarevich (loc. cit.).
It should be mentioned that the author had recently constructed some parametric families of simple (non-restricted) Lie algebras in all characteristics [Usp. Mat. Nauk 26, No. 3(159), 199–200 (1971; Zbl 0261.17009)]. These examples emphasize the importance of the paper under review and support those who choose to go along the way pointed out by Kostrikin and Shafarevich (loc. cit.).