×

A theorem on sandwich algebras. (Russian) Zbl 0729.17006

One of the main notions in Kostrikin’s theory of Engel Lie algebras is the notion of sandwich. An element \(a\) of a Lie algebra is said to be a cover of a thin sandwich if \((\text{ad}\,a)^ 2=0\) (previously such a was called just a (thin) sandwich). On the way to the theorem, which provided the positive solution of the Restricted Burnside Problem for groups of prime exponent, the second author proved [Izv. Akad. Nauk SSSR, Ser. Mat. 23, 3–34 (1959; Zbl 0090.24503) and Mat. Sb., Nov. Ser. 110(152), 3–12 (1979; Zbl 0415.17011)] that if a \((p-1)\)-Engel Lie algebra of characteristic \(p\) is generated by a finite number of covers of thin sandwiches then it is nilpotent. The first author [Mat. Sb., Nov. Ser. 112(154), 611–629 (1980; Zbl 0442.17003)] by means of Jordan algebra techniques showed that the \((p-1)\)-Engel condition here may be dropped if \(p\neq 2,3\). Now the same is proved for all \(p\) (and also for superalgebras). The proof does not use Jordan algebras, but uses some techniques from Kostrikin’s works cited above.

MSC:

17B30 Solvable, nilpotent (super)algebras
17B50 Modular Lie (super)algebras
PDFBibTeX XMLCite