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On Engel-Lie algebras. (English. Russian original) Zbl 0673.17006
Sib. Math. J. 29, No. 5, 777-781 (1988); translation from Sib. Mat. Zh. 29, No. 5(171), 112-117 (1988).
The author proves the long-standing conjecture: the nilpotency of an n- Engel Lie algebra over a field of characteristic zero. Earlier the local nilpotency of such an algebra was proved by A. I. Kostrikin [Izv. Akad. Nauk SSSR, Ser. Mat. 23, No.1, 3-34 (1959; Zbl 0090.245)] - as well as for characteristic $$p>n$$. (The latter case also gave the positive solution to the restricted Burnside problem for groups of exponent p.) Yu. P. Razmyslov [Algebra Logika 10, No.1, 33-44 (1971; Zbl 0253.17005)] showed that for positive characteristic p there exist non- solvable (p-2)-Engel Lie algebras.
The proof is based on Kostrikin’s theorem stating that any n-Engel Lie algebra of characteristic 0 (or $$p>n)$$ has a non-trivial abelian ideal. The proof also exploits $${\mathbb{Z}}/2{\mathbb{Z}}$$-graded Lie algebras and the technique of the representation theory of symmetric groups. The earlier version of this theorem of Zel’manov has already been incorporated into A.I. Kostrikin’s book “Around Burnside” (Nauka 1986; Zbl 0624.17001) but the present one is much shorter.
The author points out the following Corollary: An n-Engel Lie algebra of characteristic $$p\gg n$$ (sufficiently large with respect to n) is nilpotent. One certainly may also expect some corollaries in group theory.
Reviewer: E.I.Khukhro

##### MSC:
 17B30 Solvable, nilpotent (super)algebras 17B05 Structure theory for Lie algebras and superalgebras 17B70 Graded Lie (super)algebras
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##### References:
 [1] A. I. Kostrikin, ?On Burnside’s problem,? Izv. Akad. Nauk SSSR, Ser. Mat.,23, No. 1, 3-34 (1959). · Zbl 0090.24503 [2] A. I. Kostrikin, On Burnside [in Russian], Nauka, Moscow (1986). [3] P. J. Higgins, ?Lie rings satisfying the Engel condition,? Proc. Cambridge Phil. Soc.,50, No. 1, 8-15 (1954). · Zbl 0055.02601 [4] Yu. P. Razmyslov, ?On Engelian Lie algebras,? Algebra Logika,10, No. 1, 33-44 (1971). [5] A. I. Kostrikin, ?Lie algebras and finite groups,? Trudy Mat. Inst. im. V. A. Steklova Akad. Nauk SSSR,168, 132-154 (1984). · Zbl 0547.17004 [6] S. P. Mishchenko, ?On the Engelian problem,? Mat. Sb.,124, No. 1, 56-57 (1984). · Zbl 0546.17004 [7] Yu. A. Bakhturin, Identities in Lie Algebras [in Russian], Nauka, Moscow (1985). · Zbl 0571.17001 [8] C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York (1962). · Zbl 0131.25601
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