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Nilpotency of the Engel algebras. (English. Russian original) Zbl 1103.17300
Russ. Math. 45, No. 11, 15-18 (2001); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2001, No. 11, 17-21 (2001).
Let $$G$$ be an anticommutative algebra over a field $$K$$ and $$A(G)$$ be the associative algebra generated by all operators of right multiplication. $$G$$ is called an Engel algebra if for any $$x\in G$$ the operator $$R_x: g\to g\circ x$$, $$g\in G$$ is nilpotent.
The authors prove the following main results. (1) If $$G$$ is an Engel algebra and $$\dim_KG\leq 4$$, then $$G$$ is nilpotent. (2) For any $$n\geq 5$$, an $$n$$-dimensional Engel algebra exists which is not nilpotent. (3) For any algebra $$G$$ with basis $$e_1, e_2, ..., e_n$$, if any monomial in generators $$R_{e_i}$$ $$(i=1,2,..., n)$$ of $$A(G)$$ whose length does not exceed $$n$$ is nilpotent, then the algebra $$G$$ is nilpotent.
Notice that the proof of the last fact uses the results of V. A. Ufnarovskii [Mat. Sb., N. Ser. 128, No. 1, 124–132 (1985; Zbl 0598.15002)].
##### MSC:
 17A30 Nonassociative algebras satisfying other identities 16N40 Nil and nilpotent radicals, sets, ideals, associative rings 17A50 Free nonassociative algebras 17D10 Mal’tsev rings and algebras
##### Keywords:
anticommutative algebra; nilpotent algebra; Engel algebra