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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)].
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