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Metabelian binary-Lie algebras. (English. Russian original) Zbl 0565.17013

Algebra Logic 23, 155-160 (1984); translation from Algebra Logika 23, No. 2, 220-227 (1984).
A linear algebra A over a field F is called binary Lie if each two- generator subalgebra is a Lie algebra. We say that A is metabelian if \((A^ 2)^ 2=0\). Using an approach from a paper of Yu. A. Medvedev [Algebra Logika 17, No.6, 705-728 (1978; Zbl 0425.16016)] the author shows that over an arbitrary field F of cardinality at least 4 and of characteristic different from 3 every metabelian variety of binary Lie algebras possesses a finite basis of identities. Another result, on residual finiteness of finitely generated metabelian binary Lie algebras, is easily translated from Lie algebra language [see the reviewer, Mat. Zametki 12, 713-716 (1972; Zbl 0249.17015)].
Reviewer: Yu.A.Bakhturin

MSC:

17D99 Other nonassociative rings and algebras
17A30 Nonassociative algebras satisfying other identities
16Rxx Rings with polynomial identity
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References:

[1] Yu. A. Bakhturin, ”Approximation of Lie algebras,” Mat. Zametki,12, No. 6, 713–716 (1972).
[2] A. T. Gainov, ”Identical relations for binary-Lie rings,” Usp. Mat. Nauk,12, No. 3, 141–146 (1957).
[3] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings That Are Nearly Associative [in Russian], Nauka, Moscow (1978). · Zbl 0445.17001
[4] E. N. Kuz’min, ”The locally nilpotent radical of Mal’tsev algebras satisfying the n-th Engel condition,” Dokl. Akad. Nauk SSSR,177, No. 3, 508–510 (1967).
[5] G. P. Kukin, ”Algorithmic problems for solvable Lie algebras,” Algebra Logika,17, No. 4, 402–415 (1978). · Zbl 0426.20027 · doi:10.1007/BF01673828
[6] A. I. Mal’tsev, Classical Algebra [in Russian], Vol. 1, Nauka, Moscow (1976).
[7] Yu. A. Medvedev, ”The finite basis property for varieties with a two-term identity,” Algebra Logika,17, No. 6, 705–726 (1978).
[8] M. R. Vaughan-Lee, ”Some varieties of Lie algebras,” Ph.D. Thesis, Oxford (1968).
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