Equality problem for center-by-metabelian Lie algebras
Let \({\mathcal M}\) be a variety of Lie algebras over a field and let \(F_n({\mathcal M})\) be the \(n\)-generated relatively free algebra in \({\mathcal M}\). The \(n\)-generated algebra \(A\in {\mathcal M}\) is finitely presented (f. p.) in \({\mathcal M}\), if \(A\) is defined as a homomorphic image of \(F_n({\mathcal M})\) by a finite collection of relations. This paper deals with the word problem (or the equality problem) for a given variety \({\mathcal M}\): Does there exist an algorithm which determines whether or not two arbitrary polynomials on the generators of an f. p. algebra from \({\mathcal M}\) present the same element ? \textit{A. I. Shirshov} [Sib. Mat. Zh. 3, 292--296 (1962; Zbl 0104.26004)] has proved that this problem has an affirmative solution for \({\mathcal M}={\mathcal A}^2\). On the other hand \textit{G. P. Kukin} [Algebra Logika 17, 402--415 (1978; Zbl 0445.17010)] has answered in the negative the word problem for every variety \({\mathcal M}\) containing \({\mathcal N}_2 {\mathcal A}.\) A center-by-metabelian variety \([{\mathcal A}^ 2,{\mathcal E}]\) is defined by the polynomial identity \([[[x_ 1,x_ 2], [x_ 3,x_ 4]], x_ 5]=0\) and lies between \({\mathcal A}^2\) and \({\mathcal N}_2 {\mathcal A}\). The main result of the paper under review claims that the word problem has a positive solution for \([{\mathcal A}^2,{\mathcal E}]\). The proof is based on some ideas of Shirshov's paper cited above and uses the notion of composition for Lie algebras.
scientific article; zbMATH DE number 3955163
scientific article