Relative linear dependence problem for the variety \({\mathfrak AN}_ {\mathbf{c}}\) of Lie algebras
Let \({\mathfrak M}\) be a variety of Lie algebras over a field. The author studies the following decision problem for relative linear dependence: Is there an algorithm which determines for every finitely presented (f. p.) algebra \(L\in {\mathfrak M}\) and its subalgebra \(A\) if an arbitrary finite system of elements \(g_ 1,...,g_ n\in L\) is linearly dependent modulo A. A negative solution of this problem for the variety \({\mathfrak A}^ s\), \(s\geq 3\), of solvable of class \(\leq s\) algebras follows from the negative solution of the word problem [\textit{G. P. Kukin}, Algebra Logika 17, 402- 415 (1978; Zbl 0445.17010)]. The main result of the paper under review gives a positive solution of the relative linear dependence problem for the abelian-by-nilpotent of class c varieties \({\mathfrak AN}_ c\). The proof combines two techniques - of the verbal wreath products [\textit{A. L. Shmel'kin}, Tr. Mosk. Mat. O.- va 29, 247-260 (1973; Zbl 0286.17012)] and the composition in Lie algebras [\textit{A. I. Shirshov}, Sib. Mat. Zh. 3, 292-296 (1962; Zbl 0104.26004)]. The author also establishes a positive solution of the problem for recognizing nilpotency of a given class for f. p. solvable algebras. The following interesting result is obtained as a consequence. For any positive integers m and n every f. p. Lie algebra can be embedded in an f. p. Lie algebra \(L\) such that \(L/L^{(n)}\) is nilpotent of class \(\leq m\).
scientific article; zbMATH DE number 3959719