Abstract
Answering V. E. Shpil′răin’s question [1, Problem 13.66], E. I. Timoshenko [2] found necessary and sufficient conditions on elements of a free finitely-generated metabelian group G of rank n for every endomorphism of G to be uniquely determined from its action on these elements. Let Mn be a free metabelian Lie algebra with n free generators. The purpose of this article consists in clarifying necessary and sufficient conditions on elements b1, b2, . . . , bm of the Lie algebra Mn for every endomorphism φ of this algebra to be uniquely determined from its actions on these elements, or shortly the conditions for b1, b2, . . . , bm to determine each endomorphism of Mn uniquely. Let K be an arbitrary field and let L be a Lie algebra over K. Denote the universal enveloping algebra for L by UL. It is well known that UL is an integral domain. Denote by F the free Lie algebra over K with the set of free generators {y1, . . . , yn}. The universal enveloping algebra of F is a free associative algebra with the same set of free generators. The ideal generated by F in UF is a free right UF -module with basis y1, . . . , yn. Therefore, each element f ∈ FUF has the unique expansion
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