On a Hypothesis in the Theory of Lie Algebras

Abstract

The concepts of a free group and the free product of groups, as well as the results related to these concepts, have their analogues in the theory of algebras. It is known, for example, that any subalgebra of a free Lie algebra is also free. This result is analogous to the well-known theorem of Nielsen-Schreier in group theory. The results of A.T. Gainov [1] on subalgebras of the free commutative and free anticommutative products of algebras, are analogous to the theorem of A.G. Kurosh [2] on subgroups of the free product of groups. Under the influence of this analogy, there existed a conjecture that subalgebras of the free Lie product of Lie algebras are described by a theorem analogous to the theorem of A.T. Gainov cited above. In the present note, we prove that this is not the case. Moreover, we give here a construction of interest in its own right, which it is natural to call the free Lie product of Lie algebras with an amalgamated subalgebra.