On ideals of free and free nilpotent Lie algebras

Boral, Melih

doi:10.1090/S0002-9939-1985-0781048-6

On ideals of free and free nilpotent Lie algebras
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by Melih Boral

Proc. Amer. Math. Soc. 94 (1985), 23-28

DOI: https://doi.org/10.1090/S0002-9939-1985-0781048-6

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Abstract:

It is proved that in a free nilpotent Lie algebra there are no nonabelian ideals which are free nilpotent as subalgebras. It is also shown that, for any proper ideal $S$ of a free Lie algebra $F$, the quotient of the lower central terms ${F_m}/{S_m}$ is not finitely generated when $F \ne {F_2} + S$. If $F = {F_2} + S,\;F/S$ is finite-dimensional and $S$ is finitely generated as an ideal in $F$, then ${F_m}/{S_m}$ is finitely generated as an algebra for all $m \geqslant 1$.

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