Class, dimension and length in nilpotent Lie algebras

Abstract

The problem of finding the smallest order of a p-group of a given derived length has
a long history. Nilpotent Lie algebra versions of this and related problems are considered. Thus,
the smallest order of a p-group is replaced by the smallest dimension of a nilpotent Lie algebra. For
each length t, an upper bound for this smallest dimension is found. Also, it is shown that for each
t ≥ 5 there is a two generated Lie algebra of nilpotent class d = 21(2^t−5) whose derived length is t.
For two generated Lie algebras, this result is best possible. Results for small t are also found. The
results are obtained by constructing Lie algebras of strictly upper triangular matrices.