Proof of a conjecture of Wiegold for nilpotent Lie algebras

Abstract

Abstract Let be a nilpotent Lie algebra. By the breadth of an element of we mean the number . Vaughan-Lee showed that if the breadth of all elements of the Lie algebra is bounded by a number , then the dimension of the commutator subalgebra of the Lie algebra does not exceed . We show that if n(n+1)/2$?> for some nonnegative , then the Lie algebra is generated by the elements of breadth n$?> , and thus we prove a conjecture due to Wiegold (Question 4.69 in the Kourovka Notebook) in the case of nilpotent Lie algebras. Bibliography: 4 titles.