On generators of 𝐿/𝑅² Lie algebras

Abstract

Let L L be a free Lie algebra of finite rank n n and R R its arbitrary ideal. A necessary and sufficient condition for n n elements of the Lie algebra L / R 2 L/{R^2} to be a generating set is given. In particular, we have a criterion for n n elements of a free Lie algebra of rank n n to be a generating set which is similar to the corresponding group-theoretic result due to Birman (An inverse function theorem for free groups, Proc. Amer. Math. Soc. 41 (1973), 634-638).