On generators of 𝐿/𝑅² Lie algebras

Shpilrain, Vladimir

doi:10.1090/S0002-9939-1993-1154249-X

On generators of $L/R^ 2$ Lie algebras
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Proc. Amer. Math. Soc. 119 (1993), 1039-1043 Request permission

Abstract:

Let $L$ be a free Lie algebra of finite rank $n$ and $R$ its arbitrary ideal. A necessary and sufficient condition for $n$ elements of the Lie algebra $L/{R^2}$ to be a generating set is given. In particular, we have a criterion for $n$ elements of a free Lie algebra of rank $n$ 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).

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