Free step-N nilpotent Lie group
= Free step-N nilpotent Lie group
{title2=$G^N(V)=\exp(\mathfrak g^N(V))$}
Inside the <truncated tensor algebra>, let $\mathfrak g^N(V)$ be the span of iterated <Lie brackets> of elements of $V$ of lengths at most $N$. Its exponential is the connected simply connected free nilpotent <Lie group> of step $N$, with tensor multiplication. The bracket is the commutator in the <tensor algebra>. Its grading defines dilations that multiply degree $k$ by $\lambda^k$.