= Free graded Lie algebra
{title2=$\mathbb L(V)$}
For a positively graded <vector space> $V$, the free <graded Lie algebra> $\mathbb L(V)$ receives a <linear map> from $V$ and extends every graded <linear map> $V\to L$ uniquely to a Lie <homomorphism>. Over $\mathbb Q$, it is the Lie subalgebra generated by $V$ inside the <tensor algebra> $T(V)$, using $[u,v]=uv-(-1)^{|u||v|}vu$. Its <universal enveloping algebra> is $T(V)$: both have the same universal property for maps from $V$ into <associative algebras>. The primitives of the tensor <Hopf algebra> with primitive generators are exactly $\mathbb L(V)$.
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